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  • Committer: Package Import Robot
  • Author(s): Sylvestre Ledru
  • Date: 2011-11-29 01:48:33 UTC
  • mfrom: (5.1.1 sid)
  • Revision ID: package-import@ubuntu.com-20111129014833-3leawsns2nghyaoj
Tags: 2.1.5-1
http://bugs.debian.org/571446
http://bugs.debian.org/633175
* Team upload.
* New upstream release
* Package moved into the Debian Science team (no answer from the previous
  maintainer) and package not maintained.
* Standards-Version updated to version 3.9.2
* Vcs-Browser & Vcs-Svn updated
* Switch to mpi-default-dev (Closes: #571446)
* Get ride of .la files (Closes: #633175)
* Fix lintian warning debhelper-but-no-misc-depends
* Fix lintian warning patch-system-but-direct-changes-in-diff 
* Switch to dpkg-source 3.0 (quilt) format

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doc/fftw.ps
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%!PS-Adobe-2.0
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%%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com)
 
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%%Creator: dvips(k) 5.86e Copyright 2001 Radical Eye Software
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%%Title: fftw.dvi
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%%Pages: 73
 
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%%Pages: 74
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%%PageOrder: Ascend
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%%BoundingBox: 0 0 596 842
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%%EndComments
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%DVIPSCommandLine: dvips -Z fftw.dvi -o fftw.ps
 
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%DVIPSWebPage: (www.radicaleye.com)
 
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%DVIPSCommandLine: dvips -Z -o fftw.ps fftw.dvi
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%DVIPSParameters: dpi=600, compressed
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%DVIPSSource:  TeX output 1999.11.07:2050
 
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%DVIPSSource:  TeX output 2003.03.24:0228
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%%BeginProcSet: texc.pro
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%!
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/TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N
14
 
/X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72
15
 
mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1}
16
 
ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale
17
 
isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div
18
 
hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul
19
 
TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if}
20
 
forall round exch round exch]setmatrix}N /@landscape{/isls true N}B
21
 
/@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B
22
 
/FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{
23
 
/nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N
24
 
string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N
25
 
end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{
26
 
/sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0]
27
 
N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup
28
 
length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{
29
 
128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub
30
 
get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data
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dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N
32
 
/rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup
33
 
/base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx
34
 
0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff
35
 
setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff
36
 
.1 sub]/id ch-image N /rw ch-width 7 add 8 idiv string N /rc 0 N /gp 0 N
37
 
/cp 0 N{rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id
38
 
gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp
39
 
add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add
40
 
/gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{
41
 
dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1
42
 
adv}B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2
43
 
idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string
44
 
putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval
45
 
adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg}
46
 
{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{
47
 
adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2
48
 
chg nd}{pop nd}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{]
49
 
}if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup
50
 
length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{
51
 
cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin
52
 
0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul
53
 
add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore userdict
54
 
/eop-hook known{eop-hook}if showpage}N /@start{userdict /start-hook
55
 
known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X
56
 
/IE 256 array N 2 string 0 1 255{IE S dup 360 add 36 4 index cvrs cvn
57
 
put}for pop 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N
58
 
/RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley
59
 
X /rulex X V}B /V{}B /RV statusdict begin /product where{pop false[
60
 
(Display)(NeXT)(LaserWriter 16/600)]{dup length product length le{dup
61
 
length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}
62
 
forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false
63
 
RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1
64
 
false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform
65
 
round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg
66
 
rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail
67
 
{dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M}
68
 
B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{
69
 
4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{
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p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p
71
 
a}B /bos{/SS save N}B /eos{SS restore}B end
 
14
/TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S
 
15
N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72
 
16
mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0
 
17
0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{
 
18
landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize
 
19
mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[
 
20
matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round
 
21
exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{
 
22
statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0]
 
23
N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin
 
24
/FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array
 
25
/BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2
 
26
array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N
 
27
df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A
 
28
definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get
 
29
}B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub}
 
30
B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr
 
31
1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3
 
32
1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx
 
33
0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx
 
34
sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{
 
35
rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp
 
36
gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B
 
37
/chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{
 
38
/cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{
 
39
A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy
 
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get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse}
 
41
ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp
 
42
fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17
 
43
{2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add
 
44
chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{
 
45
1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop}
 
46
forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn
 
47
/BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put
 
48
}if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{
 
49
bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A
 
50
mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{
 
51
SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{
 
52
userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X
 
53
1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4
 
54
index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N
 
55
/p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{
 
56
/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)
 
57
(LaserWriter 16/600)]{A length product length le{A length product exch 0
 
58
exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse
 
59
end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask
 
60
grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot}
 
61
imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round
 
62
exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto
 
63
fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p
 
64
delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M}
 
65
B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{
 
66
p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S
 
67
rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end
72
68
 
73
69
%%EndProcSet
74
70
%%BeginProcSet: special.pro
75
71
%!
76
 
TeXDict begin /SDict 200 dict N SDict begin /@SpecialDefaults{/hs 612 N
77
 
/vs 792 N /ho 0 N /vo 0 N /hsc 1 N /vsc 1 N /ang 0 N /CLIP 0 N /rwiSeen
78
 
false N /rhiSeen false N /letter{}N /note{}N /a4{}N /legal{}N}B
79
 
/@scaleunit 100 N /@hscale{@scaleunit div /hsc X}B /@vscale{@scaleunit
80
 
div /vsc X}B /@hsize{/hs X /CLIP 1 N}B /@vsize{/vs X /CLIP 1 N}B /@clip{
81
 
/CLIP 2 N}B /@hoffset{/ho X}B /@voffset{/vo X}B /@angle{/ang X}B /@rwi{
82
 
10 div /rwi X /rwiSeen true N}B /@rhi{10 div /rhi X /rhiSeen true N}B
83
 
/@llx{/llx X}B /@lly{/lly X}B /@urx{/urx X}B /@ury{/ury X}B /magscale
84
 
true def end /@MacSetUp{userdict /md known{userdict /md get type
85
 
/dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup
86
 
length 20 add dict copy def}if end md begin /letter{}N /note{}N /legal{}
87
 
N /od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath
88
 
clippath mark{transform{itransform moveto}}{transform{itransform lineto}
89
 
}{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{
90
 
itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{
91
 
closepath}}pathforall newpath counttomark array astore /gc xdf pop ct 39
92
 
0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}if}N
93
 
/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1
94
 
scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get
95
 
ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip
96
 
not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0
97
 
TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{noflips{TR
98
 
pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1
99
 
-1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg
100
 
TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg
101
 
sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr
102
 
0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add
103
 
2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N /cp
104
 
{pop pop showpage pm restore}N end}if}if}N /normalscale{Resolution 72
105
 
div VResolution 72 div neg scale magscale{DVImag dup scale}if 0 setgray}
106
 
N /psfts{S 65781.76 div N}N /startTexFig{/psf$SavedState save N userdict
107
 
maxlength dict begin /magscale true def normalscale currentpoint TR
108
 
/psf$ury psfts /psf$urx psfts /psf$lly psfts /psf$llx psfts /psf$y psfts
109
 
/psf$x psfts currentpoint /psf$cy X /psf$cx X /psf$sx psf$x psf$urx
110
 
psf$llx sub div N /psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy
111
 
scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR
112
 
/showpage{}N /erasepage{}N /copypage{}N /p 3 def @MacSetUp}N /doclip{
 
72
TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N
 
73
/vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N
 
74
/rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N
 
75
/@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{
 
76
/hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho
 
77
X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B
 
78
/@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{
 
79
/urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known
 
80
{userdict/md get type/dicttype eq{userdict begin md length 10 add md
 
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maxlength ge{/md md dup length 20 add dict copy def}if end md begin
 
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/letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S
 
83
atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{
 
84
itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll
 
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transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll
 
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curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf
 
87
pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}
 
88
if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1
 
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-1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3
 
90
get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip
 
91
yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub
 
92
neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{
 
93
noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop
 
94
90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get
 
95
neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr
 
96
1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr
 
97
2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4
 
98
-1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S
 
99
TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{
 
100
Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale
 
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}if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState
 
102
save N userdict maxlength dict begin/magscale true def normalscale
 
103
currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts
 
104
/psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x
 
105
psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx
 
106
psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub
 
107
TR/showpage{}N/erasepage{}N/copypage{}N/p 3 def @MacSetUp}N/doclip{
113
108
psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2
114
109
roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath
115
 
moveto}N /endTexFig{end psf$SavedState restore}N /@beginspecial{SDict
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begin /SpecialSave save N gsave normalscale currentpoint TR
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@SpecialDefaults count /ocount X /dcount countdictstack N}N /@setspecial
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{CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto
 
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moveto}N/endTexFig{end psf$SavedState restore}N/@beginspecial{SDict
 
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begin/SpecialSave save N gsave normalscale currentpoint TR
 
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@SpecialDefaults count/ocount X/dcount countdictstack N}N/@setspecial{
 
113
CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto
119
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closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx
120
115
sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR
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116
}{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse
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CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury
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lineto closepath clip}if /showpage{}N /erasepage{}N /copypage{}N newpath
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}N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{
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end}repeat grestore SpecialSave restore end}N /@defspecial{SDict begin}
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N /@fedspecial{end}B /li{lineto}B /rl{rlineto}B /rc{rcurveto}B /np{
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yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end
 
118
lineto closepath clip}if/showpage{}N/erasepage{}N/copypage{}N newpath}N
 
119
/@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{end}
 
120
repeat grestore SpecialSave restore end}N/@defspecial{SDict begin}N
 
121
/@fedspecial{end}B/li{lineto}B/rl{rlineto}B/rc{rcurveto}B/np{/SaveX
 
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currentpoint/SaveY X N 1 setlinecap newpath}N/st{stroke SaveX SaveY
 
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moveto}N/fil{fill SaveX SaveY moveto}N/ellipse{/endangle X/startangle X
 
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/yrad X/xrad X/savematrix matrix currentmatrix N TR xrad yrad scale 0 0
 
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1 startangle endangle arc savematrix setmatrix}N end
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1057
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1076
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1077
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1078
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1081
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1082
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1083
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1084
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1085
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387C03F0381E07C0380FFF803801FC00185185BD1C>I<EA01FC12FFA3120712031201B2
2053
 
92B51280A392383FFC0016E0168093C7FC153C5D5D4A5AEC07C04A5A4AC8FC143E147F4A
2054
 
7E13FD9038FFDFC0EC9FE0140F496C7E01FC7F496C7E1401816E7E81826F7E151F826F7E
2055
 
A282486C14FEB539F07FFFE0A32B3F7EBE30>I<EA01FC12FFA3120712031201B3B3B148
2056
 
7EB512F8A3153F7DBE1A>I<2701F801FE14FF00FF902707FFC00313E0913B1E07E00F03
2057
 
F0913B7803F03C01F80007903BE001F87000FC2603F9C06D487F000101805C01FBD900FF
2058
 
147F91C75B13FF4992C7FCA2495CB3A6486C496CECFF80B5D8F87FD9FC3F13FEA347287D
2059
 
A74C>I<3901F801FE00FF903807FFC091381E07E091387803F000079038E001F82603F9
2060
 
C07F0001138001FB6D7E91C7FC13FF5BA25BB3A6486C497EB5D8F87F13FCA32E287DA733
2061
 
>I<14FF010713E090381F81F890387E007E01F8131F4848EB0F804848EB07C04848EB03
2062
 
E0000F15F04848EB01F8A2003F15FCA248C812FEA44815FFA96C15FEA36C6CEB01FCA300
2063
 
1F15F86C6CEB03F0A26C6CEB07E06C6CEB0FC06C6CEB1F80D8007EEB7E0090383F81FC90
2064
 
380FFFF0010090C7FC282A7EA82D>I<3901FC03FC00FF90381FFF8091387C0FE09039FD
2065
 
E003F03A03FFC001FC6C496C7E91C7127F49EC3F805BEE1FC017E0A2EE0FF0A3EE07F8AA
2066
 
EE0FF0A4EE1FE0A2EE3FC06D1580EE7F007F6E13FE9138C001F89039FDE007F09039FC78
2067
 
0FC0DA3FFFC7FCEC07F891C9FCAD487EB512F8A32D3A7EA733>I<02FF131C0107EBC03C
2068
 
90381F80F090397F00387C01FC131CD803F8130E4848EB0FFC150748481303121F485A15
2069
 
01485AA448C7FCAA6C7EA36C7EA2001F14036C7E15076C6C130F6C7E6C6C133DD8007E13
2070
 
7990383F81F190380FFFC1903801FE0190C7FCAD4B7E92B512F8A32D3A7DA730>I<3901
2071
 
F807E000FFEB1FF8EC787CECE1FE3807F9C100031381EA01FB1401EC00FC01FF13304913
2072
 
00A35BB3A5487EB512FEA31F287EA724>I<90383FC0603901FFF8E03807C03F381F000F
2073
 
003E1307003C1303127C0078130112F81400A27E7E7E6D1300EA7FF8EBFFC06C13F86C13
2074
 
FE6C7F6C1480000114C0D8003F13E0010313F0EB001FEC0FF800E01303A214017E1400A2
2075
 
7E15F07E14016C14E06CEB03C0903880078039F3E01F0038E0FFFC38C01FE01D2A7DA824
2076
 
>I<131CA6133CA4137CA213FCA2120112031207001FB512C0B6FCA2D801FCC7FCB3A215
2077
 
E0A912009038FE01C0A2EB7F03013F138090381F8700EB07FEEB01F81B397EB723>I<D8
2078
 
01FC14FE00FF147FA3000714030003140100011400B3A51501A31503120015076DEB06FF
2079
 
017E010E13806D4913FC90381FC078903807FFE00100903880FE002E297DA733>I<B539
2080
 
E00FFFE0A32707FE000313006C48EB00FC5E00015D7F00005DA26D13016D5CA26D6C485A
2081
 
A2ECC007011F91C7FCA290380FE00EA2ECF01E0107131CA26D6C5AA2ECFC7801011370A2
2082
 
ECFEF001005BA2EC7FC0A36E5AA26EC8FCA3140E2B287EA630>I<B53BC3FFFE03FFF8A3
2083
 
290FFE003FE00013C06C486D48EB3F806C4817006D010F141E00016F131C15076D163C00
2084
 
004A6C1338A2017F5E4B7E151DD93F805DED3DFC1538D91FC04A5AED78FE9238707E03D9
2085
 
0FE0017F5BEDE03F02F0140701070387C7FC9138F1C01F02F9148F010315CE9138FB800F
2086
 
02FF14DE6D15FCED00076D5DA24A1303027E5CA2027C1301023C5C023813003D287EA642
2087
 
>I<B539F01FFFE0A30003D9C00F1300C690388007F8D97F0013E002805BD93FC05B011F
2088
 
49C7FC90380FE00EECF01E6D6C5A01035B6D6C5A6E5AEB00FF6E5A6E5A81141F814A7E81
2089
 
147BECF1FC903801E1FEECC0FF01037F49486C7ED90F007F011E6D7E013E130F496D7E01
2090
 
FC80486C80000F4A7EB539803FFFF8A32D277FA630>I<B539E00FFFE0A32707FE000313
2091
 
006C48EB01FC6F5A00015D7F00005DA2017F495AA2EC8003013F5CA26D6C48C7FCA26E5A
2092
 
010F130EA26D6C5AA2ECF83C01031338A26D6C5AA2ECFEF001005BA2EC7FC0A36E5AA36E
2093
 
C8FCA2140EA2141E141C143C1438A2147800181370127EB45BA2495AA248485AD87E07C9
2094
 
FCEA780EEA3C3CEA1FF8EA07E02B3A7EA630>I<001FB61280A2EBE0000180140049485A
2095
 
001E495A121C4A5A003C495A141F00385C4A5A147F5D4AC7FCC6485AA2495A495A130F5C
2096
 
495A90393FC00380A2EB7F80EBFF005A5B484813071207491400485A48485BA248485B48
2097
 
48137F00FF495A90B6FCA221277EA628>I<B812F0A22C0280982D>I<BE12C0A25A028098
2098
 
5B>I E
 
1983
00A31870A418381638A41800A21678A216F81501150791B5FCA3EC8007150115001678A2
 
1984
1638A2180EA3181C93C7FCA4183C1838A21878A318F8EF01F0A21707170F173F48486CEB
 
1985
03FFB912E0A3373E7DBD3E>I<B91280A300019038C000036C6C48EB007FEF1FC0170F17
 
1986
07A21703A31701A4EF00E0A21638A31800A31678A216F81501150791B5FCA3EC80071501
 
1987
15001678A21638A693C8FCAF3801FFE0B612F0A3333E7DBD3B>I<DB3FE0130C912603FF
 
1988
FE131C021F9038FF803C913A7FF00FC07C9139FF0001F0D903FC90380078FC4948143DD9
 
1989
1FE0141F4948140F4948140701FF15034890C8FC491501485A000716005B000F177C5B00
 
1990
1F173CA2485AA2181C127FA25B95C7FC12FFAB041FB512F0127FA26D9139000FFE00EF03
 
1991
FC123FA27F121FA26C7EA212077F12036C7E7F6C7F6D6C14076D7E6D6C140FD907F8141E
 
1992
D903FEEC3C7C902600FF80EBF83C913A7FF007F01C021FB5EAC00C020391C8FC9138003F
 
1993
F03C427BBF47>I<B6D8C01FB512F8A3000101E0C7383FFC0026007F80EC0FF0B3A691B7
 
1994
FCA30280C7120FB3A92601FFE0EC3FFCB6D8C01FB512F8A33D3E7DBD44>I<B612F0A3C6
 
1995
EBF000EB3FC0B3B3B2EBFFF0B612F0A31C3E7EBD21>I<011FB512FCA3D9000713006E5A
 
1996
1401B3B3A6123FEA7F80EAFFC0A44A5A1380D87F005B007C130700385C003C495A6C495A
 
1997
6C495A2603E07EC7FC3800FFF8EB3FC026407CBD2F>I<B600C090387FFFFCA3000101E0
 
1998
C7000F138026007F80913807FE0018F818E0604D5A4DC7FC173E5F5F4C5A4C5A4C5A4C5A
 
1999
4CC8FC163E5E5E4B5A4B5AED07804B7E151F4B7E4B7E15FF913881EFF8913883C7FCEC87
 
2000
8791388F03FE91389E01FF14BCDAF8007F4A6D7E5C4A6D7E4A6D7EA2707E707EA2707E70
 
2001
7EA2707F717E84173F717E717EA2717E848419802601FFE04A13C0B600C090B6FCA3403E
 
2002
7DBD47>I<B612F8A3000101E0C9FC38007F80B3B0EF0380A517071800A45FA35FA25F5F
 
2003
5F4C5A160748486C133FB8FCA3313E7DBD39>I<B500C093B512C0A300016D4BEBE000D8
 
2004
007F1880D977F0ED03BFA3D973F8ED073FA3D971FC150EA2D970FE151CA3027F1538A36E
 
2005
6C1470A36E6C14E0A26E6CEB01C0A36E6CEB0380A36E6CEB0700A26E6C130EA36E6C5BA3
 
2006
037F5BA26F6C5AA36F6C5AA392380FE1C0A3923807F380A26FB4C7FCA36F5AA213F8486C
 
2007
6D5AD807FFEFFFE0B500F80178017FEBFFC0A34A3E7CBD53>I<B56C91B512F88080D800
 
2008
7F030713006EEC01FC6E6E5A1870EB77FCEB73FEA2EB71FF01707FA26E7E6E7EA26E7E6E
 
2009
7EA26E7E6E7EA26E7E6E7FA26F7E6F7EA26F7E6F7EA26F7E6F7EA26F7E6F1380A2EE7FC0
 
2010
EE3FE0A2EE1FF0EE0FF8A2EE07FCEE03FEA2EE01FF7013F0A2177F173FA2171F170FA217
 
2011
0701F81503487ED807FF1501B500F81400A218703D3E7DBD44>I<ED7FE0913807FFFE91
 
2012
391FC03F8091397E0007E04948EB03F8D907F0EB00FE4948147F49486E7E49486E7E49C8
 
2013
6C7E01FE6F7E00018349150300038348486F7EA248486F7EA2001F188049167F003F18C0
 
2014
A3007F18E049163FA300FF18F0AC007F18E06D167FA4003F18C0A26C6CEEFF80A36C6C4B
 
2015
1300A26C6C4B5A00035F6D150700015F6C6C4B5A6D5E6D6C4A5A6D6C4A5A6D6C4AC7FC6D
 
2016
6C14FED901FCEB03F8D9007FEB0FE091391FC03F80912607FFFEC8FC9138007FE03C427B
 
2017
BF47>I<B712F8EEFF8017E000019039C0003FF86C6C48EB07FCEE01FE707EEF7F80EF3F
 
2018
C018E0A2EF1FF0A218F8A818F0A2EF3FE0A218C0EF7F80EFFF004C5AEE07FCEE3FF091B6
 
2019
12C04CC7FC0280C9FCB3A73801FFE0B612C0A3353E7DBD3E>I<ED7FE0913807FFFE9139
 
2020
1FC03F8091397F000FE0D901FCEB03F8D907F0EB00FE4948147F49486E7E49486E7E49C8
 
2021
6C7E498248486F7E49150300038348486F7EA2000F834981001F1880A24848EE7FC0A300
 
2022
7F18E0A249163FA200FF18F0AC007F18E0A26D167FA3003F18C0A26C6CEEFF80A3000F18
 
2023
006D5D0007DA0F805B6C6C90393FE003FCED70706C6C496C485A6C6C48486C485A017FD9
 
2024
800E5BD93F819038061FC0D91FC19038073F80D90FE14AC7FCD907F1EB03FE902601FDC0
 
2025
13F8903A007EE007E091271FF03FC013180207B5FC9139007FE1E0DB0001143883711378
 
2026
A2706C13F0EFFF0318FFA27113E0A37113C0711380711300715AEF01F83D527BBF47>I<
 
2027
B712C016FCEEFF800001D9C00013E06C6C48EB1FF0EE07FCEE01FE707E84717EA2717EA2
 
2028
84A760177F606017FF95C7FCEE01FCEE07F8EE1FE0EEFF8091B500FCC8FC16F091388001
 
2029
FCED003FEE1FC0707E707E83160383160183A383A484A4F0C004190EA28218E0057F131E
 
2030
2601FFE0161CB600C0EB3FF094381FF83805071370CA3801FFE09438003F803F407DBD43
 
2031
>I<D907FC131890391FFF8038017FEBE0783901FC03F83A03F0007CF8D807C0133F4848
 
2032
130F001F140748C7FC003E1403007E1401A2007C140012FC1678A46C1538A27EA26C6C14
 
2033
007F7FEA3FF8EBFF806C13F86CEBFF806C14F06C14FC6C14FF6C15C0013F14E0010714F0
 
2034
EB007F020713F89138007FFC150FED07FE15031501ED00FFA200E0157FA3163FA27EA316
 
2035
3E7E167E6C157C6C15FC6C15F86D13016DEB03F06DEB07E0D8F9FCEB0FC03AF07F803F80
 
2036
90391FFFFE00D8E00713F839C0007FC028427BBF33>I<003FB91280A3903AF0007FE001
 
2037
018090393FC0003F48C7ED1FC0007E1707127C00781703A300701701A548EF00E0A5C816
 
2038
00B3B14B7E4B7E0107B612FEA33B3D7DBC42>I<B600C090B512F8A3000101E0C7000713
 
2039
0026007F80EC01FC715A1870B3B3A4013F16F06E5DA21701011F5E80010F15036E4A5A01
 
2040
0793C7FC6D6C5C6D6C141E6D6C5C027F14F86E6C485A91390FF00FE00203B51280020049
 
2041
C8FCED1FF03D407DBD44>I<B691380FFFFEA3000301E0020113E06C01809138007F806C
 
2042
EF3F00017F163E181C6E153C013F1638A26E1578011F1670A26D6C5DA26E140101075EA2
 
2043
6E140301035EA26D6C4AC7FCA2806D150EA26F131E027F141CA26F133C023F1438A26E6C
 
2044
5BA26F13F0020F5CA2EDF80102075CA26E6C485AA2EDFE07020191C8FCA26F5A6E130EA2
 
2045
ED7F9CA216DCED3FF8A36F5AA36F5AA26F5AA36F5A3F407EBD44>I<B500FE017FB5D880
 
2046
07B5FCA3000301C0010101E0C713F86C90C849EC3FE07148EC0F807E7215006E143F017F
 
2047
190E84A26D6C60A24D7E6D6C60A2EFE7F86D6C60A2933801C3FC6E18F001076104037F6E
 
2048
0281140101036104077F17006D6C4D5AA2040EEB7F806D6C4DC7FCA24CEB3FC0DA7F8016
 
2049
0EA24CEB1FE003C0161E023F171C047814F0DBE070010F133C021F173804F014F84C1307
 
2050
DA0FF05EA2DBF1C0EB03FCDA07F95EA2DBFB80EB01FEDA03FF6F5AA293C8FCA26E5FA24B
 
2051
157F020094C8FCA24B81037C153EA20378151E0338151C58407EBD5D>I<007FB5D8C003
 
2052
B512E0A3C649C7EBFC00D93FF8EC3FE06D48EC1F806D6C92C7FC171E6D6C141C6D6C143C
 
2053
5F6D6C14706D6D13F04C5ADA7FC05B023F13036F485ADA1FF090C8FC020F5BEDF81E9138
 
2054
07FC1C163C6E6C5A913801FF7016F06E5B6F5AA26F7E6F7EA28282153FED3BFEED71FF15
 
2055
F103E07F913801C07F0203804B6C7EEC07004A6D7E020E6D7E5C023C6D7E02386D7E1478
 
2056
4A6D7E4A6D7F130149486E7E4A6E7E130749C86C7E496F7E497ED9FFC04A7E00076DEC7F
 
2057
FFB500FC0103B512FEA33F3E7EBD44>I<B66C0103B51280A3000101F0C8EBF8006C6C48
 
2058
ED3FC0725A013F041EC7FC6D7E606D6C15386D6C1578606D6C5D6E14016D5E6D6D130360
 
2059
6E6C49C8FC6E6C5B170E6E6C131E171C6E6C5B6E6C137817706E6C13F06F5B6E13016EEB
 
2060
83C05FED7FC7DB3FE7C9FC16EFED1FFE5E150F6F5AB3A4ED1FFC020FB512FCA3413E7FBD
 
2061
44>I<EAFFFCA4EAF000B3B3B3B3ABEAFFFCA40E5B77C319>91 D<486C13C00003130101
 
2062
001380481303000EEB070048130E0018130C0038131C003013180070133800601330A300
 
2063
E01370481360A400CFEB678039FFC07FE001E013F0A3007F133FA2003F131F01C013E039
 
2064
0F0007801C1C73BE2D>I<EAFFFCA4EA003CB3B3B3B3ABEAFFFCA40E5B7FC319>I<EA0180
 
2065
120313005A120E5A12181238123012701260A312E05AA412CFEAFFC013E0A3127FA2123F
 
2066
13C0EA0F000B1C7ABE19>96 D<EB0FF8EBFFFE3903F01F8039078007E0000F6D7E9038E0
 
2067
01F8D81FF07F6E7EA3157F6C5AEA0380C8FCA4EC1FFF0103B5FC90381FF87FEB7F803801
 
2068
FC00EA07F8EA0FE0485A485AA248C7FCEE038012FEA315FFA3007F5BEC03BF3B3F80071F
 
2069
8700261FC00E13CF3A07F03C0FFE3A01FFF807FC3A003FC001F0292A7DA82D>I<EA01FC
 
2070
12FFA3120712031201B1EC03FC91381FFF8091387C07E09039FDE001F09039FFC000FC4A
 
2071
137E91C77E49158049141F17C0EE0FE0A217F0A2160717F8AA17F0A2160FA217E0161F17
 
2072
C06D1580EE3F006D5C6E13FE9039F3C001F89039F1E003F09039E0780FC09026C03FFFC7
 
2073
FCC7EA07F82D407EBE33>I<49B4FC010F13E090383F00F8017C131E4848131F4848137F
 
2074
0007ECFF80485A5B121FA24848EB7F00151C007F91C7FCA290C9FC5AAB6C7EA3003FEC01
 
2075
C07F001F140316806C6C13076C6C14000003140E6C6C131E6C6C137890383F01F090380F
 
2076
FFC0D901FEC7FC222A7DA828>I<ED01FC15FFA3150715031501B114FF010713E190381F
 
2077
80F990387E003D49131FD803F81307485A49130348481301121F123F5B127FA290C7FCA2
 
2078
5AAA7E7FA2123FA26C7E000F14037F000714076C6C497E6C6C497ED8007C017913F89038
 
2079
3F01F190380FFFC1903A01FE01FC002D407DBE33>I<EB01FE90380FFFC090383F03F090
 
2080
38FC01F848486C7E4848137E48487F000F158049131F001F15C04848130FA2127F16E090
 
2081
C7FCA25AA290B6FCA290C9FCA67EA27F123F16E06C7E1501000F15C06C6C13036DEB0780
 
2082
6C6C1400C66C131E017E5B90381F80F8903807FFE0010090C7FC232A7EA828>I<EC1FC0
 
2083
EC7FF8903801F83C903807E07E90380FC0FFEB1FC1EB3F811401137FEC00FE01FE137C15
 
2084
00AEB6FCA3C648C7FCB3AE487E007F13FFA320407EBF1C>I<167C903903F801FF903A1F
 
2085
FF078F8090397E0FDE1F9038F803F83803F001A23B07E000FC0600000F6EC7FC49137E00
 
2086
1F147FA8000F147E6D13FE00075C6C6C485AA23901F803E03903FE0FC026071FFFC8FCEB
 
2087
03F80006CAFC120EA3120FA27F7F6CB512E015FE6C6E7E6C15E06C810003813A0FC0001F
 
2088
FC48C7EA01FE003E140048157E825A82A46C5D007C153E007E157E6C5D6C6C495A6C6C49
 
2089
5AD803F0EB0FC0D800FE017FC7FC90383FFFFC010313C0293D7EA82D>I<EA01FC12FFA3
 
2090
120712031201B1EC01FE913807FFC091381E07E091387803F09138E001F8D9FDC07F1480
 
2091
01FF6D7E91C7FCA25BA25BB3A6486C497EB5D8F87F13FCA32E3F7DBE33>I<EA01E0EA07
 
2092
F8A2487EA46C5AA2EA01E0C8FCACEA01FC127FA3120712031201B3AC487EB512F0A3143E
 
2093
7DBD1A>I<1478EB01FEA2EB03FFA4EB01FEA2EB00781400AC147FEB7FFFA313017F147F
 
2094
B3B3A5123E127F38FF807E14FEA214FCEB81F8EA7F01387C03F0381E07C0380FFF803801
 
2095
FC00185185BD1C>I<EA01FC12FFA3120712031201B292B51280A392383FFC0016E01680
 
2096
93C7FC153C5D5D4A5AEC07C04A5A4AC8FC143E147F4A7E13FD9038FFDFC0EC9FE0140F49
 
2097
6C7E01FC7F496C7E1401816E7E81826F7E151F826F7EA282486C14FEB539F07FFFE0A32B
 
2098
3F7EBE30>I<EA01FC12FFA3120712031201B3B3B1487EB512F8A3153F7DBE1A>I<2701F8
 
2099
01FE14FF00FF902707FFC00313E0913B1E07E00F03F0913B7803F03C01F80007903BE001
 
2100
F87000FC2603F9C06D487F000101805C01FBD900FF147F91C75B13FF4992C7FCA2495CB3
 
2101
A6486C496CECFF80B5D8F87FD9FC3F13FEA347287DA74C>I<3901F801FE00FF903807FF
 
2102
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2109
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2110
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2124
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2125
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2126
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2130
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2134
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2137
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2138
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2139
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2140
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2171
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2174
2216
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2175
 
1 0 bop 150 2414 a Fv(FFTW)64 b(User's)h(Man)-5 b(ual)p
2176
 
150 2481 3600 34 v 2348 2577 a Fu(F)d(or)32 b(v)m(ersion)d(2.1.3,)k(7)e
2177
 
(No)m(v)m(em)m(b)s(er)g(1999)150 4958 y Ft(Matteo)46
 
2217
TeXDict begin 1 0 bop 150 2414 a Fw(FFTW)64 b(User's)h(Man)-5
 
2218
b(ual)p 150 2481 3600 34 v 2449 2577 a Fv(F)d(or)32 b(v)m(ersion)d
 
2219
(2.1.5,)k(16)e(Marc)m(h)g(2003)150 4958 y Fu(Matteo)46
2178
2220
b(F)-11 b(rigo)150 5091 y(Stev)l(en)46 b(G.)e(Johnson)p
2179
 
150 5141 3600 17 v eop
 
2221
150 5141 3600 17 v eop end
2180
2222
%%Page: 2 2
2181
 
2 1 bop 150 4371 a Fu(Cop)m(yrigh)m(t)602 4368 y(c)577
2182
 
4371 y Fs(\015)30 b Fu(1997{1999)35 b(Massac)m(h)m(usetts)d(Institute)e
2183
 
(of)g(T)-8 b(ec)m(hnology)g(.)150 4505 y(P)m(ermission)30
 
2223
TeXDict begin 2 1 bop 150 4371 a Fv(Cop)m(yrigh)m(t)602
 
2224
4368 y(c)577 4371 y Ft(\015)30 b Fv(1997{1999)35 b(Massac)m(h)m(usetts)
 
2225
d(Institute)e(of)g(T)-8 b(ec)m(hnology)g(.)150 4505 y(P)m(ermission)30
2184
2226
b(is)i(gran)m(ted)h(to)f(mak)m(e)i(and)d(distribute)f(v)m(erbatim)i
2185
2227
(copies)g(of)g(this)f(man)m(ual)h(pro)m(vided)f(the)150
2186
2228
4615 y(cop)m(yrigh)m(t)g(notice)f(and)g(this)f(p)s(ermission)f(notice)i
2197
2239
(di\014ed)d(v)m(ersions,)j(except)g(that)h(this)d(p)s(ermission)f
2198
2240
(notice)150 5322 y(ma)m(y)31 b(b)s(e)f(stated)h(in)e(a)i(translation)e
2199
2241
(appro)m(v)m(ed)i(b)m(y)f(the)h(F)-8 b(ree)31 b(Soft)m(w)m(are)h(F)-8
2200
 
b(oundation.)p eop
 
2242
b(oundation.)p eop end
2201
2243
%%Page: 1 3
2202
 
1 2 bop 150 -116 a Fu(Chapter)30 b(1:)41 b(In)m(tro)s(duction)2591
2203
 
b(1)150 299 y Fr(1)80 b(In)l(tro)t(duction)275 518 y
2204
 
Fu(This)33 b(man)m(ual)i(do)s(cumen)m(ts)h(v)m(ersion)f(2.1.3)j(of)d
2205
 
(FFTW,)i(the)f Fq(F)-7 b(astest)39 b(F)-7 b(ourier)38
2206
 
b(T)-7 b(r)i(ansform)40 b(in)e(the)150 628 y(West)p Fu(.)56
2207
 
b(FFTW)36 b(is)e(a)i(comprehensiv)m(e)f(collection)g(of)h(fast)g(C)f
2208
 
(routines)f(for)h(computing)g(the)g(discrete)150 737
2209
 
y(F)-8 b(ourier)34 b(transform)g(\(DFT\))i(in)d(one)i(or)g(more)g
 
2244
TeXDict begin 1 2 bop 150 -116 a Fv(Chapter)30 b(1:)41
 
2245
b(In)m(tro)s(duction)2591 b(1)150 299 y Fs(1)80 b(In)l(tro)t(duction)
 
2246
275 518 y Fv(This)33 b(man)m(ual)i(do)s(cumen)m(ts)h(v)m(ersion)f
 
2247
(2.1.5)j(of)d(FFTW,)i(the)f Fr(F)-7 b(astest)39 b(F)-7
 
2248
b(ourier)38 b(T)-7 b(r)i(ansform)40 b(in)e(the)150 628
 
2249
y(West)p Fv(.)56 b(FFTW)36 b(is)e(a)i(comprehensiv)m(e)f(collection)g
 
2250
(of)h(fast)g(C)f(routines)f(for)h(computing)g(the)g(discrete)150
 
2251
737 y(F)-8 b(ourier)34 b(transform)g(\(DFT\))i(in)d(one)i(or)g(more)g
2210
2252
(dimensions,)e(of)i(b)s(oth)f(real)g(and)g(complex)h(data,)i(and)150
2211
2253
847 y(of)47 b(arbitrary)e(input)f(size.)89 b(FFTW)47
2212
2254
b(also)f(includes)e(parallel)g(transforms)i(for)g(b)s(oth)g(shared-)g
2215
2257
(familiar)d(with)150 1066 y(the)28 b(prop)s(erties)d(and)i(uses)g(of)g
2216
2258
(the)h(DFT)g(that)g(are)f(relev)-5 b(an)m(t)28 b(to)g(her)f
2217
2259
(application.)38 b(Otherwise,)27 b(see)h(e.g.)150 1176
2218
 
y Fp(The)f(F)-8 b(ast)29 b(F)-8 b(ourier)27 b(T)-8 b(ransform)27
2219
 
b Fu(b)m(y)g(E.)g(O.)h(Brigham)f(\(Pren)m(tice-Hall,)h(Englew)m(o)s(o)s
 
2260
y Fq(The)f(F)-8 b(ast)29 b(F)-8 b(ourier)27 b(T)-8 b(ransform)27
 
2261
b Fv(b)m(y)g(E.)g(O.)h(Brigham)f(\(Pren)m(tice-Hall,)h(Englew)m(o)s(o)s
2220
2262
(d)f(Cli\013s,)f(NJ,)i(1974\).)150 1285 y(Our)h(w)m(eb)h(page)i(\()p
2221
 
Fo(http://www.fftw.org)p Fu(\))25 b(also)31 b(has)f(links)e(to)j
 
2263
Fp(http://www.fftw.org)p Fv(\))25 b(also)31 b(has)f(links)e(to)j
2222
2264
(FFT-related)g(information)e(online.)275 1417 y(FFTW)46
2223
2265
b(is)e(usually)f(faster)j(\(and)f(sometimes)h(m)m(uc)m(h)f(faster\))i
2224
2266
(than)e(all)f(other)i(freely-a)m(v)-5 b(ailable)150 1526
2228
2270
(the)i(FFT)g(co)s(des)g(in)e(Sun's)g(P)m(erformance)i(Library)e(and)h
2229
2271
(IBM's)150 1745 y(ESSL)k(library)-8 b(,)41 b(whic)m(h)f(are)h(targeted)
2230
2272
h(at)f(sp)s(eci\014c)e(mac)m(hines.)71 b(Moreo)m(v)m(er,)45
2231
 
b(FFTW's)d(p)s(erformance)150 1855 y(is)e Fq(p)-5 b(ortable)p
2232
 
Fu(.)73 b(Indeed,)43 b(FFTW)e(is)f(unique)e(in)i(that)h(it)f
 
2273
b(FFTW's)d(p)s(erformance)150 1855 y(is)e Fr(p)-5 b(ortable)p
 
2274
Fv(.)73 b(Indeed,)43 b(FFTW)e(is)f(unique)e(in)i(that)h(it)f
2233
2275
(automatically)g(adapts)h(itself)f(to)h(y)m(our)f(ma-)150
2234
2276
1965 y(c)m(hine,)47 b(y)m(our)c(cac)m(he,)49 b(the)43
2235
2277
b(size)h(of)f(y)m(our)h(memory)-8 b(,)47 b(the)d(n)m(um)m(b)s(er)e(of)i
2241
2283
b(transform)150 2293 y(co)s(des)35 b(has)f(b)s(een)g(made.)53
2242
2284
b(The)34 b(results)f(are)i(a)m(v)-5 b(ailable)34 b(on)g(the)h(W)-8
2243
2285
b(eb)35 b(at)g(the)g(b)s(enc)m(hFFT)f(home)g(page)150
2244
 
2403 y(\()p Fo(http://theory.lcs.mit.edu/)o(~be)o(nchf)o(ft)p
2245
 
Fu(\).)275 2534 y(In)27 b(order)g(to)i(use)e(FFTW)i(e\013ectiv)m(ely)-8
 
2286
2403 y(\()p Fp(http://theory.lcs.mit.edu/)o(~be)o(nchf)o(ft)p
 
2287
Fv(\).)275 2534 y(In)27 b(order)g(to)i(use)e(FFTW)i(e\013ectiv)m(ely)-8
2246
2288
b(,)30 b(y)m(ou)e(need)f(to)i(understand)d(one)i(basic)f(concept)i(of)f
2247
2289
(FFTW's)150 2644 y(in)m(ternal)35 b(structure.)60 b(FFTW)37
2248
2290
b(do)s(es)f(not)h(used)f(a)h(\014xed)f(algorithm)g(for)h(computing)e
2251
2293
(order)g(to)150 2863 y(ac)m(hiev)m(e)27 b(b)s(est)e(p)s(erformance.)38
2252
2294
b(Hence,)27 b(the)f(computation)f(of)h(the)f(transform)g(is)g(split)e
2253
2295
(in)m(to)i(t)m(w)m(o)i(phases.)150 2973 y(First,)k(FFTW's)h
2254
 
Fp(planner)j Fu(is)c(called,)f(whic)m(h)g(\\learns")h(the)g(fastest)h
 
2296
Fq(planner)j Fv(is)c(called,)f(whic)m(h)g(\\learns")h(the)g(fastest)h
2255
2297
(w)m(a)m(y)g(to)g(compute)f(the)h(transform)150 3082
2256
2298
y(on)j(y)m(our)f(mac)m(hine.)54 b(The)34 b(planner)f(pro)s(duces)g(a)i
2257
 
(data)g(structure)g(called)e(a)j Fp(plan)d Fu(that)i(con)m(tains)g
 
2299
(data)g(structure)g(called)e(a)j Fq(plan)d Fv(that)i(con)m(tains)g
2258
2300
(this)150 3192 y(information.)61 b(Subsequen)m(tly)-8
2259
 
b(,)38 b(the)g(plan)f(is)g(passed)g(to)h(FFTW's)h Fp(executor)p
2260
 
Fu(,)i(along)c(with)g(an)g(arra)m(y)150 3302 y(of)c(input)d(data.)48
 
2301
b(,)38 b(the)g(plan)f(is)g(passed)g(to)h(FFTW's)h Fq(executor)p
 
2302
Fv(,)i(along)c(with)g(an)g(arra)m(y)150 3302 y(of)c(input)d(data.)48
2261
2303
b(The)32 b(executor)i(computes)f(the)f(actual)h(transform,)g(as)g
2262
2304
(dictated)f(b)m(y)h(the)g(plan.)45 b(The)150 3411 y(plan)39
2263
2305
b(can)h(b)s(e)g(reused)f(as)i(man)m(y)f(times)g(as)g(needed.)70
2291
2333
s(duce)e(fast)h(C)g(programs)f(for)h(an)m(y)h(particular)d(arra)m(y)j
2292
2334
(size)f(y)m(ou)150 4989 y(ma)m(y)25 b(care)g(ab)s(out.)38
2293
2335
b(F)-8 b(or)25 b(example,)g(if)e(y)m(ou)i(need)f(transforms)f(of)h
2294
 
(size)g(513)j(=)e(19)8 b Fs(\001)g Fu(3)2946 4956 y Fn(3)2984
2295
 
4989 y Fu(,)26 b(y)m(ou)e(can)h(customize)150 5099 y(FFTW)31
 
2336
(size)g(513)j(=)e(19)8 b Ft(\001)g Fv(3)2946 4956 y Fo(3)2984
 
2337
4989 y Fv(,)26 b(y)m(ou)e(can)h(customize)150 5099 y(FFTW)31
2296
2338
b(to)g(supp)s(ort)e(the)h(factor)i(19)f(e\016cien)m(tly)-8
2297
2339
b(.)275 5230 y(FFTW)33 b(can)f(exploit)g(m)m(ultiple)e(pro)s(cessors)i
2298
2340
(if)f(y)m(ou)i(ha)m(v)m(e)g(them.)47 b(FFTW)33 b(comes)g(with)f(a)g
2299
2341
(shared-)150 5340 y(memory)i(implemen)m(tation)f(on)h(top)g(of)g(POSIX)
2300
2342
g(\(and)f(similar\))f(threads,)j(as)g(w)m(ell)e(as)h(a)h(distributed-)p
2301
 
eop
 
2343
eop end
2302
2344
%%Page: 2 4
2303
 
2 3 bop 150 -116 a Fu(2)3277 b(FFTW)150 299 y(memory)23
2304
 
b(implemen)m(tation)e(based)i(on)g(MPI.)g(W)-8 b(e)24
2305
 
b(also)e(pro)m(vide)g(an)h(exp)s(erimen)m(tal)f(parallel)f(implemen-)
2306
 
150 408 y(tation)34 b(written)f(in)g(Cilk,)g Fq(the)j(sup)-5
2307
 
b(erior)37 b(pr)-5 b(o)g(gr)g(amming)40 b(to)-5 b(ol)37
2308
 
b(of)f(choic)-5 b(e)37 b(for)f(discriminating)h(hackers)150
2309
 
518 y Fu(\(Olin)28 b(Shiv)m(ers\).)40 b(\(See)31 b(the)g(Cilk)d(home)i
2310
 
(page)h(\()p Fo(http://supertech.lcs.mit.ed)o(u/ci)o(lk)p
2311
 
Fu(\).\))275 653 y(F)-8 b(or)31 b(more)g(information)e(regarding)g
2312
 
(FFTW,)j(see)f(the)g(pap)s(er,)f(\\The)g(F)-8 b(astest)33
2313
 
b(F)-8 b(ourier)30 b(T)-8 b(ransform)150 762 y(in)29
2314
 
b(the)i(W)-8 b(est,")33 b(b)m(y)e(M.)g(F)-8 b(rigo)32
2315
 
b(and)e(S.)g(G.)i(Johnson,)e(whic)m(h)f(is)h(the)h(tec)m(hnical)g(rep)s
2316
 
(ort)f(MIT-LCS-TR-)150 872 y(728)f(\(Sep.)40 b('97\).)i(See)28
2317
 
b(also,)h(\\FFTW:)g(An)f(Adaptiv)m(e)g(Soft)m(w)m(are)h(Arc)m
2318
 
(hitecture)g(for)f(the)g(FFT,")h(b)m(y)f(M.)150 981 y(F)-8
2319
 
b(rigo)24 b(and)f(S.)g(G.)h(Johnson,)g(whic)m(h)e(app)s(eared)h(in)f
2320
 
(the)i(23rd)g(In)m(ternational)e(Conference)i(on)f(Acoustics,)150
2321
 
1091 y(Sp)s(eec)m(h,)40 b(and)d(Signal)f(Pro)s(cessing)h(\()p
2322
 
Fp(Pro)s(c.)63 b(ICASSP)36 b(1998)47 b Fm(3)p Fu(,)40
2323
 
b(p.)63 b(1381\).)i(The)37 b(co)s(de)h(generator)h(is)150
 
2345
TeXDict begin 2 3 bop 150 -116 a Fv(2)3277 b(FFTW)150
 
2346
299 y(memory)23 b(implemen)m(tation)e(based)i(on)g(MPI.)g(W)-8
 
2347
b(e)24 b(also)e(pro)m(vide)g(an)h(exp)s(erimen)m(tal)f(parallel)f
 
2348
(implemen-)150 408 y(tation)34 b(written)f(in)g(Cilk,)g
 
2349
Fr(the)j(sup)-5 b(erior)37 b(pr)-5 b(o)g(gr)g(amming)40
 
2350
b(to)-5 b(ol)37 b(of)f(choic)-5 b(e)37 b(for)f(discriminating)h
 
2351
(hackers)150 518 y Fv(\(Olin)28 b(Shiv)m(ers\).)40 b(\(See)31
 
2352
b(the)g(Cilk)d(home)i(page)h(\()p Fp(http://supertech.lcs.mit.ed)o
 
2353
(u/ci)o(lk)p Fv(\).\))275 653 y(F)-8 b(or)31 b(more)g(information)e
 
2354
(regarding)g(FFTW,)j(see)f(the)g(pap)s(er,)f(\\The)g(F)-8
 
2355
b(astest)33 b(F)-8 b(ourier)30 b(T)-8 b(ransform)150
 
2356
762 y(in)29 b(the)i(W)-8 b(est,")33 b(b)m(y)e(M.)g(F)-8
 
2357
b(rigo)32 b(and)e(S.)g(G.)i(Johnson,)e(whic)m(h)f(is)h(the)h(tec)m
 
2358
(hnical)g(rep)s(ort)f(MIT-LCS-TR-)150 872 y(728)f(\(Sep.)40
 
2359
b('97\).)i(See)28 b(also,)h(\\FFTW:)g(An)f(Adaptiv)m(e)g(Soft)m(w)m
 
2360
(are)h(Arc)m(hitecture)g(for)f(the)g(FFT,")h(b)m(y)f(M.)150
 
2361
981 y(F)-8 b(rigo)24 b(and)f(S.)g(G.)h(Johnson,)g(whic)m(h)e(app)s
 
2362
(eared)h(in)f(the)i(23rd)g(In)m(ternational)e(Conference)i(on)f
 
2363
(Acoustics,)150 1091 y(Sp)s(eec)m(h,)40 b(and)d(Signal)f(Pro)s(cessing)
 
2364
h(\()p Fq(Pro)s(c.)63 b(ICASSP)36 b(1998)47 b Fn(3)p
 
2365
Fv(,)40 b(p.)63 b(1381\).)i(The)37 b(co)s(de)h(generator)h(is)150
2324
2366
1200 y(describ)s(ed)d(in)g(the)i(pap)s(er)e(\\A)i(F)-8
2325
2367
b(ast)39 b(F)-8 b(ourier)37 b(T)-8 b(ransform)37 b(Compiler",)h(b)m(y)g
2326
2368
(M.)g(F)-8 b(rigo,)40 b(to)e(app)s(ear)f(in)150 1310
2327
 
y(the)25 b Fp(Pro)s(ceedings)f(of)h(the)g(1999)h(A)m(CM)g(SIGPLAN)e
 
2369
y(the)25 b Fq(Pro)s(ceedings)f(of)h(the)g(1999)h(A)m(CM)g(SIGPLAN)e
2328
2370
(Conference)h(on)f(Programming)g(Language)i(Design)150
2329
2371
1420 y(and)36 b(Implemen)m(tation)h(\(PLDI\),)h(A)m(tlan)m(ta,)i
2330
 
(Georgia,)g(Ma)m(y)e(1999)p Fu(.)62 b(These)37 b(pap)s(ers,)h(along)f
 
2372
(Georgia,)g(Ma)m(y)e(1999)p Fv(.)62 b(These)37 b(pap)s(ers,)h(along)f
2331
2373
(with)f(the)150 1529 y(latest)28 b(v)m(ersion)e(of)h(FFTW,)h(the)f(F)
2332
2374
-10 b(A)m(Q,)28 b(b)s(enc)m(hmarks,)f(and)g(other)g(links,)e(are)j(a)m
2333
2375
(v)-5 b(ailable)26 b(at)i(the)f(FFTW)150 1639 y(home)22
2334
 
b(page)h(\()p Fo(http://www.fftw.org)p Fu(\).)33 b(The)21
 
2376
b(page)h(\()p Fp(http://www.fftw.org)p Fv(\).)33 b(The)21
2335
2377
b(curren)m(t)h(v)m(ersion)f(of)h(FFTW)h(incorp)s(orates)e(man)m(y)h(go)
2336
2378
s(o)s(d)150 1748 y(ideas)30 b(from)f(the)i(past)f(thirt)m(y)g(y)m(ears)
2337
2379
h(of)f(FFT)g(literature.)40 b(In)30 b(one)g(w)m(a)m(y)h(or)f(another,)h
2370
2412
(Chapter)g(8)h([License)g(and)f(Cop)m(yrigh)m(t],)i(page)g(63.)52
2371
2413
b(Finally)-8 b(,)33 b(w)m(e)h(thank)g(all)150 3527 y(the)d(p)s(eople)e
2372
2414
(who)h(help)s(ed)e(us)i(in)f(Chapter)h(7)h([Ac)m(kno)m(wledgmen)m(ts],)
2373
 
h(page)f(61.)p eop
 
2415
h(page)f(61.)p eop end
2374
2416
%%Page: 3 5
2375
 
3 4 bop 150 -116 a Fu(Chapter)30 b(2:)41 b(T)-8 b(utorial)2768
2376
 
b(3)150 299 y Fr(2)80 b(T)-13 b(utorial)275 535 y Fu(This)24
2377
 
b(c)m(hapter)i(describ)s(es)f(the)h(basic)f(usage)i(of)f(FFTW,)h(i.e.,)
2378
 
g(ho)m(w)f(to)h(compute)f(the)g(F)-8 b(ourier)26 b(trans-)150
2379
 
645 y(form)33 b(of)h(a)h(single)d(arra)m(y)-8 b(.)52
2380
 
b(This)32 b(c)m(hapter)i(tells)f(the)h(truth,)g(but)g(not)g(the)g
2381
 
Fq(whole)42 b Fu(truth.)50 b(Sp)s(eci\014cally)-8 b(,)150
 
2417
TeXDict begin 3 4 bop 150 -116 a Fv(Chapter)30 b(2:)41
 
2418
b(T)-8 b(utorial)2768 b(3)150 299 y Fs(2)80 b(T)-13 b(utorial)275
 
2419
535 y Fv(This)24 b(c)m(hapter)i(describ)s(es)f(the)h(basic)f(usage)i
 
2420
(of)f(FFTW,)h(i.e.,)g(ho)m(w)f(to)h(compute)f(the)g(F)-8
 
2421
b(ourier)26 b(trans-)150 645 y(form)33 b(of)h(a)h(single)d(arra)m(y)-8
 
2422
b(.)52 b(This)32 b(c)m(hapter)i(tells)f(the)h(truth,)g(but)g(not)g(the)
 
2423
g Fr(whole)42 b Fv(truth.)50 b(Sp)s(eci\014cally)-8 b(,)150
2382
2424
754 y(FFTW)26 b(implemen)m(ts)f(additional)e(routines)i(and)g(\015ags,)
2383
2425
i(pro)m(viding)d(extra)i(functionalit)m(y)-8 b(,)26 b(that)h(are)f(not)
2384
2426
150 864 y(do)s(cumen)m(ted)32 b(here.)45 b(See)33 b(Chapter)e(3)h
2391
2433
(installation)f(of)j(FFTW.)h(In)d(some)i(installations)d(\(particulary)
2392
2434
h(from)150 1328 y(binary)e(pac)m(k)-5 b(ages\),)28 b(the)d(FFTW)g
2393
2435
(header)g(\014les)e(and)i(libraries)c(are)k(pre\014xed)f(with)f(`)p
2394
 
Fo(d)p Fu(')i(or)g(`)p Fo(s)p Fu(')f(to)i(indicate)150
 
2436
Fp(d)p Fv(')i(or)g(`)p Fp(s)p Fv(')f(to)i(indicate)150
2395
2437
1438 y(v)m(ersions)k(in)f(double)g(or)i(single)e(precision,)g(resp)s
2396
2438
(ectiv)m(ely)-8 b(.)42 b(The)30 b(usage)h(of)g(FFTW)g(in)e(that)i(case)
2397
 
h(is)e(the)150 1547 y(same,)e(except)h(that)e Fo(#include)e
2398
 
Fu(directiv)m(es)i(and)f(link)g(commands)g(m)m(ust)h(use)g(the)h
 
2439
h(is)e(the)150 1547 y(same,)e(except)h(that)e Fp(#include)e
 
2440
Fv(directiv)m(es)i(and)f(link)g(commands)g(m)m(ust)h(use)g(the)h
2399
2441
(appropriate)e(pre\014x.)150 1657 y(See)35 b(Section)g(6.3)i
2400
2442
([Installing)c(FFTW)j(in)e(b)s(oth)g(single)g(and)h(double)e
2401
2443
(precision],)i(page)h(57,)i(for)d(more)150 1766 y(information.)275
2423
2465
(ulti-dimensional)d(arra)m(ys)k(in)e(memory)-8 b(.)39
2424
2466
b(Section)27 b(2.6)150 2997 y([W)-8 b(ords)31 b(of)f(Wisdom],)g(page)i
2425
2467
(13)f(sho)m(ws)f(ho)m(w)g(y)m(ou)h(can)g(sa)m(v)m(e)h(FFTW's)f(plans)e
2426
 
(for)h(future)f(use.)150 3257 y Ft(2.1)68 b(Complex)46
 
2468
(for)h(future)f(use.)150 3257 y Fu(2.1)68 b(Complex)46
2427
2469
b(One-dimensional)g(T)-11 b(ransforms)45 b(T)-11 b(utorial)275
2428
 
3450 y Fu(The)29 b(basic)h(usage)h(of)g(FFTW)g(is)e(simple.)39
 
2470
3450 y Fv(The)29 b(basic)h(usage)h(of)g(FFTW)g(is)e(simple.)39
2429
2471
b(A)30 b(t)m(ypical)g(call)g(to)h(FFTW)g(lo)s(oks)f(lik)m(e:)390
2430
 
3579 y Fo(#include)46 b(<fftw.h>)390 3683 y(...)390 3787
 
2472
3579 y Fp(#include)46 b(<fftw.h>)390 3683 y(...)390 3787
2431
2473
y({)629 3891 y(fftw_complex)e(in[N],)i(out[N];)629 3994
2432
2474
y(fftw_plan)f(p;)629 4098 y(...)629 4202 y(p)i(=)h
2433
2475
(fftw_create_plan\(N,)42 b(FFTW_FORWARD,)i(FFTW_ESTIMATE\);)629
2434
2476
4306 y(...)629 4410 y(fftw_one\(p,)g(in,)j(out\);)629
2435
2477
4513 y(...)629 4617 y(fftw_destroy_plan\(p\);)390 4721
2436
 
y(})275 4856 y Fu(The)33 b(\014rst)g(thing)g(w)m(e)h(do)g(is)f(to)i
2437
 
(create)g(a)f Fp(plan)p Fu(,)g(whic)m(h)e(is)h(an)h(ob)5
 
2478
y(})275 4856 y Fv(The)33 b(\014rst)g(thing)g(w)m(e)h(do)g(is)f(to)i
 
2479
(create)g(a)f Fq(plan)p Fv(,)g(whic)m(h)e(is)h(an)h(ob)5
2438
2480
b(ject)35 b(that)f(con)m(tains)g(all)f(the)h(data)150
2439
2481
4966 y(that)d(FFTW)g(needs)f(to)h(compute)g(the)f(FFT,)h(using)e(the)i
2440
 
(follo)m(wing)e(function:)390 5095 y Fo(fftw_plan)45
 
2482
(follo)m(wing)e(function:)390 5095 y Fp(fftw_plan)45
2441
2483
b(fftw_create_plan\(int)e(n,)k(fftw_direction)d(dir,)i(int)h(flags\);)
2442
 
275 5230 y Fu(The)26 b(\014rst)h(argumen)m(t,)i Fo(n)p
2443
 
Fu(,)e(is)g(the)g(size)h(of)f(the)h(transform)e(y)m(ou)i(are)g(trying)e
2444
 
(to)i(compute.)40 b(The)27 b(size)g Fo(n)150 5340 y Fu(can)c(b)s(e)f
 
2484
275 5230 y Fv(The)26 b(\014rst)h(argumen)m(t,)i Fp(n)p
 
2485
Fv(,)e(is)g(the)g(size)h(of)f(the)h(transform)e(y)m(ou)i(are)g(trying)e
 
2486
(to)i(compute.)40 b(The)27 b(size)g Fp(n)150 5340 y Fv(can)c(b)s(e)f
2445
2487
(an)m(y)h(p)s(ositiv)m(e)e(in)m(teger,)k(but)d(sizes)g(that)h(are)g
2446
2488
(pro)s(ducts)f(of)g(small)f(factors)j(are)f(transformed)f(most)p
2447
 
eop
 
2489
eop end
2448
2490
%%Page: 4 6
2449
 
4 5 bop 150 -116 a Fu(4)3277 b(FFTW)150 299 y(e\016cien)m(tly)-8
2450
 
b(.)40 b(The)26 b(second)h(argumen)m(t,)i Fo(dir)p Fu(,)e(can)g(b)s(e)f
2451
 
(either)h Fo(FFTW_FORWARD)c Fu(or)k Fo(FFTW_BACKWARD)p
2452
 
Fu(,)e(and)150 408 y(indicates)35 b(the)i(direction)e(of)h(the)h
2453
 
(transform)f(y)m(ou)g(are)h(in)m(terested)g(in.)57 b(Alternativ)m(ely)
2454
 
-8 b(,)38 b(y)m(ou)f(can)f(use)150 518 y(the)c(sign)g(of)g(the)g(exp)s
2455
 
(onen)m(t)h(in)d(the)j(transform,)f Fs(\000)p Fu(1)g(or)h(+)o(1,)h
2456
 
(whic)m(h)d(corresp)s(onds)f(to)j Fo(FFTW_FORWARD)150
2457
 
628 y Fu(or)43 b Fo(FFTW_BACKWARD)c Fu(resp)s(ectiv)m(ely)-8
2458
 
b(.)78 b(The)42 b Fo(flags)g Fu(argumen)m(t)h(is)f(either)g
2459
 
Fo(FFTW_MEASURE)e Fu(or)i Fo(FFTW_)150 737 y(ESTIMATE)p
2460
 
Fu(.)68 b Fo(FFTW_MEASURE)37 b Fu(means)j(that)h(FFTW)f(actually)g
2461
 
(runs)f(and)g(measures)h(the)h(execution)150 847 y(time)f(of)g(sev)m
2462
 
(eral)g(FFTs)h(in)d(order)i(to)h(\014nd)d(the)i(b)s(est)g(w)m(a)m(y)h
2463
 
(to)g(compute)f(the)h(transform)e(of)h(size)g Fo(n)p
2464
 
Fu(.)150 956 y(This)27 b(ma)m(y)j(tak)m(e)h(some)e(time,)g(dep)s
2465
 
(ending)e(on)i(y)m(our)g(installation)e(and)i(on)g(the)g(precision)e
2466
 
(of)i(the)h(timer)150 1066 y(in)39 b(y)m(our)h(mac)m(hine.)71
2467
 
b Fo(FFTW_ESTIMATE)p Fu(,)39 b(on)h(the)h(con)m(trary)-8
 
2491
TeXDict begin 4 5 bop 150 -116 a Fv(4)3277 b(FFTW)150
 
2492
299 y(e\016cien)m(tly)-8 b(.)40 b(The)26 b(second)h(argumen)m(t,)i
 
2493
Fp(dir)p Fv(,)e(can)g(b)s(e)f(either)h Fp(FFTW_FORWARD)c
 
2494
Fv(or)k Fp(FFTW_BACKWARD)p Fv(,)e(and)150 408 y(indicates)35
 
2495
b(the)i(direction)e(of)h(the)h(transform)f(y)m(ou)g(are)h(in)m
 
2496
(terested)g(in.)57 b(Alternativ)m(ely)-8 b(,)38 b(y)m(ou)f(can)f(use)
 
2497
150 518 y(the)c(sign)g(of)g(the)g(exp)s(onen)m(t)h(in)d(the)j
 
2498
(transform,)f Ft(\000)p Fv(1)g(or)h(+)o(1,)h(whic)m(h)d(corresp)s(onds)
 
2499
f(to)j Fp(FFTW_FORWARD)150 628 y Fv(or)43 b Fp(FFTW_BACKWARD)c
 
2500
Fv(resp)s(ectiv)m(ely)-8 b(.)78 b(The)42 b Fp(flags)g
 
2501
Fv(argumen)m(t)h(is)f(either)g Fp(FFTW_MEASURE)e Fv(or)i
 
2502
Fp(FFTW_)150 737 y(ESTIMATE)p Fv(.)68 b Fp(FFTW_MEASURE)37
 
2503
b Fv(means)j(that)h(FFTW)f(actually)g(runs)f(and)g(measures)h(the)h
 
2504
(execution)150 847 y(time)f(of)g(sev)m(eral)g(FFTs)h(in)d(order)i(to)h
 
2505
(\014nd)d(the)i(b)s(est)g(w)m(a)m(y)h(to)g(compute)f(the)h(transform)e
 
2506
(of)h(size)g Fp(n)p Fv(.)150 956 y(This)27 b(ma)m(y)j(tak)m(e)h(some)e
 
2507
(time,)g(dep)s(ending)e(on)i(y)m(our)g(installation)e(and)i(on)g(the)g
 
2508
(precision)e(of)i(the)h(timer)150 1066 y(in)39 b(y)m(our)h(mac)m(hine.)
 
2509
71 b Fp(FFTW_ESTIMATE)p Fv(,)39 b(on)h(the)h(con)m(trary)-8
2468
2510
b(,)44 b(do)s(es)c(not)h(run)e(an)m(y)i(computation,)i(and)150
2469
2511
1176 y(just)31 b(builds)e(a)j(reasonable)g(plan,)f(whic)m(h)g(ma)m(y)h
2470
2512
(b)s(e)g(sub-optimal.)43 b(In)31 b(other)h(w)m(ords,)g(if)f(y)m(our)h
2471
2513
(program)150 1285 y(p)s(erforms)37 b(man)m(y)i(transforms)f(of)h(the)g
2472
2514
(same)g(size)f(and)g(initialization)e(time)i(is)g(not)h(imp)s(ortan)m
2473
 
(t,)h(use)150 1395 y Fo(FFTW_MEASURE)p Fu(;)h(otherwise)d(use)h(the)h
 
2515
(t,)h(use)150 1395 y Fp(FFTW_MEASURE)p Fv(;)h(otherwise)d(use)h(the)h
2474
2516
(estimate.)68 b(\(A)40 b(compromise)f(b)s(et)m(w)m(een)h(these)g(t)m(w)
2475
2517
m(o)g(extremes)150 1504 y(exists.)g(See)31 b(Section)f(2.6)i([W)-8
2476
2518
b(ords)30 b(of)h(Wisdom],)f(page)h(13.\))275 1639 y(Once)d(the)g(plan)f
2478
2520
(times)e(as)i(y)m(ou)f(lik)m(e)g(for)g(transforms)150
2479
2521
1749 y(on)k(arra)m(ys)h(of)f(the)g(same)h(size.)46 b(When)32
2480
2522
b(y)m(ou)g(are)h(done)f(with)f(the)h(plan,)g(y)m(ou)g(deallo)s(cate)h
2481
 
(it)e(b)m(y)h(calling)150 1858 y Fo(fftw_destroy_plan\(plan\))p
2482
 
Fu(.)275 1993 y(The)g(transform)h(itself)g(is)f(computed)h(b)m(y)h
 
2523
(it)e(b)m(y)h(calling)150 1858 y Fp(fftw_destroy_plan\(plan\))p
 
2524
Fv(.)275 1993 y(The)g(transform)h(itself)g(is)f(computed)h(b)m(y)h
2483
2525
(passing)e(the)i(plan)e(along)h(with)f(the)i(input)d(and)i(output)150
2484
 
2102 y(arra)m(ys)e(to)g Fo(fftw_one)p Fu(:)390 2231 y
2485
 
Fo(void)47 b(fftw_one\(fftw_plan)42 b(plan,)47 b(fftw_complex)d(*in,)j
2486
 
(fftw_complex)d(*out\);)275 2366 y Fu(Note)h(that)f(the)g(transform)f
2487
 
(is)f(out)i(of)g(place:)68 b Fo(in)43 b Fu(and)g Fo(out)g
2488
 
Fu(m)m(ust)g(p)s(oin)m(t)g(to)h(distinct)e(arra)m(ys.)150
 
2526
2102 y(arra)m(ys)e(to)g Fp(fftw_one)p Fv(:)390 2231 y
 
2527
Fp(void)47 b(fftw_one\(fftw_plan)42 b(plan,)47 b(fftw_complex)d(*in,)j
 
2528
(fftw_complex)d(*out\);)275 2366 y Fv(Note)h(that)f(the)g(transform)f
 
2529
(is)f(out)i(of)g(place:)68 b Fp(in)43 b Fv(and)g Fp(out)g
 
2530
Fv(m)m(ust)g(p)s(oin)m(t)g(to)h(distinct)e(arra)m(ys.)150
2489
2531
2475 y(It)i(op)s(erates)h(on)f(data)h(of)g(t)m(yp)s(e)f
2490
 
Fo(fftw_complex)p Fu(,)h(a)f(data)h(structure)f(with)f(real)h(\()p
2491
 
Fo(in[i].re)p Fu(\))f(and)150 2585 y(imaginary)31 b(\()p
2492
 
Fo(in[i].im)p Fu(\))g(\015oating-p)s(oin)m(t)h(comp)s(onen)m(ts.)47
2493
 
b(The)32 b Fo(in)g Fu(and)g Fo(out)g Fu(arra)m(ys)g(should)f(ha)m(v)m
 
2532
Fp(fftw_complex)p Fv(,)h(a)f(data)h(structure)f(with)f(real)h(\()p
 
2533
Fp(in[i].re)p Fv(\))f(and)150 2585 y(imaginary)31 b(\()p
 
2534
Fp(in[i].im)p Fv(\))g(\015oating-p)s(oin)m(t)h(comp)s(onen)m(ts.)47
 
2535
b(The)32 b Fp(in)g Fv(and)g Fp(out)g Fv(arra)m(ys)g(should)f(ha)m(v)m
2494
2536
(e)j(the)150 2694 y(length)h(sp)s(eci\014ed)e(when)i(the)g(plan)f(w)m
2495
2537
(as)i(created.)56 b(An)35 b(alternativ)m(e)h(function,)f
2496
 
Fo(fftw)p Fu(,)h(allo)m(ws)f(y)m(ou)g(to)150 2804 y(e\016cien)m(tly)g
 
2538
Fp(fftw)p Fv(,)h(allo)m(ws)f(y)m(ou)g(to)150 2804 y(e\016cien)m(tly)g
2497
2539
(p)s(erform)f(m)m(ultiple)g(and/or)h(strided)f(transforms)h(\(see)h
2498
2540
(Chapter)f(3)h([FFTW)h(Reference],)150 2914 y(page)31
2499
2541
b(17\).)275 3048 y(The)39 b(DFT)i(results)e(are)i(stored)g(in-order)e
2500
 
(in)g(the)h(arra)m(y)h Fo(out)p Fu(,)h(with)e(the)g(zero-frequency)h
2501
 
(\(DC\))150 3158 y(comp)s(onen)m(t)46 b(in)e Fo(out[0])p
2502
 
Fu(.)84 b(The)45 b(arra)m(y)h Fo(in)f Fu(is)f(not)i(mo)s(di\014ed.)83
 
2542
(in)g(the)h(arra)m(y)h Fp(out)p Fv(,)h(with)e(the)g(zero-frequency)h
 
2543
(\(DC\))150 3158 y(comp)s(onen)m(t)46 b(in)e Fp(out[0])p
 
2544
Fv(.)84 b(The)45 b(arra)m(y)h Fp(in)f Fv(is)f(not)i(mo)s(di\014ed.)83
2503
2545
b(Users)46 b(should)d(note)j(that)g(FFTW)150 3267 y(computes)27
2504
2546
b(an)h(unnormalized)c(DFT,)k(the)g(sign)e(of)h(whose)h(exp)s(onen)m(t)f
2505
 
(is)f(giv)m(en)h(b)m(y)h(the)f Fo(dir)f Fu(parameter)150
2506
 
3377 y(of)35 b Fo(fftw_create_plan)p Fu(.)50 b(Th)m(us,)35
 
2547
(is)f(giv)m(en)h(b)m(y)h(the)f Fp(dir)f Fv(parameter)150
 
2548
3377 y(of)35 b Fp(fftw_create_plan)p Fv(.)50 b(Th)m(us,)35
2507
2549
b(computing)f(a)h(forw)m(ard)f(follo)m(w)m(ed)g(b)m(y)h(a)g(bac)m(kw)m
2508
2550
(ard)g(transform)g(\(or)150 3486 y(vice)29 b(v)m(ersa\))i(results)d(in)
2509
 
g(the)h(original)f(arra)m(y)h(scaled)h(b)m(y)f Fo(n)p
2510
 
Fu(.)40 b(See)29 b(Section)g(3.2.5)j([What)e(FFTW)g(Really)150
 
2551
g(the)h(original)f(arra)m(y)h(scaled)h(b)m(y)f Fp(n)p
 
2552
Fv(.)40 b(See)29 b(Section)g(3.2.5)j([What)e(FFTW)g(Really)150
2511
2553
3596 y(Computes],)g(page)h(21,)h(for)e(the)h(de\014nition)d(of)i(DFT.)
2512
2554
275 3731 y(A)37 b(program)g(using)f(FFTW)h(should)f(b)s(e)g(link)m(ed)g
2513
 
(with)g Fo(-lfftw)28 b(-lm)37 b Fu(on)g(Unix)f(systems,)j(or)e(with)150
 
2555
(with)g Fp(-lfftw)28 b(-lm)37 b Fv(on)g(Unix)f(systems,)j(or)e(with)150
2514
2556
3840 y(the)31 b(FFTW)g(and)e(standard)h(math)g(libraries)e(in)h
2515
 
(general.)150 4098 y Ft(2.2)68 b(Complex)46 b(Multi-dimensional)g(T)-11
2516
 
b(ransforms)45 b(T)-11 b(utorial)275 4290 y Fu(FFTW)34
 
2557
(general.)150 4098 y Fu(2.2)68 b(Complex)46 b(Multi-dimensional)g(T)-11
 
2558
b(ransforms)45 b(T)-11 b(utorial)275 4290 y Fv(FFTW)34
2517
2559
b(can)h(also)f(compute)g(transforms)f(of)i(an)m(y)f(n)m(um)m(b)s(er)f
2518
 
(of)h(dimensions)d(\()p Fp(rank)6 b Fu(\).)52 b(The)33
 
2560
(of)h(dimensions)d(\()p Fq(rank)6 b Fv(\).)52 b(The)33
2519
2561
b(syn)m(tax)150 4400 y(is)i(similar)e(to)j(that)g(for)f(the)h
2520
 
(one-dimensional)e(transforms,)i(with)e(`)p Fo(fftw_)p
2521
 
Fu(')h(replaced)g(b)m(y)g(`)p Fo(fftwnd_)p Fu(')150 4510
2522
 
y(\(whic)m(h)29 b(stands)h(for)g(\\)p Fo(fftw)g Fu(in)f
2523
 
Fo(N)h Fu(dimensions"\).)275 4644 y(As)24 b(b)s(efore,)i(w)m(e)g
2524
 
Fo(#include)i(<fftw.h>)22 b Fu(and)j(create)h(a)f(plan)f(for)h(the)g
 
2562
(one-dimensional)e(transforms,)i(with)e(`)p Fp(fftw_)p
 
2563
Fv(')h(replaced)g(b)m(y)g(`)p Fp(fftwnd_)p Fv(')150 4510
 
2564
y(\(whic)m(h)29 b(stands)h(for)g(\\)p Fp(fftw)g Fv(in)f
 
2565
Fp(N)h Fv(dimensions"\).)275 4644 y(As)24 b(b)s(efore,)i(w)m(e)g
 
2566
Fp(#include)i(<fftw.h>)22 b Fv(and)j(create)h(a)f(plan)f(for)h(the)g
2525
2567
(transforms,)g(this)f(time)h(of)g(t)m(yp)s(e)150 4754
2526
 
y Fo(fftwnd_plan)p Fu(:)390 4882 y Fo(fftwnd_plan)45
 
2568
y Fp(fftwnd_plan)p Fv(:)390 4882 y Fp(fftwnd_plan)45
2527
2569
b(fftwnd_create_plan\(int)c(rank,)47 b(const)f(int)h(*n,)1870
2528
2570
4986 y(fftw_direction)c(dir,)k(int)g(flags\);)275 5121
2529
 
y(rank)30 b Fu(is)g(the)h(dimensionalit)m(y)d(of)k(the)f(arra)m(y)-8
 
2571
y(rank)30 b Fv(is)g(the)h(dimensionalit)m(y)d(of)k(the)f(arra)m(y)-8
2530
2572
b(,)33 b(and)d(can)i(b)s(e)e(an)m(y)i(non-negativ)m(e)g(in)m(teger.)43
2531
 
b(The)31 b(next)150 5230 y(argumen)m(t,)i Fo(n)p Fu(,)f(is)f(a)h(p)s
 
2573
b(The)31 b(next)150 5230 y(argumen)m(t,)i Fp(n)p Fv(,)f(is)f(a)h(p)s
2532
2574
(oin)m(ter)f(to)h(an)g(in)m(teger)g(arra)m(y)h(of)f(length)f
2533
 
Fo(rank)f Fu(con)m(taining)i(the)g(\(p)s(ositiv)m(e\))f(sizes)150
 
2575
Fp(rank)f Fv(con)m(taining)i(the)g(\(p)s(ositiv)m(e\))f(sizes)150
2534
2576
5340 y(of)e(eac)m(h)g(dimension)d(of)i(the)h(arra)m(y)-8
2535
2577
b(.)41 b(\(Note)30 b(that)f(the)f(arra)m(y)h(will)d(b)s(e)i(stored)g
2536
 
(in)f(ro)m(w-ma)5 b(jor)29 b(order.)40 b(See)p eop
 
2578
(in)f(ro)m(w-ma)5 b(jor)29 b(order.)40 b(See)p eop end
2537
2579
%%Page: 5 7
2538
 
5 6 bop 150 -116 a Fu(Chapter)30 b(2:)41 b(T)-8 b(utorial)2768
2539
 
b(5)150 299 y(Section)21 b(2.5)i([Multi-dimensional)17
2540
 
b(Arra)m(y)22 b(F)-8 b(ormat],)25 b(page)d(11,)i(for)d(information)f
2541
 
(on)h(ro)m(w-ma)5 b(jor)22 b(order.\))150 408 y(The)27
2542
 
b(last)h(t)m(w)m(o)h(parameters)f(are)g(the)f(same)h(as)g(in)f
2543
 
Fo(fftw_create_plan)p Fu(.)35 b(W)-8 b(e)29 b(no)m(w,)f(ho)m(w)m(ev)m
2544
 
(er,)i(ha)m(v)m(e)f(an)150 518 y(additional)35 b(p)s(ossible)g(\015ag,)
2545
 
k Fo(FFTW_IN_PLACE)p Fu(,)d(since)g Fo(fftwnd)g Fu(supp)s(orts)f(true)i
2546
 
(in-place)g(transforms.)150 628 y(Multiple)25 b(\015ags)j(are)f(com)m
2547
 
(bined)g(using)f(a)h(bit)m(wise)f Fp(or)34 b Fu(\(`)p
2548
 
Fo(|)p Fu('\).)41 b(\(An)27 b Fp(in-place)32 b Fu(transform)26
 
2580
TeXDict begin 5 6 bop 150 -116 a Fv(Chapter)30 b(2:)41
 
2581
b(T)-8 b(utorial)2768 b(5)150 299 y(Section)21 b(2.5)i
 
2582
([Multi-dimensional)17 b(Arra)m(y)22 b(F)-8 b(ormat],)25
 
2583
b(page)d(11,)i(for)d(information)f(on)h(ro)m(w-ma)5 b(jor)22
 
2584
b(order.\))150 408 y(The)27 b(last)h(t)m(w)m(o)h(parameters)f(are)g
 
2585
(the)f(same)h(as)g(in)f Fp(fftw_create_plan)p Fv(.)35
 
2586
b(W)-8 b(e)29 b(no)m(w,)f(ho)m(w)m(ev)m(er,)i(ha)m(v)m(e)f(an)150
 
2587
518 y(additional)35 b(p)s(ossible)g(\015ag,)k Fp(FFTW_IN_PLACE)p
 
2588
Fv(,)d(since)g Fp(fftwnd)g Fv(supp)s(orts)f(true)i(in-place)g
 
2589
(transforms.)150 628 y(Multiple)25 b(\015ags)j(are)f(com)m(bined)g
 
2590
(using)f(a)h(bit)m(wise)f Fq(or)34 b Fv(\(`)p Fp(|)p
 
2591
Fv('\).)41 b(\(An)27 b Fq(in-place)32 b Fv(transform)26
2549
2592
b(is)h(one)g(in)f(whic)m(h)150 737 y(the)i(output)g(data)g(o)m(v)m
2550
2593
(erwrite)h(the)f(input)e(data.)40 b(It)29 b(th)m(us)e(requires)g(half)f
2551
2594
(as)j(m)m(uc)m(h)f(memory)f(as|and)h(is)150 847 y(often)j(faster)g
2552
 
(than|its)e(opp)s(osite,)h(an)g Fp(out-of-place)37 b
2553
 
Fu(transform.\))275 1005 y(F)-8 b(or)25 b(t)m(w)m(o-)g(and)f
 
2595
(than|its)e(opp)s(osite,)h(an)g Fq(out-of-place)37 b
 
2596
Fv(transform.\))275 1005 y(F)-8 b(or)25 b(t)m(w)m(o-)g(and)f
2554
2597
(three-dimensional)e(transforms,)j(FFTWND)g(pro)m(vides)f(alternativ)m
2555
2598
(e)g(routines)f(that)150 1114 y(accept)37 b(the)e(sizes)h(of)f(eac)m(h)
2556
2599
i(dimension)c(directly)-8 b(,)36 b(rather)f(than)g(indirectly)e
2557
2600
(through)i(a)g(rank)g(and)g(an)150 1224 y(arra)m(y)h(of)f(sizes.)55
2558
 
b(These)35 b(are)h(otherwise)f(iden)m(tical)f(to)i Fo
2559
 
(fftwnd_create_plan)p Fu(,)c(and)i(are)i(sometimes)150
2560
 
1333 y(more)31 b(con)m(v)m(enien)m(t:)390 1485 y Fo(fftwnd_plan)45
 
2601
b(These)35 b(are)h(otherwise)f(iden)m(tical)f(to)i Fp
 
2602
(fftwnd_create_plan)p Fv(,)c(and)i(are)i(sometimes)150
 
2603
1333 y(more)31 b(con)m(v)m(enien)m(t:)390 1485 y Fp(fftwnd_plan)45
2561
2604
b(fftw2d_create_plan\(int)c(nx,)47 b(int)g(ny,)1870 1589
2562
2605
y(fftw_direction)c(dir,)k(int)g(flags\);)390 1693 y(fftwnd_plan)e
2563
2606
(fftw3d_create_plan\(int)c(nx,)47 b(int)g(ny,)g(int)g(nz,)1870
2564
2607
1797 y(fftw_direction)c(dir,)k(int)g(flags\);)275 1954
2565
 
y Fu(Once)24 b(the)h(plan)f(has)g(b)s(een)g(created,)j(y)m(ou)f(can)f
 
2608
y Fv(Once)24 b(the)h(plan)f(has)g(b)s(een)g(created,)j(y)m(ou)f(can)f
2566
2609
(use)f(it)g(an)m(y)i(n)m(um)m(b)s(er)d(of)i(times)f(for)h(transforms)f
2567
2610
(of)h(the)150 2064 y(same)33 b(size.)49 b(When)33 b(y)m(ou)g(do)g(not)g
2568
2611
(need)g(a)g(plan)f(an)m(ymore,)j(y)m(ou)e(can)g(deallo)s(cate)h(the)f
2569
 
(plan)f(b)m(y)g(calling)150 2174 y Fo(fftwnd_destroy_plan\(plan)o(\))p
2570
 
Fu(.)275 2331 y(Giv)m(en)e(a)h(plan,)e(y)m(ou)i(can)f(compute)h(the)f
 
2612
(plan)f(b)m(y)g(calling)150 2174 y Fp(fftwnd_destroy_plan\(plan)o(\))p
 
2613
Fv(.)275 2331 y(Giv)m(en)e(a)h(plan,)e(y)m(ou)i(can)f(compute)h(the)f
2571
2614
(transform)g(of)h(an)f(arra)m(y)h(of)f(data)h(b)m(y)g(calling:)390
2572
 
2483 y Fo(void)47 b(fftwnd_one\(fftwnd_plan)41 b(plan,)47
 
2615
2483 y Fp(void)47 b(fftwnd_one\(fftwnd_plan)41 b(plan,)47
2573
2616
b(fftw_complex)d(*in,)j(fftw_complex)d(*out\);)275 2641
2574
 
y Fu(Here,)k Fo(in)43 b Fu(and)h Fo(out)f Fu(p)s(oin)m(t)g(to)i(m)m
 
2617
y Fv(Here,)k Fp(in)43 b Fv(and)h Fp(out)f Fv(p)s(oin)m(t)g(to)i(m)m
2575
2618
(ulti-dimensional)40 b(arra)m(ys)45 b(in)e(ro)m(w-ma)5
2576
2619
b(jor)44 b(order,)j(of)e(the)f(size)150 2751 y(sp)s(eci\014ed)22
2577
2620
b(when)g(the)h(plan)f(w)m(as)i(created.)40 b(In)22 b(the)i(case)g(of)g
2578
 
(an)f(in-place)f(transform,)i(the)g Fo(out)e Fu(parameter)150
 
2621
(an)f(in-place)f(transform,)i(the)g Fp(out)e Fv(parameter)150
2579
2622
2860 y(is)31 b(ignored)g(and)g(the)h(output)g(data)g(are)h(stored)f(in)
2580
 
e(the)i Fo(in)g Fu(arra)m(y)-8 b(.)46 b(The)31 b(results)g(are)h
 
2623
e(the)i Fp(in)g Fv(arra)m(y)-8 b(.)46 b(The)31 b(results)g(are)h
2581
2624
(stored)g(in-order,)150 2970 y(unnormalized,)44 b(with)d(the)i
2582
 
(zero-frequency)g(comp)s(onen)m(t)g(in)f Fo(out[0])p
2583
 
Fu(.)76 b(A)42 b(forw)m(ard)h(follo)m(w)m(ed)f(b)m(y)h(a)150
 
2625
(zero-frequency)g(comp)s(onen)m(t)g(in)f Fp(out[0])p
 
2626
Fv(.)76 b(A)42 b(forw)m(ard)h(follo)m(w)m(ed)f(b)m(y)h(a)150
2584
2627
3079 y(bac)m(kw)m(ard)c(transform)g(\(or)g(vice-v)m(ersa\))h(yields)d
2585
2628
(the)j(original)d(data)i(m)m(ultiplied)d(b)m(y)j(the)g(size)g(of)g(the)
2586
2629
150 3189 y(arra)m(y)k(\(i.e.)79 b(the)43 b(pro)s(duct)f(of)h(the)h
2589
2632
(what)g(FFTWND)i(computes.)275 3456 y(F)-8 b(or)31 b(example,)h(co)s
2590
2633
(de)f(to)h(p)s(erform)e(an)h(in-place)f(FFT)h(of)g(a)h
2591
2634
(three-dimensional)d(arra)m(y)i(migh)m(t)g(lo)s(ok)150
2592
 
3566 y(lik)m(e:)390 3718 y Fo(#include)46 b(<fftw.h>)390
 
2635
3566 y(lik)m(e:)390 3718 y Fp(#include)46 b(<fftw.h>)390
2593
2636
3822 y(...)390 3925 y({)629 4029 y(fftw_complex)e(in[L][M][N];)629
2594
2637
4133 y(fftwnd_plan)g(p;)629 4237 y(...)629 4340 y(p)j(=)h
2595
2638
(fftw3d_create_plan\(L,)42 b(M,)47 b(N,)g(FFTW_FORWARD,)1726
2596
2639
4444 y(FFTW_MEASURE)e(|)i(FFTW_IN_PLACE\);)629 4548 y(...)629
2597
2640
4652 y(fftwnd_one\(p,)d(&in[0][0][0],)g(NULL\);)629 4755
2598
2641
y(...)629 4859 y(fftwnd_destroy_plan\(p\);)390 4963 y(})275
2599
 
5121 y Fu(Note)36 b(that)f Fo(in)g Fu(is)f(a)h(statically-declared)f
 
2642
5121 y Fv(Note)36 b(that)f Fp(in)g Fv(is)f(a)h(statically-declared)f
2600
2643
(arra)m(y)-8 b(,)38 b(whic)m(h)33 b(is)h(automatically)h(in)f(ro)m
2601
2644
(w-ma)5 b(jor)35 b(order,)150 5230 y(but)42 b(w)m(e)h(m)m(ust)f(tak)m
2602
2645
(e)i(the)f(address)e(of)i(the)g(\014rst)e(elemen)m(t)i(in)e(order)h(to)
2603
2646
i(\014t)e(the)h(t)m(yp)s(e)f(exp)s(ected)h(b)m(y)150
2604
 
5340 y Fo(fftwnd_one)p Fu(.)38 b(\(See)31 b(Section)f(2.5)i
 
2647
5340 y Fp(fftwnd_one)p Fv(.)38 b(\(See)31 b(Section)f(2.5)i
2605
2648
([Multi-dimensional)26 b(Arra)m(y)31 b(F)-8 b(ormat],)32
2606
 
b(page)f(11.\))p eop
 
2649
b(page)f(11.\))p eop end
2607
2650
%%Page: 6 8
2608
 
6 7 bop 150 -116 a Fu(6)3277 b(FFTW)150 299 y Ft(2.3)68
2609
 
b(Real)46 b(One-dimensional)g(T)-11 b(ransforms)45 b(T)-11
2610
 
b(utorial)275 497 y Fu(If)29 b(the)h(input)e(data)j(are)g(purely)d
2611
 
(real,)i(y)m(ou)g(can)g(sa)m(v)m(e)i(roughly)d(a)h(factor)h(of)f(t)m(w)
2612
 
m(o)i(in)d(b)s(oth)g(time)h(and)150 606 y(storage)d(b)m(y)e(using)f
2613
 
(the)h Fp(r\013t)m(w)33 b Fu(transforms,)26 b(whic)m(h)e(are)i(FFTs)f
2614
 
(sp)s(ecialized)f(for)h(real)g(data.)40 b(The)24 b(output)150
2615
 
716 y(of)h(a)f(suc)m(h)h(a)f(transform)g(is)g(a)g Fp(halfcomplex)30
2616
 
b Fu(arra)m(y)-8 b(,)27 b(whic)m(h)c(consists)h(of)g(only)g(half)f(of)i
2617
 
(the)f(complex)h(DFT)150 826 y(amplitudes)20 b(\(since)j(the)g(negativ)
2618
 
m(e-frequency)h(amplitudes)c(for)j(real)f(data)h(are)g(the)g(complex)g
2619
 
(conjugate)150 935 y(of)31 b(the)f(p)s(ositiv)m(e-frequency)f
2620
 
(amplitudes\).)275 1075 y(In)k(exc)m(hange)j(for)e(these)h(sp)s(eed)e
2621
 
(and)h(space)h(adv)-5 b(an)m(tages,)37 b(the)e(user)e(sacri\014ces)h
2622
 
(some)h(of)f(the)h(sim-)150 1184 y(plicit)m(y)g(of)h(FFTW's)h(complex)f
2623
 
(transforms.)58 b(First)35 b(of)i(all,)g(to)g(allo)m(w)e(maxim)m(um)g
2624
 
(p)s(erformance,)j(the)150 1294 y(output)i(format)g(of)g(the)h
2625
 
(one-dimensional)d(real)h(transforms)h(is)f(di\013eren)m(t)g(from)h
2626
 
(that)g(used)g(b)m(y)g(the)150 1404 y(m)m(ulti-dimensional)27
2627
 
b(transforms.)42 b(Second,)32 b(the)f(in)m(v)m(erse)g(transform)g
2628
 
(\(halfcomplex)f(to)i(real\))f(has)g(the)150 1513 y(side-e\013ect)22
2629
 
b(of)g(destro)m(ying)f(its)g(input)e(arra)m(y)-8 b(.)39
2630
 
b(Neither)21 b(of)h(these)g(incon)m(v)m(eniences)f(should)e(p)s(ose)i
2631
 
(a)h(serious)150 1623 y(problem)29 b(for)h(users,)g(but)f(it)h(is)g
2632
 
(imp)s(ortan)m(t)f(to)i(b)s(e)f(a)m(w)m(are)i(of)e(them.)41
2633
 
b(\(Both)32 b(the)e(incon)m(v)m(enien)m(t)g(output)150
2634
 
1732 y(format)h(and)g(the)g(side-e\013ect)h(of)f(the)g(in)m(v)m(erse)g
2635
 
(transform)g(can)g(b)s(e)f(ameliorated)h(for)g(one-dimensional)150
2636
 
1842 y(transforms,)40 b(at)f(the)g(exp)s(ense)f(of)g(some)h(p)s
2637
 
(erformance,)h(b)m(y)f(using)e(instead)g(the)i(m)m(ulti-dimensional)150
2638
 
1952 y(transform)30 b(routines)f(with)g(a)i(rank)f(of)g(one.\))275
2639
 
2091 y(The)k(computation)g(of)h(the)g(plan)e(is)h(similar)e(to)j(that)g
2640
 
(for)g(the)f(complex)h(transforms.)52 b(First,)36 b(y)m(ou)150
2641
 
2201 y Fo(#include)28 b(<rfftw.h>)p Fu(.)38 b(Then,)30
2642
 
b(y)m(ou)h(create)h(a)e(plan)f(\(of)i(t)m(yp)s(e)g Fo(rfftw_plan)p
2643
 
Fu(\))d(b)m(y)i(calling:)390 2335 y Fo(rfftw_plan)45
 
2651
TeXDict begin 6 7 bop 150 -116 a Fv(6)3277 b(FFTW)150
 
2652
299 y Fu(2.3)68 b(Real)46 b(One-dimensional)g(T)-11 b(ransforms)45
 
2653
b(T)-11 b(utorial)275 498 y Fv(If)29 b(the)h(input)e(data)j(are)g
 
2654
(purely)d(real,)i(y)m(ou)g(can)g(sa)m(v)m(e)i(roughly)d(a)h(factor)h
 
2655
(of)f(t)m(w)m(o)i(in)d(b)s(oth)g(time)h(and)150 607 y(storage)d(b)m(y)e
 
2656
(using)f(the)h Fq(r\013t)m(w)33 b Fv(transforms,)26 b(whic)m(h)e(are)i
 
2657
(FFTs)f(sp)s(ecialized)f(for)h(real)g(data.)40 b(The)24
 
2658
b(output)150 717 y(of)h(a)f(suc)m(h)h(a)f(transform)g(is)g(a)g
 
2659
Fq(halfcomplex)30 b Fv(arra)m(y)-8 b(,)27 b(whic)m(h)c(consists)h(of)g
 
2660
(only)g(half)f(of)i(the)f(complex)h(DFT)150 827 y(amplitudes)20
 
2661
b(\(since)j(the)g(negativ)m(e-frequency)h(amplitudes)c(for)j(real)f
 
2662
(data)h(are)g(the)g(complex)g(conjugate)150 936 y(of)31
 
2663
b(the)f(p)s(ositiv)m(e-frequency)f(amplitudes\).)275
 
2664
1077 y(In)k(exc)m(hange)j(for)e(these)h(sp)s(eed)e(and)h(space)h(adv)-5
 
2665
b(an)m(tages,)37 b(the)e(user)e(sacri\014ces)h(some)h(of)f(the)h(sim-)
 
2666
150 1187 y(plicit)m(y)g(of)h(FFTW's)h(complex)f(transforms.)58
 
2667
b(First)35 b(of)i(all,)g(to)g(allo)m(w)e(maxim)m(um)g(p)s(erformance,)j
 
2668
(the)150 1296 y(output)i(format)g(of)g(the)h(one-dimensional)d(real)h
 
2669
(transforms)h(is)f(di\013eren)m(t)g(from)h(that)g(used)g(b)m(y)g(the)
 
2670
150 1406 y(m)m(ulti-dimensional)27 b(transforms.)42 b(Second,)32
 
2671
b(the)f(in)m(v)m(erse)g(transform)g(\(halfcomplex)f(to)i(real\))f(has)g
 
2672
(the)150 1515 y(side-e\013ect)22 b(of)g(destro)m(ying)f(its)g(input)e
 
2673
(arra)m(y)-8 b(.)39 b(Neither)21 b(of)h(these)g(incon)m(v)m(eniences)f
 
2674
(should)e(p)s(ose)i(a)h(serious)150 1625 y(problem)29
 
2675
b(for)h(users,)g(but)f(it)h(is)g(imp)s(ortan)m(t)f(to)i(b)s(e)f(a)m(w)m
 
2676
(are)i(of)e(them.)41 b(\(Both)32 b(the)e(incon)m(v)m(enien)m(t)g
 
2677
(output)150 1735 y(format)h(and)g(the)g(side-e\013ect)h(of)f(the)g(in)m
 
2678
(v)m(erse)g(transform)g(can)g(b)s(e)f(ameliorated)h(for)g
 
2679
(one-dimensional)150 1844 y(transforms,)40 b(at)f(the)g(exp)s(ense)f
 
2680
(of)g(some)h(p)s(erformance,)h(b)m(y)f(using)e(instead)g(the)i(m)m
 
2681
(ulti-dimensional)150 1954 y(transform)30 b(routines)f(with)g(a)i(rank)
 
2682
f(of)g(one.\))275 2095 y(The)k(computation)g(of)h(the)g(plan)e(is)h
 
2683
(similar)e(to)j(that)g(for)g(the)f(complex)h(transforms.)52
 
2684
b(First,)36 b(y)m(ou)150 2204 y Fp(#include)28 b(<rfftw.h>)p
 
2685
Fv(.)38 b(Then,)30 b(y)m(ou)h(create)h(a)e(plan)f(\(of)i(t)m(yp)s(e)g
 
2686
Fp(rfftw_plan)p Fv(\))d(b)m(y)i(calling:)390 2339 y Fp(rfftw_plan)45
2644
2687
b(rfftw_create_plan\(int)d(n,)47 b(fftw_direction)d(dir,)j(int)g
2645
 
(flags\);)275 2475 y(n)37 b Fu(is)g(the)h(length)f(of)h(the)h
2646
 
Fq(r)-5 b(e)g(al)48 b Fu(arra)m(y)39 b(in)d(the)i(transform)g(\(ev)m
2647
 
(en)h(for)e(halfcomplex-to-real)h(trans-)150 2584 y(forms\),)32
 
2688
(flags\);)275 2480 y(n)37 b Fv(is)g(the)h(length)f(of)h(the)h
 
2689
Fr(r)-5 b(e)g(al)48 b Fv(arra)m(y)39 b(in)d(the)i(transform)g(\(ev)m
 
2690
(en)h(for)e(halfcomplex-to-real)h(trans-)150 2590 y(forms\),)32
2648
2691
b(and)f(can)i(b)s(e)e(an)m(y)h(p)s(ositiv)m(e)f(in)m(teger)h
2649
2692
(\(although)f(sizes)h(with)f(small)f(factors)j(are)f(transformed)150
2650
 
2694 y(more)39 b(e\016cien)m(tly\).)65 b Fo(dir)37 b
2651
 
Fu(is)h(either)g Fo(FFTW_REAL_TO_COMPLEX)33 b Fu(or)38
2652
 
b Fo(FFTW_COMPLEX_TO_REAL)p Fu(.)59 b(The)150 2803 y
2653
 
Fo(flags)29 b Fu(parameter)i(is)e(the)i(same)g(as)f(in)f
2654
 
Fo(fftw_create_plan)p Fu(.)275 2943 y(Once)d(created,)j(a)e(plan)f(can)
 
2693
2699 y(more)39 b(e\016cien)m(tly\).)65 b Fp(dir)37 b
 
2694
Fv(is)h(either)g Fp(FFTW_REAL_TO_COMPLEX)33 b Fv(or)38
 
2695
b Fp(FFTW_COMPLEX_TO_REAL)p Fv(.)59 b(The)150 2809 y
 
2696
Fp(flags)29 b Fv(parameter)i(is)e(the)i(same)g(as)f(in)f
 
2697
Fp(fftw_create_plan)p Fv(.)275 2950 y(Once)d(created,)j(a)e(plan)f(can)
2655
2698
h(b)s(e)f(used)g(for)g(an)m(y)h(n)m(um)m(b)s(er)f(of)g(transforms,)h
2656
 
(and)g(is)e(deallo)s(cated)i(when)150 3053 y(y)m(ou)k(are)f(done)h
2657
 
(with)e(it)h(b)m(y)g(calling)f Fo(rfftw_destroy_plan\(plan)o(\))p
2658
 
Fu(.)275 3192 y(Giv)m(en)h(a)h(plan,)e(a)i(real-to-complex)g(or)f
 
2699
(and)g(is)e(deallo)s(cated)i(when)150 3060 y(y)m(ou)k(are)f(done)h
 
2700
(with)e(it)h(b)m(y)g(calling)f Fp(rfftw_destroy_plan\(plan)o(\))p
 
2701
Fv(.)275 3200 y(Giv)m(en)h(a)h(plan,)e(a)i(real-to-complex)g(or)f
2659
2702
(complex-to-real)h(transform)f(is)f(computed)i(b)m(y)f(calling:)390
2660
 
3326 y Fo(void)47 b(rfftw_one\(rfftw_plan)42 b(plan,)k(fftw_real)g
2661
 
(*in,)g(fftw_real)f(*out\);)275 3466 y Fu(\(Note)31 b(that)g
2662
 
Fo(fftw_real)d Fu(is)h(an)i(alias)e(for)h(the)g(\015oating-p)s(oin)m(t)
 
2703
3335 y Fp(void)47 b(rfftw_one\(rfftw_plan)42 b(plan,)k(fftw_real)g
 
2704
(*in,)g(fftw_real)f(*out\);)275 3476 y Fv(\(Note)31 b(that)g
 
2705
Fp(fftw_real)d Fv(is)h(an)i(alias)e(for)h(the)g(\015oating-p)s(oin)m(t)
2663
2706
g(t)m(yp)s(e)g(for)g(whic)m(h)f(FFTW)i(w)m(as)g(com-)150
2664
 
3576 y(piled.\))52 b(Dep)s(ending)34 b(up)s(on)f(the)i(direction)e(of)i
 
2707
3586 y(piled.\))52 b(Dep)s(ending)34 b(up)s(on)f(the)i(direction)e(of)i
2665
2708
(the)g(plan,)g(either)f(the)h(input)e(or)i(the)g(output)g(arra)m(y)g
2666
 
(is)150 3685 y(halfcomplex,)29 b(and)h(is)f(stored)i(in)e(the)i(follo)m
2667
 
(wing)d(format:)1222 3943 y Fl(r)1263 3957 y Fn(0)1301
2668
 
3943 y Fl(;)15 b(r)1382 3957 y Fn(1)1419 3943 y Fl(;)g(r)1500
2669
 
3957 y Fn(2)1538 3943 y Fl(;)g(:)g(:)g(:)h(;)f(r)1780
2670
 
3957 y Fk(n=)p Fn(2)1893 3943 y Fl(;)g(i)1964 3957 y
2671
 
Fn(\()p Fk(n)p Fn(+1\))p Fk(=)p Fn(2)p Fj(\000)p Fn(1)2298
2672
 
3943 y Fl(;)g(:)g(:)g(:)i(;)e(i)2531 3957 y Fn(2)2569
2673
 
3943 y Fl(;)g(i)2640 3957 y Fn(1)275 4148 y Fu(Here,)28
2674
 
b Fl(r)553 4162 y Fk(k)621 4148 y Fu(is)e(the)h(real)g(part)g(of)g(the)
2675
 
g Fl(k)s Fu(th)g(output,)g(and)g Fl(i)2171 4162 y Fk(k)2239
2676
 
4148 y Fu(is)f(the)h(imaginary)f(part.)39 b(\(W)-8 b(e)29
2677
 
b(follo)m(w)d(here)150 4258 y(the)33 b(C)g(con)m(v)m(en)m(tion)h(that)g
 
2709
(is)150 3696 y(halfcomplex,)29 b(and)h(is)f(stored)i(in)e(the)i(follo)m
 
2710
(wing)d(format:)1222 3956 y Fm(r)1263 3970 y Fo(0)1301
 
2711
3956 y Fm(;)15 b(r)1382 3970 y Fo(1)1419 3956 y Fm(;)g(r)1500
 
2712
3970 y Fo(2)1538 3956 y Fm(;)g(:)g(:)g(:)h(;)f(r)1780
 
2713
3970 y Fl(n=)p Fo(2)1893 3956 y Fm(;)g(i)1964 3970 y
 
2714
Fo(\()p Fl(n)p Fo(+1\))p Fl(=)p Fo(2)p Fk(\000)p Fo(1)2298
 
2715
3956 y Fm(;)g(:)g(:)g(:)i(;)e(i)2531 3970 y Fo(2)2569
 
2716
3956 y Fm(;)g(i)2640 3970 y Fo(1)275 4164 y Fv(Here,)28
 
2717
b Fm(r)553 4178 y Fl(k)621 4164 y Fv(is)e(the)h(real)g(part)g(of)g(the)
 
2718
g Fm(k)s Fv(th)g(output,)g(and)g Fm(i)2171 4178 y Fl(k)2239
 
2719
4164 y Fv(is)f(the)h(imaginary)f(part.)39 b(\(W)-8 b(e)29
 
2720
b(follo)m(w)d(here)150 4274 y(the)33 b(C)g(con)m(v)m(en)m(tion)h(that)g
2678
2721
(in)m(teger)f(division)d(is)i(rounded)f(do)m(wn,)j(e.g.)49
2679
 
b(7)p Fl(=)p Fu(2)31 b(=)f(3.\))49 b(F)-8 b(or)34 b(a)f(halfcomplex)150
2680
 
4367 y(arra)m(y)f Fo(hc[])p Fu(,)f(the)g Fl(k)s Fu(th)h(comp)s(onen)m
2681
 
(t)g(has)f(its)g(real)g(part)g(in)f Fo(hc[k])g Fu(and)h(its)g
2682
 
(imaginary)f(part)h(in)f Fo(hc[n-)150 4477 y(k])p Fu(,)35
2683
 
b(with)e(the)i(exception)f(of)h Fo(k)f(==)g(0)g Fu(or)g
2684
 
Fo(n/2)f Fu(\(the)i(latter)g(only)f(if)f(n)h(is)f(ev)m(en\)|in)h(these)
2685
 
h(t)m(w)m(o)h(cases,)150 4587 y(the)k(imaginary)f(part)h(is)f(zero)i
 
2722
b(7)p Fm(=)p Fv(2)31 b(=)f(3.\))49 b(F)-8 b(or)34 b(a)f(halfcomplex)150
 
2723
4383 y(arra)m(y)f Fp(hc[])p Fv(,)f(the)g Fm(k)s Fv(th)h(comp)s(onen)m
 
2724
(t)g(has)f(its)g(real)g(part)g(in)f Fp(hc[k])g Fv(and)h(its)g
 
2725
(imaginary)f(part)h(in)f Fp(hc[n-)150 4493 y(k])p Fv(,)35
 
2726
b(with)e(the)i(exception)f(of)h Fp(k)f(==)g(0)g Fv(or)g
 
2727
Fp(n/2)f Fv(\(the)i(latter)g(only)f(if)f(n)h(is)f(ev)m(en\)|in)h(these)
 
2728
h(t)m(w)m(o)h(cases,)150 4603 y(the)k(imaginary)f(part)h(is)f(zero)i
2686
2729
(due)e(to)i(symmetries)f(of)g(the)g(real-complex)g(transform,)i(and)e
2687
 
(is)f(not)150 4696 y(stored.)58 b(Th)m(us,)37 b(the)f(transform)g(of)g
2688
 
Fo(n)g Fu(real)g(v)-5 b(alues)35 b(is)g(a)i(halfcomplex)d(arra)m(y)j
2689
 
(of)f(length)g Fo(n)p Fu(,)h(and)f(vice)150 4806 y(v)m(ersa.)377
2690
 
4773 y Fn(1)475 4806 y Fu(This)f(is)g(actually)i(only)f(half)f(of)i
 
2730
(is)f(not)150 4712 y(stored.)58 b(Th)m(us,)37 b(the)f(transform)g(of)g
 
2731
Fp(n)g Fv(real)g(v)-5 b(alues)35 b(is)g(a)i(halfcomplex)d(arra)m(y)j
 
2732
(of)f(length)g Fp(n)p Fv(,)h(and)f(vice)150 4822 y(v)m(ersa.)377
 
2733
4789 y Fo(1)475 4822 y Fv(This)f(is)g(actually)i(only)f(half)f(of)i
2691
2734
(the)g(DFT)g(sp)s(ectrum)f(of)h(the)f(data.)61 b(Although)36
2692
 
b(the)h(other)150 4915 y(half)29 b(can)i(b)s(e)f(obtained)g(b)m(y)g
 
2735
b(the)h(other)150 4931 y(half)29 b(can)i(b)s(e)f(obtained)g(b)m(y)g
2693
2736
(complex)h(conjugation,)f(it)h(is)e(not)i(required)e(b)m(y)h(man)m(y)h
2694
 
(applications)d(suc)m(h)150 5025 y(as)j(con)m(v)m(olution)f(and)g
2695
 
(\014ltering.)p 150 5131 1200 4 v 192 5197 a Fn(1)275
2696
 
5230 y Fu(The)21 b(output)g(for)g(the)h(m)m(ulti-dimensional)17
2697
 
b(r\013t)m(w)22 b(is)f(a)h(more-con)m(v)m(en)m(tional)h(arra)m(y)f(of)f
2698
 
Fo(fftw_complex)275 5340 y Fu(v)-5 b(alues,)29 b(but)h(the)h(format)f
2699
 
(here)h(p)s(ermitted)e(us)g(greater)j(e\016ciency)e(in)f(one)i
2700
 
(dimension.)p eop
 
2737
(applications)d(suc)m(h)150 5041 y(as)j(con)m(v)m(olution)f(and)g
 
2738
(\014ltering.)p 150 5149 1200 4 v 199 5217 a Fo(1)275
 
2739
5249 y Fj(The)22 b(output)f(for)h(the)g(m)n(ulti-dimensional)e(r\013t)n
 
2740
(w)i(is)h(a)f(more-con)n(v)n(en)n(tional)f(arra)n(y)h(of)h
 
2741
Fi(fftw_complex)h Fj(v)l(alues,)f(but)e(the)275 5340
 
2742
y(format)k(here)h(p)r(ermitted)e(us)i(greater)h(e\016ciency)e(in)h(one)
 
2743
g(dimension.)p eop end
2701
2744
%%Page: 7 9
2702
 
7 8 bop 150 -116 a Fu(Chapter)30 b(2:)41 b(T)-8 b(utorial)2768
2703
 
b(7)275 299 y(Lik)m(e)36 b(the)g(complex)g(transforms,)h(the)f(RFFTW)h
2704
 
(transforms)e(are)i(unnormalized,)e(so)i(a)f(forw)m(ard)150
2705
 
408 y(follo)m(w)m(ed)42 b(b)m(y)g(a)h(bac)m(kw)m(ard)f(transform)g
2706
 
(\(or)h(vice-v)m(ersa\))g(yields)e(the)h(original)f(data)i(scaled)f(b)m
2707
 
(y)g(the)150 518 y(length)30 b(of)g(the)h(arra)m(y)-8
2708
 
b(,)31 b Fo(n)p Fu(.)275 646 y(Let)41 b(us)f(reiterate)h(here)f(our)g
2709
 
(w)m(arning)g(that)h(an)f Fo(FFTW_COMPLEX_TO_REAL)c Fu(transform)j(has)
 
2745
TeXDict begin 7 8 bop 150 -116 a Fv(Chapter)30 b(2:)41
 
2746
b(T)-8 b(utorial)2768 b(7)275 299 y(Lik)m(e)36 b(the)g(complex)g
 
2747
(transforms,)h(the)f(RFFTW)h(transforms)e(are)i(unnormalized,)e(so)i(a)
 
2748
f(forw)m(ard)150 408 y(follo)m(w)m(ed)42 b(b)m(y)g(a)h(bac)m(kw)m(ard)f
 
2749
(transform)g(\(or)h(vice-v)m(ersa\))g(yields)e(the)h(original)f(data)i
 
2750
(scaled)f(b)m(y)g(the)150 518 y(length)30 b(of)g(the)h(arra)m(y)-8
 
2751
b(,)31 b Fp(n)p Fv(.)275 646 y(Let)41 b(us)f(reiterate)h(here)f(our)g
 
2752
(w)m(arning)g(that)h(an)f Fp(FFTW_COMPLEX_TO_REAL)c Fv(transform)j(has)
2710
2753
i(the)150 756 y(side-e\013ect)d(of)g(destro)m(ying)f(its)f
2711
 
(\(halfcomplex\))h(input.)60 b(The)37 b Fo(FFTW_REAL_TO_COMPLEX)31
2712
 
b Fu(transform,)150 865 y(ho)m(w)m(ev)m(er,)h(lea)m(v)m(es)g(its)d
 
2754
(\(halfcomplex\))h(input.)60 b(The)37 b Fp(FFTW_REAL_TO_COMPLEX)31
 
2755
b Fv(transform,)150 865 y(ho)m(w)m(ev)m(er,)h(lea)m(v)m(es)g(its)d
2713
2756
(\(real\))i(input)e(un)m(touc)m(hed,)h(just)g(as)h(y)m(ou)f(w)m(ould)f
2714
2757
(hop)s(e.)275 994 y(As)g(an)h(example,)f(here)h(is)e(an)i(outline)e(of)
2715
2758
i(ho)m(w)f(y)m(ou)h(migh)m(t)f(use)h(RFFTW)g(to)g(compute)g(the)g(p)s
2716
2759
(o)m(w)m(er)150 1103 y(sp)s(ectrum)f(of)i(a)g(real)f(arra)m(y)g(\(i.e.)
2717
2760
41 b(the)31 b(squares)f(of)g(the)h(absolute)f(v)-5 b(alues)30
2718
 
b(of)g(the)h(DFT)g(amplitudes\):)390 1225 y Fo(#include)46
 
2761
b(of)g(the)h(DFT)g(amplitudes\):)390 1225 y Fp(#include)46
2719
2762
b(<rfftw.h>)390 1329 y(...)390 1433 y({)629 1537 y(fftw_real)f(in[N],)h
2720
2763
(out[N],)g(power_spectrum[N/2+1];)629 1641 y(rfftw_plan)f(p;)629
2721
2764
1744 y(int)i(k;)629 1848 y(...)629 1952 y(p)g(=)h
2729
2772
h(is)f(even)g(*/)867 2678 y(power_spectrum[N/2])c(=)k
2730
2773
(out[N/2]*out[N/2];)91 b(/*)47 b(Nyquist)f(freq.)g(*/)629
2731
2774
2782 y(...)629 2886 y(rfftw_destroy_plan\(p\);)390 2990
2732
 
y(})275 3118 y Fu(Programs)39 b(using)f(RFFTW)i(should)e(link)f(with)i
2733
 
Fo(-lrfftw)28 b(-lfftw)h(-lm)38 b Fu(on)i(Unix,)h(or)e(with)g(the)150
 
2775
y(})275 3118 y Fv(Programs)39 b(using)f(RFFTW)i(should)e(link)f(with)i
 
2776
Fp(-lrfftw)28 b(-lfftw)h(-lm)38 b Fv(on)i(Unix,)h(or)e(with)g(the)150
2734
2777
3227 y(FFTW,)31 b(RFFTW,)h(and)d(math)i(libraries)c(in)i(general.)150
2735
 
3466 y Ft(2.4)68 b(Real)46 b(Multi-dimensional)h(T)-11
2736
 
b(ransforms)44 b(T)-11 b(utorial)275 3652 y Fu(FFTW)39
 
2778
3466 y Fu(2.4)68 b(Real)46 b(Multi-dimensional)h(T)-11
 
2779
b(ransforms)44 b(T)-11 b(utorial)275 3652 y Fv(FFTW)39
2737
2780
b(includes)d(m)m(ulti-dimensional)e(transforms)k(for)g(real)g(data)h
2738
2781
(of)g(an)m(y)g(rank.)64 b(As)39 b(with)e(the)150 3762
2739
2782
y(one-dimensional)j(real)j(transforms,)i(they)d(sa)m(v)m(e)j(roughly)c
2747
2790
4200 y(\(complex)e(to)i(real\))e(transforms)f(ha)m(v)m(e)j(the)f
2748
2791
(side-e\013ect)g(of)f(o)m(v)m(erwriting)g(their)g(input)e(data.)275
2749
2792
4328 y(T)-8 b(o)35 b(use)g(the)g(real)f(m)m(ulti-dimensional)d
2750
 
(transforms,)36 b(y)m(ou)f(\014rst)f Fo(#include)28 b(<rfftw.h>)33
2751
 
b Fu(and)h(then)150 4438 y(create)e(a)f(plan)e(for)h(the)g(size)h(and)e
 
2793
(transforms,)36 b(y)m(ou)f(\014rst)f Fp(#include)28 b(<rfftw.h>)33
 
2794
b Fv(and)h(then)150 4438 y(create)e(a)f(plan)e(for)h(the)g(size)h(and)e
2752
2795
(direction)g(of)i(transform)f(that)h(y)m(ou)f(are)h(in)m(terested)g
2753
 
(in:)390 4560 y Fo(rfftwnd_plan)44 b(rfftwnd_create_plan\(int)e(rank,)k
 
2796
(in:)390 4560 y Fp(rfftwnd_plan)44 b(rfftwnd_create_plan\(int)e(rank,)k
2754
2797
(const)h(int)g(*n,)1965 4664 y(fftw_direction)d(dir,)j(int)f(flags\);)
2755
 
275 4792 y Fu(The)35 b(\014rst)f(t)m(w)m(o)j(parameters)f(describ)s(e)e
 
2798
275 4792 y Fv(The)35 b(\014rst)f(t)m(w)m(o)j(parameters)f(describ)s(e)e
2756
2799
(the)i(size)g(of)f(the)h(real)f(data)i(\(not)f(the)g(halfcomplex)e
2757
2800
(data,)150 4902 y(whic)m(h)f(will)f(ha)m(v)m(e)k(di\013eren)m(t)e
2758
2801
(dimensions\).)51 b(The)34 b(last)h(t)m(w)m(o)h(parameters)e(are)h(the)
2759
 
g(same)g(as)g(those)g(for)150 5011 y Fo(rfftw_create_plan)p
2760
 
Fu(.)55 b(Just)36 b(as)h(for)g(\013t)m(wnd,)h(there)f(are)g(t)m(w)m(o)h
 
2802
g(same)g(as)g(those)g(for)150 5011 y Fp(rfftw_create_plan)p
 
2803
Fv(.)55 b(Just)36 b(as)h(for)g(\013t)m(wnd,)h(there)f(are)g(t)m(w)m(o)h
2761
2804
(alternate)g(v)m(ersions)e(of)h(this)f(routine,)150 5121
2762
 
y Fo(rfftw2d_create_plan)19 b Fu(and)k Fo(rfftw3d_create_plan)p
2763
 
Fu(,)d(that)k(are)g(sometimes)g(more)g(con)m(v)m(enien)m(t)h(for)150
 
2805
y Fp(rfftw2d_create_plan)19 b Fv(and)k Fp(rfftw3d_create_plan)p
 
2806
Fv(,)d(that)k(are)g(sometimes)g(more)g(con)m(v)m(enien)m(t)h(for)150
2764
2807
5230 y(t)m(w)m(o-)38 b(and)d(three-dimensional)f(transforms.)58
2765
2808
b(Also)36 b(as)g(in)f(\013t)m(wnd,)i(r\013t)m(wnd)f(supp)s(orts)e(true)
2766
2809
i(in-place)150 5340 y(transforms,)30 b(sp)s(eci\014ed)e(b)m(y)j
2767
 
(including)c Fo(FFTW_IN_PLACE)f Fu(in)j(the)i(\015ags.)p
2768
 
eop
 
2810
(including)c Fp(FFTW_IN_PLACE)f Fv(in)j(the)i(\015ags.)p
 
2811
eop end
2769
2812
%%Page: 8 10
2770
 
8 9 bop 150 -116 a Fu(8)3277 b(FFTW)275 299 y(Once)34
2771
 
b(created,)i(a)e(plan)f(can)i(b)s(e)e(used)g(for)h(an)m(y)h(n)m(um)m(b)
2772
 
s(er)e(of)h(transforms,)g(and)g(is)f(deallo)s(cated)h(b)m(y)150
2773
 
408 y(calling)29 b Fo(rfftwnd_destroy_plan\(pla)o(n\))p
2774
 
Fu(.)275 545 y(Giv)m(en)h(a)h(plan,)e(the)h(transform)g(is)f(computed)i
 
2813
TeXDict begin 8 9 bop 150 -116 a Fv(8)3277 b(FFTW)275
 
2814
299 y(Once)34 b(created,)i(a)e(plan)f(can)i(b)s(e)e(used)g(for)h(an)m
 
2815
(y)h(n)m(um)m(b)s(er)e(of)h(transforms,)g(and)g(is)f(deallo)s(cated)h
 
2816
(b)m(y)150 408 y(calling)29 b Fp(rfftwnd_destroy_plan\(pla)o(n\))p
 
2817
Fv(.)275 545 y(Giv)m(en)h(a)h(plan,)e(the)h(transform)g(is)f(computed)i
2775
2818
(b)m(y)f(calling)f(one)h(of)h(the)g(follo)m(wing)d(t)m(w)m(o)k
2776
 
(routines:)390 676 y Fo(void)47 b(rfftwnd_one_real_to_com)o(plex)o
 
2819
(routines:)390 676 y Fp(void)47 b(rfftwnd_one_real_to_com)o(plex)o
2777
2820
(\(rf)o(ftwn)o(d_pl)o(an)41 b(plan,)1965 779 y(fftw_real)k(*in,)i
2778
2821
(fftw_complex)d(*out\);)390 883 y(void)j(rfftwnd_one_complex_to_)o
2779
2822
(real)o(\(rf)o(ftwn)o(d_pl)o(an)41 b(plan,)1965 987 y(fftw_complex)j
2780
 
(*in,)j(fftw_real)e(*out\);)275 1124 y Fu(As)25 b(is)g(clear)g(from)g
 
2823
(*in,)j(fftw_real)e(*out\);)275 1124 y Fv(As)25 b(is)g(clear)g(from)g
2781
2824
(their)g(names)g(and)g(parameter)h(t)m(yp)s(es,)h(the)f(former)f
2782
 
(function)f(is)g(for)i Fo(FFTW_REAL_)150 1233 y(TO_COMPLEX)c
2783
 
Fu(transforms)i(and)h(the)g(latter)g(is)f(for)h Fo
2784
 
(FFTW_COMPLEX_TO_REAL)19 b Fu(transforms.)38 b(\(W)-8
 
2825
(function)f(is)g(for)i Fp(FFTW_REAL_)150 1233 y(TO_COMPLEX)c
 
2826
Fv(transforms)i(and)h(the)g(latter)g(is)f(for)h Fp
 
2827
(FFTW_COMPLEX_TO_REAL)19 b Fv(transforms.)38 b(\(W)-8
2785
2828
b(e)27 b(could)150 1343 y(ha)m(v)m(e)j(used)e(only)g(a)i(single)d
2786
2829
(routine,)i(since)f(the)h(direction)f(of)h(the)g(transform)g(is)f(enco)
2787
2830
s(ded)g(in)g(the)h(plan,)150 1452 y(but)34 b(w)m(e)h(w)m(an)m(ted)g(to)
2790
2833
y(as)e(w)m(e)g(discuss)e(elsewhere,)i(has)g(the)g(side-e\013ect)g(of)g
2791
2834
(o)m(v)m(erwriting)g(its)f(input)f(\(except)j(when)d(the)i(rank)150
2792
2835
1672 y(of)e(the)f(arra)m(y)h(is)e(one\).)42 b(In)29 b(b)s(oth)h(cases,)
2793
 
i(the)e Fo(out)g Fu(parameter)g(is)g(ignored)f(for)i(in-place)e
 
2836
i(the)e Fp(out)g Fv(parameter)g(is)g(ignored)f(for)i(in-place)e
2794
2837
(transforms.)275 1808 y(The)d(format)i(of)f(the)h(complex)f(arra)m(ys)g
2795
2838
(deserv)m(es)h(careful)f(atten)m(tion.)40 b(Supp)s(ose)26
2796
2839
b(that)i(the)f(real)g(data)150 1918 y(has)41 b(dimensions)e
2797
 
Fl(n)855 1932 y Fn(1)919 1918 y Fs(\002)28 b Fl(n)1073
2798
 
1932 y Fn(2)1137 1918 y Fs(\002)f(\001)15 b(\001)g(\001)29
2799
 
b(\002)e Fl(n)1522 1932 y Fk(d)1602 1918 y Fu(\(in)40
 
2840
Fm(n)855 1932 y Fo(1)919 1918 y Ft(\002)28 b Fm(n)1073
 
2841
1932 y Fo(2)1137 1918 y Ft(\002)f(\001)15 b(\001)g(\001)29
 
2842
b(\002)e Fm(n)1522 1932 y Fl(d)1602 1918 y Fv(\(in)40
2800
2843
b(ro)m(w-ma)5 b(jor)42 b(order\).)74 b(Then,)43 b(after)f(a)g
2801
2844
(real-to-complex)150 2027 y(transform,)28 b(the)f(output)h(is)e(an)i
2802
 
Fl(n)1307 2041 y Fn(1)1358 2027 y Fs(\002)15 b Fl(n)1499
2803
 
2041 y Fn(2)1550 2027 y Fs(\002)g(\001)g(\001)g(\001)f(\002)h
2804
 
Fu(\()p Fl(n)1931 2041 y Fk(d)1969 2027 y Fl(=)p Fu(2)g(+)g(1\))28
2805
 
b(arra)m(y)g(of)g Fo(fftw_complex)c Fu(v)-5 b(alues)27
 
2845
Fm(n)1307 2041 y Fo(1)1358 2027 y Ft(\002)15 b Fm(n)1499
 
2846
2041 y Fo(2)1550 2027 y Ft(\002)g(\001)g(\001)g(\001)f(\002)h
 
2847
Fv(\()p Fm(n)1931 2041 y Fl(d)1969 2027 y Fm(=)p Fv(2)g(+)g(1\))28
 
2848
b(arra)m(y)g(of)g Fp(fftw_complex)c Fv(v)-5 b(alues)27
2806
2849
b(in)g(ro)m(w-)150 2137 y(ma)5 b(jor)27 b(order,)g(corresp)s(onding)e
2807
2850
(to)j(sligh)m(tly)d(o)m(v)m(er)j(half)e(of)h(the)g(output)f(of)h(the)g
2808
2851
(corresp)s(onding)e(complex)150 2246 y(transform.)39
2825
2868
(data|t)m(w)m(o)h(extra)g(if)e(the)h(last)g(dimension)d(is)150
2826
2869
3260 y(ev)m(en)k(and)f(one)h(if)f(it)g(is)g(o)s(dd.)37
2827
2870
b(That)24 b(is,)g(the)g(last)f(dimension)e(of)j(the)g(real)f(data)i(m)m
2828
 
(ust)e(ph)m(ysically)f(con)m(tain)150 3369 y(2\()p Fl(n)285
2829
 
3383 y Fk(d)324 3369 y Fl(=)p Fu(2)j(+)g(1\))37 b Fo(fftw_real)d
2830
 
Fu(v)-5 b(alues)36 b(\(exactly)i(enough)f(to)g(hold)e(the)i(complex)g
 
2871
(ust)e(ph)m(ysically)f(con)m(tain)150 3369 y(2\()p Fm(n)285
 
2872
3383 y Fl(d)324 3369 y Fm(=)p Fv(2)j(+)g(1\))37 b Fp(fftw_real)d
 
2873
Fv(v)-5 b(alues)36 b(\(exactly)i(enough)f(to)g(hold)e(the)i(complex)g
2831
2874
(data\).)60 b(This)35 b(ph)m(ysical)150 3479 y(arra)m(y)42
2832
2875
b(size)g(do)s(es)f(not,)k(ho)m(w)m(ev)m(er,)i(c)m(hange)c(the)e
2833
 
Fq(lo)-5 b(gic)g(al)53 b Fu(arra)m(y)42 b(size|only)f
2834
 
Fl(n)2909 3493 y Fk(d)2989 3479 y Fu(v)-5 b(alues)41
 
2876
Fr(lo)-5 b(gic)g(al)53 b Fv(arra)m(y)42 b(size|only)f
 
2877
Fm(n)2909 3493 y Fl(d)2989 3479 y Fv(v)-5 b(alues)41
2835
2878
b(are)h(actually)150 3588 y(stored)27 b(in)e(the)i(last)g(dimension,)e
2836
 
(and)h Fl(n)1524 3602 y Fk(d)1589 3588 y Fu(is)g(the)h(last)g
2837
 
(dimension)d(passed)i(to)i Fo(rfftwnd_create_plan)p Fu(.)275
 
2879
(and)h Fm(n)1524 3602 y Fl(d)1589 3588 y Fv(is)g(the)h(last)g
 
2880
(dimension)d(passed)i(to)i Fp(rfftwnd_create_plan)p Fv(.)275
2838
2881
3725 y(F)-8 b(or)37 b(example,)h(consider)d(the)i(transform)f(of)g(a)h
2839
2882
(t)m(w)m(o-dimensional)f(real)g(arra)m(y)h(of)g(size)f
2840
 
Fo(nx)g Fu(b)m(y)g Fo(ny)p Fu(.)150 3835 y(The)28 b(output)h(of)g(the)g
2841
 
Fo(rfftwnd)e Fu(transform)i(is)f(a)h(t)m(w)m(o-dimensional)f(real)h
2842
 
(arra)m(y)g(of)g(size)g Fo(nx)g Fu(b)m(y)f Fo(ny/2+1)p
2843
 
Fu(,)150 3944 y(where)k(the)h Fo(y)g Fu(dimension)d(has)j(b)s(een)f
2844
 
(cut)h(nearly)f(in)f(half)h(b)s(ecause)h(of)g(redundancies)e(in)g(the)i
2845
 
(output.)150 4054 y(Because)j Fo(fftw_complex)c Fu(is)i(t)m(wice)h(the)
2846
 
g(size)g(of)g Fo(fftw_real)p Fu(,)f(the)h(output)g(arra)m(y)g(is)f
2847
 
(sligh)m(tly)f(bigger)150 4163 y(than)27 b(the)h(input)d(arra)m(y)-8
2848
 
b(.)40 b(Th)m(us,)28 b(if)e(w)m(e)i(w)m(an)m(t)g(to)g(compute)f(the)h
2849
 
(transform)e(in)g(place,)i(w)m(e)g(m)m(ust)f Fq(p)-5
2850
 
b(ad)39 b Fu(the)150 4273 y(input)25 b(arra)m(y)i(so)g(that)h(it)e(is)g
2851
 
(of)h(size)g Fo(nx)f Fu(b)m(y)h Fo(2*\(ny/2+1\))p Fu(.)37
2852
 
b(If)26 b Fo(ny)g Fu(is)g(ev)m(en,)j(then)d(there)h(are)h(t)m(w)m(o)g
2853
 
(padding)150 4382 y(elemen)m(ts)36 b(at)g(the)f(end)g(of)g(eac)m(h)i
2854
 
(ro)m(w)e(\(whic)m(h)g(need)g(not)g(b)s(e)g(initialized,)e(as)j(they)g
2855
 
(are)f(only)g(used)f(for)150 4492 y(output\).)275 4629
2856
 
y(Figure)h(1)h(depicts)f(the)h(input)e(and)h(output)g(arra)m(ys)i(just)
2857
 
e(describ)s(ed,)g(for)h(b)s(oth)f(the)h(out-of-place)150
 
2883
Fp(nx)g Fv(b)m(y)g Fp(ny)p Fv(.)150 3835 y(The)k(output)g(of)h(the)g
 
2884
Fp(rfftwnd)d Fv(transform)i(is)g(a)g(t)m(w)m(o-dimensional)g(complex)g
 
2885
(arra)m(y)h(of)g(size)f Fp(nx)g Fv(b)m(y)150 3944 y Fp(ny/2+1)p
 
2886
Fv(,)30 b(where)h(the)h Fp(y)f Fv(dimension)e(has)j(b)s(een)f(cut)g
 
2887
(nearly)g(in)f(half)h(b)s(ecause)g(of)h(redundancies)d(in)i(the)150
 
2888
4054 y(output.)40 b(Because)32 b Fp(fftw_complex)27 b
 
2889
Fv(is)j(t)m(wice)g(the)h(size)f(of)h Fp(fftw_real)p Fv(,)d(the)i
 
2890
(output)g(arra)m(y)h(is)f(sligh)m(tly)150 4163 y(bigger)g(than)h(the)g
 
2891
(input)e(arra)m(y)-8 b(.)43 b(Th)m(us,)30 b(if)g(w)m(e)h(w)m(an)m(t)h
 
2892
(to)g(compute)f(the)g(transform)f(in)g(place,)h(w)m(e)g(m)m(ust)150
 
2893
4273 y Fr(p)-5 b(ad)39 b Fv(the)28 b(input)e(arra)m(y)j(so)f(that)g(it)
 
2894
g(is)f(of)h(size)g Fp(nx)f Fv(b)m(y)h Fp(2*\(ny/2+1\))p
 
2895
Fv(.)37 b(If)28 b Fp(ny)f Fv(is)g(ev)m(en,)i(then)f(there)g(are)h(t)m
 
2896
(w)m(o)150 4382 y(padding)34 b(elemen)m(ts)h(at)i(the)e(end)g(of)h(eac)
 
2897
m(h)h(ro)m(w)e(\(whic)m(h)g(need)g(not)h(b)s(e)f(initialized,)e(as)j
 
2898
(they)g(are)g(only)150 4492 y(used)30 b(for)g(output\).)275
 
2899
4629 y(Figure)35 b(1)h(depicts)f(the)h(input)e(and)h(output)g(arra)m
 
2900
(ys)i(just)e(describ)s(ed,)g(for)h(b)s(oth)f(the)h(out-of-place)150
2858
2901
4738 y(and)30 b(in-place)f(transforms)h(\(with)f(the)i(arro)m(ws)f
2859
2902
(indicating)e(consecutiv)m(e)k(memory)e(lo)s(cations\).)275
2860
2903
4875 y(The)41 b(RFFTWND)i(transforms)e(are)h(unnormalized,)h(so)f(a)g
2866
2909
(illustrate)f(the)i(use)f(of)g(RFFTWND)i(b)m(y)e(sho)m(wing)g(ho)m(w)g
2867
2910
(y)m(ou)h(migh)m(t)f(use)g(it)g(to)i(compute)150 5340
2868
2911
y(the)35 b(\(cyclic\))h(con)m(v)m(olution)f(of)h(t)m(w)m(o-dimensional)
2869
 
e(real)h(arra)m(ys)g Fo(a)g Fu(and)g Fo(b)g Fu(\(using)f(the)h(iden)m
2870
 
(tit)m(y)g(that)h(a)p eop
 
2912
e(real)h(arra)m(ys)g Fp(a)g Fv(and)g Fp(b)g Fv(\(using)f(the)h(iden)m
 
2913
(tit)m(y)g(that)h(a)p eop end
2871
2914
%%Page: 9 11
2872
 
9 10 bop 150 -116 a Fu(Chapter)30 b(2:)41 b(T)-8 b(utorial)2768
2873
 
b(9)607 4188 y @beginspecial @setspecial
 
2915
TeXDict begin 9 10 bop 150 -116 a Fv(Chapter)30 b(2:)41
 
2916
b(T)-8 b(utorial)2768 b(9)607 4188 y @beginspecial @setspecial
2874
2917
%%BeginDocument: rfftwnd.eps
2875
2918
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2876
2919
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2877
2920
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2878
 
%%CreationDate: Sun Nov  7 20:50:16 1999
 
2921
%%CreationDate: Wed Jul 28 13:17:30 1999
2879
2922
%%For: stevenj@superjdj (Steven G. Johnson)
2880
2923
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2881
2924
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4610
4653
%%EndDocument
4611
4654
 @endspecial 275 4522 a(Figure)29 b(1:)42 b(Illustration)27
4612
4655
b(of)k(the)g(data)g(la)m(y)m(out)g(for)f(real)g(to)h(complex)f
4613
 
(transforms.)p eop
 
4656
(transforms.)p eop end
4614
4657
%%Page: 10 12
4615
 
10 11 bop 150 -116 a Fu(10)3232 b(FFTW)150 299 y(con)m(v)m(olution)41
4616
 
b(corresp)s(onds)e(to)i(a)g(p)s(oin)m(t)m(wise)e(pro)s(duct)h(of)h(the)
4617
 
f(F)-8 b(ourier)41 b(transforms\).)71 b(F)-8 b(or)41
4618
 
b(v)-5 b(ariet)m(y)d(,)150 408 y(in-place)25 b(transforms)h(are)g(used)
4619
 
g(for)g(the)g(forw)m(ard)g(FFTs)h(and)e(an)h(out-of-place)i(transform)d
4620
 
(is)h(used)f(for)150 518 y(the)31 b(in)m(v)m(erse)f(transform.)390
4621
 
692 y Fo(#include)46 b(<rfftw.h>)390 796 y(...)390 900
4622
 
y({)629 1003 y(fftw_real)f(a[M][2*\(N/2+1\)],)e(b[M][2*\(N/2+1\)],)h
4623
 
(c[M][N];)629 1107 y(fftw_complex)g(*A,)j(*B,)g(C[M][N/2+1];)629
4624
 
1211 y(rfftwnd_plan)d(p,)j(pinv;)629 1315 y(fftw_real)e(scale)h(=)i
4625
 
(1.0)f(/)g(\(M)g(*)h(N\);)629 1418 y(int)f(i,)g(j;)629
4626
 
1522 y(...)629 1626 y(p)190 b(=)48 b(rfftw2d_create_plan\(M,)41
4627
 
b(N,)48 b(FFTW_REAL_TO_COMPLEX,)1917 1730 y(FFTW_ESTIMATE)c(|)k
 
4658
TeXDict begin 10 11 bop 150 -116 a Fv(10)3232 b(FFTW)150
 
4659
299 y(con)m(v)m(olution)41 b(corresp)s(onds)e(to)i(a)g(p)s(oin)m(t)m
 
4660
(wise)e(pro)s(duct)h(of)h(the)f(F)-8 b(ourier)41 b(transforms\).)71
 
4661
b(F)-8 b(or)41 b(v)-5 b(ariet)m(y)d(,)150 408 y(in-place)25
 
4662
b(transforms)h(are)g(used)g(for)g(the)g(forw)m(ard)g(FFTs)h(and)e(an)h
 
4663
(out-of-place)i(transform)d(is)h(used)f(for)150 518 y(the)31
 
4664
b(in)m(v)m(erse)f(transform.)390 692 y Fp(#include)46
 
4665
b(<rfftw.h>)390 796 y(...)390 900 y({)629 1003 y(fftw_real)f
 
4666
(a[M][2*\(N/2+1\)],)e(b[M][2*\(N/2+1\)],)h(c[M][N];)629
 
4667
1107 y(fftw_complex)g(*A,)j(*B,)g(C[M][N/2+1];)629 1211
 
4668
y(rfftwnd_plan)d(p,)j(pinv;)629 1315 y(fftw_real)e(scale)h(=)i(1.0)f(/)
 
4669
g(\(M)g(*)h(N\);)629 1418 y(int)f(i,)g(j;)629 1522 y(...)629
 
4670
1626 y(p)190 b(=)48 b(rfftw2d_create_plan\(M,)41 b(N,)48
 
4671
b(FFTW_REAL_TO_COMPLEX,)1917 1730 y(FFTW_ESTIMATE)c(|)k
4628
4672
(FFTW_IN_PLACE\);)629 1834 y(pinv)e(=)i(rfftw2d_create_plan\(M,)41
4629
4673
b(N,)48 b(FFTW_COMPLEX_TO_REAL,)1917 1937 y(FFTW_ESTIMATE\);)629
4630
4674
2145 y(/*)f(aliases)f(for)h(accessing)e(complex)h(transform)f(outputs:)
4648
4692
o(real)o(\(pi)o(nv,)41 b(&C[0][0],)46 b(&c[0][0]\);)629
4649
4693
4739 y(...)629 4843 y(rfftwnd_destroy_plan\(p\))o(;)629
4650
4694
4947 y(rfftwnd_destroy_plan\(pi)o(nv\);)390 5051 y(})275
4651
 
5230 y Fu(W)-8 b(e)27 b(access)h(the)f(complex)f(outputs)g(of)g(the)h
 
4695
5230 y Fv(W)-8 b(e)27 b(access)h(the)f(complex)f(outputs)g(of)g(the)h
4652
4696
(in-place)e(transforms)h(b)m(y)g(casting)h(eac)m(h)g(real)f(arra)m(y)h
4653
 
(to)h(a)150 5340 y Fo(fftw_complex)d Fu(p)s(oin)m(ter.)39
 
4697
(to)h(a)150 5340 y Fp(fftw_complex)d Fv(p)s(oin)m(ter.)39
4654
4698
b(Because)30 b(this)d(is)g(a)i(\\\015at")h(p)s(oin)m(ter,)e(w)m(e)h(ha)
4655
4699
m(v)m(e)g(to)g(compute)g(the)f(ro)m(w-ma)5 b(jor)p eop
 
4700
end
4656
4701
%%Page: 11 13
4657
 
11 12 bop 150 -116 a Fu(Chapter)30 b(2:)41 b(T)-8 b(utorial)2722
4658
 
b(11)150 299 y(index)28 b Fo(ij)i Fu(explicitly)d(in)i(the)h(con)m(v)m
4659
 
(olution)f(pro)s(duct)g(lo)s(op.)40 b(In)29 b(order)g(to)i(normalize)e
4660
 
(the)h(con)m(v)m(olution,)150 408 y(w)m(e)23 b(m)m(ust)f(m)m(ultiply)d
4661
 
(b)m(y)k(a)f(scale)h(factor|w)m(e)h(can)e(do)g(so)h(either)f(b)s(efore)
4662
 
g(or)g(after)h(the)f(in)m(v)m(erse)g(transform,)150 518
4663
 
y(and)34 b(c)m(ho)s(ose)h(the)g(former)f(b)s(ecause)g(it)g(ob)m(viates)
4664
 
h(the)g(necessit)m(y)f(of)h(an)f(additional)e(lo)s(op.)52
4665
 
b(Notice)36 b(the)150 628 y(limits)28 b(of)j(the)f(lo)s(ops)f(and)h
4666
 
(the)h(dimensions)c(of)k(the)g(v)-5 b(arious)29 b(arra)m(ys.)275
4667
 
765 y(As)38 b(with)g(the)h(one-dimensional)e(RFFTW,)j(an)f
4668
 
(out-of-place)h Fo(FFTW_COMPLEX_TO_REAL)33 b Fu(trans-)150
 
4702
TeXDict begin 11 12 bop 150 -116 a Fv(Chapter)30 b(2:)41
 
4703
b(T)-8 b(utorial)2722 b(11)150 299 y(index)28 b Fp(ij)i
 
4704
Fv(explicitly)d(in)i(the)h(con)m(v)m(olution)f(pro)s(duct)g(lo)s(op.)40
 
4705
b(In)29 b(order)g(to)i(normalize)e(the)h(con)m(v)m(olution,)150
 
4706
408 y(w)m(e)23 b(m)m(ust)f(m)m(ultiply)d(b)m(y)k(a)f(scale)h(factor|w)m
 
4707
(e)h(can)e(do)g(so)h(either)f(b)s(efore)g(or)g(after)h(the)f(in)m(v)m
 
4708
(erse)g(transform,)150 518 y(and)34 b(c)m(ho)s(ose)h(the)g(former)f(b)s
 
4709
(ecause)g(it)g(ob)m(viates)h(the)g(necessit)m(y)f(of)h(an)f(additional)
 
4710
e(lo)s(op.)52 b(Notice)36 b(the)150 628 y(limits)28 b(of)j(the)f(lo)s
 
4711
(ops)f(and)h(the)h(dimensions)c(of)k(the)g(v)-5 b(arious)29
 
4712
b(arra)m(ys.)275 765 y(As)38 b(with)g(the)h(one-dimensional)e(RFFTW,)j
 
4713
(an)f(out-of-place)h Fp(FFTW_COMPLEX_TO_REAL)33 b Fv(trans-)150
4669
4714
875 y(form)e(has)h(the)g(side-e\013ect)h(of)f(o)m(v)m(erwriting)g(its)f
4670
4715
(input)f(arra)m(y)-8 b(.)47 b(\(The)31 b(real-to-complex)i(transform,)f
4671
4716
(on)150 984 y(the)k(other)f(hand,)h(lea)m(v)m(es)h(its)e(input)e(arra)m
4679
4724
b(RFFTWND)d(in)e(one)h(dimension)d(is)i(sligh)m(tly)g(slo)m(w)m(er)150
4680
4725
1423 y(than)d(RFFTW)h(b)s(ecause)f(RFFTWND)i(uses)e(an)g(extra)h
4681
4726
(bu\013er)f(arra)m(y)h(in)m(ternally)-8 b(.)150 1688
4682
 
y Ft(2.5)68 b(Multi-dimensional)46 b(Arra)l(y)f(F)-11
4683
 
b(ormat)275 1883 y Fu(This)28 b(section)i(describ)s(es)f(the)h(format)h
 
4727
y Fu(2.5)68 b(Multi-dimensional)46 b(Arra)l(y)f(F)-11
 
4728
b(ormat)275 1883 y Fv(This)28 b(section)i(describ)s(es)f(the)h(format)h
4684
4729
(in)e(whic)m(h)g(m)m(ulti-dimensional)d(arra)m(ys)k(are)h(stored.)41
4685
4730
b(W)-8 b(e)31 b(felt)150 1993 y(that)f(a)h(detailed)d(discussion)f(of)j
4686
4731
(this)f(topic)h(w)m(as)g(necessary)-8 b(,)31 b(since)e(it)g(is)g(often)
4687
4732
h(a)g(source)g(of)g(confusion)150 2103 y(among)c(users)e(and)g(sev)m
4688
4733
(eral)h(di\013eren)m(t)g(formats)g(are)h(common.)39 b(Although)24
4689
4734
b(the)h(commen)m(ts)h(b)s(elo)m(w)e(refer)150 2212 y(to)31
4690
 
b Fo(fftwnd)p Fu(,)e(they)i(are)f(also)h(applicable)d(to)j(the)g
4691
 
Fo(rfftwnd)d Fu(routines.)150 2443 y Fi(2.5.1)63 b(Ro)m(w-ma)7
4692
 
b(jor)38 b(F)-10 b(ormat)275 2638 y Fu(The)38 b(m)m(ulti-dimensional)d
4693
 
(arra)m(ys)k(passed)f(to)h Fo(fftwnd)e Fu(are)j(exp)s(ected)f(to)g(b)s
 
4735
b Fp(fftwnd)p Fv(,)e(they)i(are)f(also)h(applicable)d(to)j(the)g
 
4736
Fp(rfftwnd)d Fv(routines.)150 2443 y Fh(2.5.1)63 b(Ro)m(w-ma)7
 
4737
b(jor)38 b(F)-10 b(ormat)275 2638 y Fv(The)38 b(m)m(ulti-dimensional)d
 
4738
(arra)m(ys)k(passed)f(to)h Fp(fftwnd)e Fv(are)j(exp)s(ected)f(to)g(b)s
4694
4739
(e)f(stored)h(as)g(a)g(single)150 2748 y(con)m(tiguous)c(blo)s(c)m(k)f
4695
 
(in)g Fp(ro)m(w-ma)5 b(jor)42 b Fu(order)34 b(\(sometimes)i(called)e
 
4740
(in)g Fq(ro)m(w-ma)5 b(jor)42 b Fv(order)34 b(\(sometimes)i(called)e
4696
4741
(\\C)h(order"\).)54 b(Basically)-8 b(,)36 b(this)e(means)150
4697
4742
2857 y(that)26 b(as)f(y)m(ou)h(step)f(through)f(adjacen)m(t)j(memory)e
4698
4743
(lo)s(cations,)h(the)f(\014rst)g(dimension's)d(index)i(v)-5
4699
4744
b(aries)25 b(most)150 2967 y(slo)m(wly)k(and)h(the)h(last)f
4700
4745
(dimension's)e(index)h(v)-5 b(aries)29 b(most)i(quic)m(kly)-8
4701
4746
b(.)275 3105 y(T)g(o)31 b(b)s(e)g(more)g(explicit,)f(let)i(us)e
4702
 
(consider)g(an)i(arra)m(y)f(of)h(rank)f Fl(d)g Fu(whose)g(dimensions)e
4703
 
(are)j Fl(n)3418 3119 y Fn(1)3475 3105 y Fs(\002)21 b
4704
 
Fl(n)3622 3119 y Fn(2)3679 3105 y Fs(\002)150 3214 y
4705
 
Fl(n)205 3228 y Fn(3)256 3214 y Fs(\002)15 b(\001)g(\001)g(\001)g(\002)
4706
 
g Fl(n)603 3228 y Fk(d)641 3214 y Fu(.)39 b(No)m(w,)29
 
4747
(consider)g(an)i(arra)m(y)f(of)h(rank)f Fm(d)g Fv(whose)g(dimensions)e
 
4748
(are)j Fm(n)3418 3119 y Fo(1)3475 3105 y Ft(\002)21 b
 
4749
Fm(n)3622 3119 y Fo(2)3679 3105 y Ft(\002)150 3214 y
 
4750
Fm(n)205 3228 y Fo(3)256 3214 y Ft(\002)15 b(\001)g(\001)g(\001)g(\002)
 
4751
g Fm(n)603 3228 y Fl(d)641 3214 y Fv(.)39 b(No)m(w,)29
4707
4752
b(w)m(e)f(sp)s(ecify)f(a)g(lo)s(cation)h(in)e(the)i(arra)m(y)g(b)m(y)f
4708
4753
(a)h(sequence)g(of)f(\(zero-based\))j(indices,)150 3324
4709
 
y(one)36 b(for)g(eac)m(h)i(dimension:)49 b(\()p Fl(i)1220
4710
 
3338 y Fn(1)1258 3324 y Fl(;)15 b(i)1329 3338 y Fn(2)1367
4711
 
3324 y Fl(;)g(i)1438 3338 y Fn(3)1476 3324 y Fl(;)g(:)g(:)g(:)i(;)e(i)
4712
 
1709 3338 y Fk(d)1748 3324 y Fu(\).)58 b(If)36 b(the)g(arra)m(y)h(is)e
 
4754
y(one)36 b(for)g(eac)m(h)i(dimension:)49 b(\()p Fm(i)1220
 
4755
3338 y Fo(1)1258 3324 y Fm(;)15 b(i)1329 3338 y Fo(2)1367
 
4756
3324 y Fm(;)g(i)1438 3338 y Fo(3)1476 3324 y Fm(;)g(:)g(:)g(:)i(;)e(i)
 
4757
1709 3338 y Fl(d)1748 3324 y Fv(\).)58 b(If)36 b(the)g(arra)m(y)h(is)e
4713
4758
(stored)h(in)f(ro)m(w-ma)5 b(jor)37 b(order,)g(then)150
4714
4759
3433 y(this)29 b(elemen)m(t)i(is)f(lo)s(cated)g(at)h(the)g(p)s(osition)
4715
 
d Fl(i)1715 3447 y Fk(d)1775 3433 y Fu(+)19 b Fl(n)1920
4716
 
3447 y Fk(d)1959 3433 y Fu(\()p Fl(i)2025 3447 y Fk(d)p
4717
 
Fj(\000)p Fn(1)2169 3433 y Fu(+)h Fl(n)2315 3447 y Fk(d)p
4718
 
Fj(\000)p Fn(1)2439 3433 y Fu(\()p Fl(:)15 b(:)g(:)21
4719
 
b Fu(+)f Fl(n)2746 3447 y Fn(2)2783 3433 y Fl(i)2814
4720
 
3447 y Fn(1)2851 3433 y Fu(\)\).)275 3571 y(Note)32 b(that)g(eac)m(h)h
 
4760
d Fm(i)1715 3447 y Fl(d)1775 3433 y Fv(+)19 b Fm(n)1920
 
4761
3447 y Fl(d)1959 3433 y Fv(\()p Fm(i)2025 3447 y Fl(d)p
 
4762
Fk(\000)p Fo(1)2169 3433 y Fv(+)h Fm(n)2315 3447 y Fl(d)p
 
4763
Fk(\000)p Fo(1)2439 3433 y Fv(\()p Fm(:)15 b(:)g(:)21
 
4764
b Fv(+)f Fm(n)2746 3447 y Fo(2)2783 3433 y Fm(i)2814
 
4765
3447 y Fo(1)2851 3433 y Fv(\)\).)275 3571 y(Note)32 b(that)g(eac)m(h)h
4721
4766
(elemen)m(t)f(of)f(the)h(arra)m(y)g(m)m(ust)f(b)s(e)g(of)g(t)m(yp)s(e)h
4722
 
Fo(fftw_complex)p Fu(;)c(i.e.)44 b(a)32 b(\(real,)g(imagi-)150
 
4767
Fp(fftw_complex)p Fv(;)c(i.e.)44 b(a)32 b(\(real,)g(imagi-)150
4723
4768
3681 y(nary\))g(pair)f(of)h(\(double-precision\))e(n)m(um)m(b)s(ers.)44
4724
 
b(Note)34 b(also)e(that,)h(in)e Fo(fftwnd)p Fu(,)g(the)h(expression)f
 
4769
b(Note)34 b(also)e(that,)h(in)e Fp(fftwnd)p Fv(,)g(the)h(expression)f
4725
4770
(ab)s(o)m(v)m(e)150 3790 y(is)g(m)m(ultiplied)d(b)m(y)k(the)g(stride)f
4726
4771
(to)h(get)h(the)f(actual)h(arra)m(y)f(index|this)d(is)i(useful)f(in)h
4727
4772
(situations)f(where)150 3900 y(eac)m(h)35 b(elemen)m(t)f(of)f(the)h(m)m
4729
4774
(or)f(another)h(arra)m(y)-8 b(,)150 4009 y(and)28 b(y)m(ou)g(just)g(w)m
4730
4775
(an)m(t)h(to)g(transform)e(a)i(single)e(\014eld.)39 b(In)27
4731
4776
b(most)i(cases,)g(ho)m(w)m(ev)m(er,)i(y)m(ou)d(use)g(a)h(stride)e(of)h
4732
 
(1.)150 4349 y Fi(2.5.2)63 b(Column-ma)7 b(jor)39 b(F)-10
4733
 
b(ormat)275 4545 y Fu(Readers)27 b(from)g(the)h(F)-8
 
4777
(1.)150 4349 y Fh(2.5.2)63 b(Column-ma)7 b(jor)39 b(F)-10
 
4778
b(ormat)275 4545 y Fv(Readers)27 b(from)g(the)h(F)-8
4734
4779
b(ortran)28 b(w)m(orld)f(are)h(used)e(to)j(arra)m(ys)f(stored)f(in)g
4735
 
Fp(column-ma)5 b(jor)33 b Fu(order)27 b(\(some-)150 4654
 
4780
Fq(column-ma)5 b(jor)33 b Fv(order)27 b(\(some-)150 4654
4736
4781
y(times)k(called)g(\\F)-8 b(ortran)33 b(order"\).)45
4737
4782
b(This)30 b(is)h(essen)m(tially)g(the)h(exact)h(opp)s(osite)e(of)h(ro)m
4738
4783
(w-ma)5 b(jor)32 b(order)f(in)150 4764 y(that,)g(here,)g(the)f
4739
 
Fq(\014rst)40 b Fu(dimension's)28 b(index)h(v)-5 b(aries)30
 
4784
Fr(\014rst)40 b Fv(dimension's)28 b(index)h(v)-5 b(aries)30
4740
4785
b(most)g(quic)m(kly)-8 b(.)275 4902 y(If)22 b(y)m(ou)h(ha)m(v)m(e)h(an)
4741
4786
e(arra)m(y)h(stored)g(in)f(column-ma)5 b(jor)21 b(order)i(and)f(wish)f
4742
 
(to)i(transform)f(it)g(using)f Fo(fftwnd)p Fu(,)150 5011
 
4787
(to)i(transform)f(it)g(using)f Fp(fftwnd)p Fv(,)150 5011
4743
4788
y(it)31 b(is)g(quite)g(easy)i(to)f(do.)45 b(When)32 b(creating)g(the)g
4744
4789
(plan,)f(simply)e(pass)j(the)g(dimensions)d(of)j(the)g(arra)m(y)g(to)
4745
 
150 5121 y Fo(fftwnd_create_plan)d Fu(in)k Fq(r)-5 b(everse)37
4746
 
b(or)-5 b(der)p Fu(.)54 b(F)-8 b(or)35 b(example,)g(if)e(y)m(our)h
4747
 
(arra)m(y)h(is)e(a)i(rank)f(three)g Fo(N)c(x)g(M)g(x)150
4748
 
5230 y(L)i Fu(matrix)f(in)g(column-ma)5 b(jor)32 b(order,)g(y)m(ou)h
 
4790
150 5121 y Fp(fftwnd_create_plan)d Fv(in)k Fr(r)-5 b(everse)37
 
4791
b(or)-5 b(der)p Fv(.)54 b(F)-8 b(or)35 b(example,)g(if)e(y)m(our)h
 
4792
(arra)m(y)h(is)e(a)i(rank)f(three)g Fp(N)c(x)g(M)g(x)150
 
4793
5230 y(L)i Fv(matrix)f(in)g(column-ma)5 b(jor)32 b(order,)g(y)m(ou)h
4749
4794
(should)d(pass)i(the)g(dimensions)e(of)i(the)g(arra)m(y)h(as)g(if)e(it)
4750
 
g(w)m(ere)150 5340 y(an)39 b Fo(L)30 b(x)g(M)g(x)g(N)39
4751
 
b Fu(matrix)f(\(whic)m(h)g(it)h(is,)i(from)d(the)h(p)s(ersp)s(ectiv)m
4752
 
(e)g(of)g Fo(fftwnd)p Fu(\).)65 b(This)38 b(is)g(done)h(for)g(y)m(ou)p
4753
 
eop
 
4795
g(w)m(ere)150 5340 y(an)39 b Fp(L)30 b(x)g(M)g(x)g(N)39
 
4796
b Fv(matrix)f(\(whic)m(h)g(it)h(is,)i(from)d(the)h(p)s(ersp)s(ectiv)m
 
4797
(e)g(of)g Fp(fftwnd)p Fv(\).)65 b(This)38 b(is)g(done)h(for)g(y)m(ou)p
 
4798
eop end
4754
4799
%%Page: 12 14
4755
 
12 13 bop 150 -116 a Fu(12)3232 b(FFTW)150 299 y(automatically)27
4756
 
b(b)m(y)f(the)h(FFTW)h(F)-8 b(ortran)27 b(wrapp)s(er)e(routines)h
4757
 
(\(see)i(Chapter)e(5)h([Calling)e(FFTW)i(from)150 408
4758
 
y(F)-8 b(ortran],)32 b(page)f(51\).)150 635 y Fi(2.5.3)63
4759
 
b(Static)39 b(Arra)m(ys)i(in)g(C)275 829 y Fu(Multi-dimensional)35
4760
 
b(arra)m(ys)k(declared)f(statically)h(\(that)h(is,)g(at)g(compile)e
4761
 
(time,)j(not)e(necessarily)150 938 y(with)29 b(the)h
4762
 
Fo(static)e Fu(k)m(eyw)m(ord\))j(in)d(C)i(are)g Fq(alr)-5
4763
 
b(e)g(ady)41 b Fu(in)29 b(ro)m(w-ma)5 b(jor)30 b(order.)40
4764
 
b(Y)-8 b(ou)31 b(don't)f(ha)m(v)m(e)h(to)f(do)g(an)m(y-)150
 
4800
TeXDict begin 12 13 bop 150 -116 a Fv(12)3232 b(FFTW)150
 
4801
299 y(automatically)27 b(b)m(y)f(the)h(FFTW)h(F)-8 b(ortran)27
 
4802
b(wrapp)s(er)e(routines)h(\(see)i(Chapter)e(5)h([Calling)e(FFTW)i(from)
 
4803
150 408 y(F)-8 b(ortran],)32 b(page)f(51\).)150 635 y
 
4804
Fh(2.5.3)63 b(Static)39 b(Arra)m(ys)i(in)g(C)275 829
 
4805
y Fv(Multi-dimensional)35 b(arra)m(ys)k(declared)f(statically)h(\(that)
 
4806
h(is,)g(at)g(compile)e(time,)j(not)e(necessarily)150
 
4807
938 y(with)29 b(the)h Fp(static)e Fv(k)m(eyw)m(ord\))j(in)d(C)i(are)g
 
4808
Fr(alr)-5 b(e)g(ady)41 b Fv(in)29 b(ro)m(w-ma)5 b(jor)30
 
4809
b(order.)40 b(Y)-8 b(ou)31 b(don't)f(ha)m(v)m(e)h(to)f(do)g(an)m(y-)150
4765
4810
1048 y(thing)g(sp)s(ecial)f(to)j(transform)e(them.)42
4766
4811
b(\(See)31 b(Section)g(2.2)h([Complex)e(Multi-dimensional)d(T)-8
4767
4812
b(ransforms)150 1157 y(T)g(utorial],)30 b(page)h(4,)g(for)f(an)g
4768
4813
(example)g(of)h(this)e(sort)i(of)f(co)s(de.\))150 1384
4769
 
y Fi(2.5.4)63 b(Dynamic)39 b(Arra)m(ys)h(in)i(C)275 1578
4770
 
y Fu(Often,)49 b(esp)s(ecially)43 b(for)j(large)f(arra)m(ys,)50
 
4814
y Fh(2.5.4)63 b(Dynamic)39 b(Arra)m(ys)h(in)i(C)275 1578
 
4815
y Fv(Often,)49 b(esp)s(ecially)43 b(for)j(large)f(arra)m(ys,)50
4771
4816
b(it)45 b(is)f(desirable)g(to)i(allo)s(cate)g(the)g(arra)m(ys)g
4772
4817
(dynamically)-8 b(,)150 1687 y(at)44 b(run)m(time.)76
4773
4818
b(This)41 b(isn't)h(to)s(o)i(hard)e(to)h(do,)j(although)d(it)f(is)g
4775
4820
1797 y(dimensional)28 b(arra)m(ys)i(as)h(it)f(is)f(for)h
4776
4821
(one-dimensional)e(arra)m(ys.)275 1932 y(Creating)35
4777
4822
b(the)i(arra)m(y)g(is)e(simple:)51 b(using)35 b(a)h(dynamic-allo)s
4778
 
(cation)f(routine)g(lik)m(e)h Fo(malloc)p Fu(,)g(allo)s(cate)150
 
4823
(cation)f(routine)g(lik)m(e)h Fp(malloc)p Fv(,)g(allo)s(cate)150
4779
4824
2042 y(an)27 b(arra)m(y)h(big)e(enough)h(to)h(store)g(N)g
4780
 
Fo(fftw_complex)23 b Fu(v)-5 b(alues,)28 b(where)f(N)g(is)f(the)i(pro)s
 
4825
Fp(fftw_complex)23 b Fv(v)-5 b(alues,)28 b(where)f(N)g(is)f(the)i(pro)s
4781
4826
(duct)e(of)h(the)h(sizes)f(of)150 2152 y(the)32 b(arra)m(y)h
4782
4827
(dimensions)c(\(i.e.)45 b(the)32 b(total)h(n)m(um)m(b)s(er)e(of)h
4783
4828
(complex)f(v)-5 b(alues)31 b(in)g(the)h(arra)m(y\).)46
4784
4829
b(F)-8 b(or)33 b(example,)150 2261 y(here)d(is)g(co)s(de)g(to)h(allo)s
4785
4830
(cate)g(a)g(5x12x27)h(rank)e(3)h(arra)m(y:)390 2391 y
4786
 
Fo(fftw_complex)44 b(*an_array;)390 2598 y(an_array)i(=)h
 
4831
Fp(fftw_complex)44 b(*an_array;)390 2598 y(an_array)i(=)h
4787
4832
(\(fftw_complex)d(*\))j(malloc\(5)f(*)h(12)h(*)f(27)g(*)h
4788
 
(sizeof\(fftw_complex\)\);)275 2734 y Fu(Accessing)38
 
4833
(sizeof\(fftw_complex\)\);)275 2734 y Fv(Accessing)38
4789
4834
b(the)f(arra)m(y)i(elemen)m(ts,)h(ho)m(w)m(ev)m(er,)h(is)c(more)h(tric)
4790
4835
m(ky|y)m(ou)g(can't)h(simply)c(use)i(m)m(ultiple)150
4791
 
2844 y(applications)k(of)i(the)h(`)p Fo([])p Fu(')f(op)s(erator)g(lik)m
 
4836
2844 y(applications)k(of)i(the)h(`)p Fp([])p Fv(')f(op)s(erator)g(lik)m
4792
4837
(e)f(y)m(ou)i(could)e(for)h(static)g(arra)m(ys.)79 b(Instead,)47
4793
4838
b(y)m(ou)c(ha)m(v)m(e)h(to)150 2953 y(explicitly)33 b(compute)j(the)g
4794
4839
(o\013set)h(in)m(to)f(the)f(arra)m(y)i(using)d(the)i(form)m(ula)f(giv)m
4795
4840
(en)g(earlier)g(for)g(ro)m(w-ma)5 b(jor)150 3063 y(arra)m(ys.)48
4796
4841
b(F)-8 b(or)33 b(example,)h(to)f(reference)g(the)g(\()p
4797
 
Fl(i;)15 b(j;)g(k)s Fu(\)-th)35 b(elemen)m(t)e(of)g(the)g(arra)m(y)g
 
4842
Fm(i;)15 b(j;)g(k)s Fv(\)-th)35 b(elemen)m(t)e(of)g(the)g(arra)m(y)g
4798
4843
(allo)s(cated)g(ab)s(o)m(v)m(e,)i(y)m(ou)150 3172 y(w)m(ould)29
4799
 
b(use)h(the)h(expression)e Fo(an_array[k)e(+)k(27)e(*)h(\(j)g(+)g(12)g
4800
 
(*)g(i\)])p Fu(.)275 3308 y(This)h(pain)h(can)h(b)s(e)g(alleviated)f
 
4844
b(use)h(the)h(expression)e Fp(an_array[k)e(+)k(27)e(*)h(\(j)g(+)g(12)g
 
4845
(*)g(i\)])p Fv(.)275 3308 y(This)h(pain)h(can)h(b)s(e)g(alleviated)f
4801
4846
(somewhat)i(b)m(y)f(de\014ning)e(appropriate)h(macros,)j(or,)f(in)e(C)p
4802
 
Fo(++)p Fu(,)h(cre-)150 3418 y(ating)d(a)h(class)f(and)g(o)m(v)m
4803
 
(erloading)h(the)f(`)p Fo(\(\))p Fu(')g(op)s(erator.)150
4804
 
3644 y Fi(2.5.5)63 b(Dynamic)39 b(Arra)m(ys)h(in)i(C|The)e(W)-10
4805
 
b(rong)42 b(W)-10 b(a)m(y)275 3838 y Fu(A)31 b(di\013eren)m(t)h(metho)s
 
4847
Fp(++)p Fv(,)h(cre-)150 3418 y(ating)d(a)h(class)f(and)g(o)m(v)m
 
4848
(erloading)h(the)f(`)p Fp(\(\))p Fv(')g(op)s(erator.)150
 
4849
3644 y Fh(2.5.5)63 b(Dynamic)39 b(Arra)m(ys)h(in)i(C|The)e(W)-10
 
4850
b(rong)42 b(W)-10 b(a)m(y)275 3838 y Fv(A)31 b(di\013eren)m(t)h(metho)s
4806
4851
(d)f(for)h(allo)s(cating)f(m)m(ulti-dimensional)d(arra)m(ys)k(in)e(C)i
4807
4852
(is)f(often)h(suggested)g(that)150 3947 y(is)e(incompatible)g(with)g
4808
 
Fo(fftwnd)p Fu(:)41 b Fq(using)33 b(it)g(wil)5 b(l)34
 
4853
Fp(fftwnd)p Fv(:)41 b Fr(using)33 b(it)g(wil)5 b(l)34
4809
4854
b(c)-5 b(ause)34 b(FFTW)g(to)g(die)g(a)f(p)-5 b(ainful)35
4810
 
b(de)-5 b(ath)p Fu(.)45 b(W)-8 b(e)33 b(discuss)150 4057
 
4855
b(de)-5 b(ath)p Fv(.)45 b(W)-8 b(e)33 b(discuss)150 4057
4811
4856
y(the)h(tec)m(hnique)g(here,)i(ho)m(w)m(ev)m(er,)g(b)s(ecause)f(it)e
4812
4857
(is)h(so)g(commonly)g(kno)m(wn)g(and)f(used.)52 b(This)32
4813
4858
b(metho)s(d)i(is)150 4167 y(to)d(create)g(arra)m(ys)g(of)f(p)s(oin)m
4814
4859
(ters)f(of)h(arra)m(ys)h(of)f(p)s(oin)m(ters)f(of)36
4815
4860
b(.)22 b(.)g(.)11 b(etcetera.)43 b(F)-8 b(or)31 b(example,)f(the)h
4816
4861
(analogue)150 4276 y(in)e(this)g(metho)s(d)h(to)h(the)g(example)f(ab)s
4817
 
(o)m(v)m(e)h(is:)390 4406 y Fo(int)47 b(i,j;)390 4510
 
4862
(o)m(v)m(e)h(is:)390 4406 y Fp(int)47 b(i,j;)390 4510
4818
4863
y(fftw_complex)d(***a_bad_array;)92 b(/*)47 b(another)f(way)h(to)g
4819
4864
(make)g(a)g(5x12x27)f(array)g(*/)390 4717 y(a_bad_array)f(=)i
4820
4865
(\(fftw_complex)d(***\))j(malloc\(5)f(*)h(sizeof\(fftw_complex)c
4823
4868
(malloc\(12)e(*)j(sizeof\(fftw_complex)42 b(*\)\);)629
4824
4869
5132 y(for)47 b(\(j)g(=)g(0;)g(j)h(<)f(12;)g(++j\))867
4825
4870
5236 y(a_bad_array[i][j])c(=)1154 5340 y(\(fftw_complex)h(*\))j
4826
 
(malloc\(27)e(*)j(sizeof\(fftw_complex\)\);)p eop
 
4871
(malloc\(27)e(*)j(sizeof\(fftw_complex\)\);)p eop end
4827
4872
%%Page: 13 15
4828
 
13 14 bop 150 -116 a Fu(Chapter)30 b(2:)41 b(T)-8 b(utorial)2722
4829
 
b(13)390 299 y Fo(})275 443 y Fu(As)34 b(y)m(ou)i(can)f(see,)i(this)d
4830
 
(sort)h(of)g(arra)m(y)g(is)f(incon)m(v)m(enien)m(t)h(to)h(allo)s(cate)f
4831
 
(\(and)g(deallo)s(cate\).)55 b(On)34 b(the)150 552 y(other)g(hand,)h
4832
 
(it)f(has)g(the)g(adv)-5 b(an)m(tage)36 b(that)f(the)f(\()p
4833
 
Fl(i;)15 b(j;)g(k)s Fu(\)-th)37 b(elemen)m(t)e(can)g(b)s(e)e
4834
 
(referenced)h(simply)e(b)m(y)150 662 y Fo(a_bad_array[i][j][k])p
4835
 
Fu(.)275 806 y(If)e(y)m(ou)g(lik)m(e)g(this)f(tec)m(hnique)i(and)e(w)m
4836
 
(an)m(t)j(to)f(maximize)f(con)m(v)m(enience)h(in)e(accessing)i(the)g
4837
 
(arra)m(y)-8 b(,)32 b(but)150 915 y(still)39 b(w)m(an)m(t)k(to)f(pass)f
4838
 
(the)h(arra)m(y)g(to)g(FFTW,)h(y)m(ou)e(can)h(use)f(a)h(h)m(ybrid)e
4839
 
(metho)s(d.)73 b(Allo)s(cate)42 b(the)g(ar-)150 1025
4840
 
y(ra)m(y)36 b(as)f(one)h(con)m(tiguous)f(blo)s(c)m(k,)i(but)d(also)i
4841
 
(declare)f(an)g(arra)m(y)h(of)g(arra)m(ys)f(of)h(p)s(oin)m(ters)e(that)
4842
 
i(p)s(oin)m(t)e(to)150 1134 y(appropriate)h(places)i(in)e(the)h(blo)s
4843
 
(c)m(k.)58 b(That)37 b(sort)f(of)h(tric)m(k)f(is)f(b)s(ey)m(ond)h(the)g
4844
 
(scop)s(e)h(of)f(this)g(do)s(cumen-)150 1244 y(tation;)f(for)e(more)h
4845
 
(information)d(on)i(m)m(ulti-dimensional)c(arra)m(ys)34
4846
 
b(in)e(C,)h(see)h(the)f Fo(comp.lang.c)d Fu(F)-10 b(A)m(Q)150
4847
 
1354 y(\()p Fo(http://www.eskimo.com/~scs)o(/C-)o(faq/)o(s6.h)o(tml)o
4848
 
Fu(\).)150 1634 y Ft(2.6)68 b(W)-11 b(ords)45 b(of)g(Wisdom)275
4849
 
1836 y Fu(FFTW)29 b(implemen)m(ts)f(a)h(metho)s(d)g(for)f(sa)m(ving)h
 
4873
TeXDict begin 13 14 bop 150 -116 a Fv(Chapter)30 b(2:)41
 
4874
b(T)-8 b(utorial)2722 b(13)390 299 y Fp(})275 443 y Fv(As)34
 
4875
b(y)m(ou)i(can)f(see,)i(this)d(sort)h(of)g(arra)m(y)g(is)f(incon)m(v)m
 
4876
(enien)m(t)h(to)h(allo)s(cate)f(\(and)g(deallo)s(cate\).)55
 
4877
b(On)34 b(the)150 552 y(other)g(hand,)h(it)f(has)g(the)g(adv)-5
 
4878
b(an)m(tage)36 b(that)f(the)f(\()p Fm(i;)15 b(j;)g(k)s
 
4879
Fv(\)-th)37 b(elemen)m(t)e(can)g(b)s(e)e(referenced)h(simply)e(b)m(y)
 
4880
150 662 y Fp(a_bad_array[i][j][k])p Fv(.)275 806 y(If)e(y)m(ou)g(lik)m
 
4881
(e)g(this)f(tec)m(hnique)i(and)e(w)m(an)m(t)j(to)f(maximize)f(con)m(v)m
 
4882
(enience)h(in)e(accessing)i(the)g(arra)m(y)-8 b(,)32
 
4883
b(but)150 915 y(still)39 b(w)m(an)m(t)k(to)f(pass)f(the)h(arra)m(y)g
 
4884
(to)g(FFTW,)h(y)m(ou)e(can)h(use)f(a)h(h)m(ybrid)e(metho)s(d.)73
 
4885
b(Allo)s(cate)42 b(the)g(ar-)150 1025 y(ra)m(y)36 b(as)f(one)h(con)m
 
4886
(tiguous)f(blo)s(c)m(k,)i(but)d(also)i(declare)f(an)g(arra)m(y)h(of)g
 
4887
(arra)m(ys)f(of)h(p)s(oin)m(ters)e(that)i(p)s(oin)m(t)e(to)150
 
4888
1134 y(appropriate)h(places)i(in)e(the)h(blo)s(c)m(k.)58
 
4889
b(That)37 b(sort)f(of)h(tric)m(k)f(is)f(b)s(ey)m(ond)h(the)g(scop)s(e)h
 
4890
(of)f(this)g(do)s(cumen-)150 1244 y(tation;)f(for)e(more)h(information)
 
4891
d(on)i(m)m(ulti-dimensional)c(arra)m(ys)34 b(in)e(C,)h(see)h(the)f
 
4892
Fp(comp.lang.c)d Fv(F)-10 b(A)m(Q)150 1354 y(\()p Fp
 
4893
(http://www.eskimo.com/~scs)o(/C-)o(faq/)o(s6.h)o(tml)o
 
4894
Fv(\).)150 1634 y Fu(2.6)68 b(W)-11 b(ords)45 b(of)g(Wisdom)275
 
4895
1836 y Fv(FFTW)29 b(implemen)m(ts)f(a)h(metho)s(d)g(for)f(sa)m(ving)h
4850
4896
(plans)f(to)i(disk)d(and)i(restoring)f(them.)40 b(In)29
4851
4897
b(fact,)h(what)150 1946 y(FFTW)k(do)s(es)e(is)h(more)g(general)g(than)g
4852
4898
(just)f(sa)m(ving)h(and)g(loading)e(plans.)48 b(The)32
4853
 
b(mec)m(hanism)h(is)f(called)150 2055 y Fo(wisdom)p Fu(.)47
 
4899
b(mec)m(hanism)h(is)f(called)150 2055 y Fp(wisdom)p Fv(.)47
4854
4900
b(Here,)35 b(w)m(e)e(describ)s(e)f(this)g(feature)h(at)h(a)g(high)e
4855
4901
(lev)m(el.)48 b(See)34 b(Chapter)e(3)i([FFTW)g(Reference],)150
4856
4902
2165 y(page)d(17,)h(for)e(a)g(less)g(casual)g(\(but)g(more)h
4857
 
(complete\))g(discussion)d(of)i(ho)m(w)h(to)g(use)f Fo(wisdom)f
4858
 
Fu(in)g(FFTW.)275 2309 y(Plans)34 b(created)i(with)f(the)g
4859
 
Fo(FFTW_MEASURE)e Fu(option)h(pro)s(duce)h(near-optimal)g(FFT)g(p)s
 
4903
(complete\))g(discussion)d(of)i(ho)m(w)h(to)g(use)f Fp(wisdom)f
 
4904
Fv(in)g(FFTW.)275 2309 y(Plans)34 b(created)i(with)f(the)g
 
4905
Fp(FFTW_MEASURE)e Fv(option)h(pro)s(duce)h(near-optimal)g(FFT)g(p)s
4860
4906
(erformance,)150 2418 y(but)d(it)h(can)h(tak)m(e)h(a)e(long)g(time)g
4861
4907
(to)h(compute)f(a)h(plan)e(b)s(ecause)h(FFTW)h(m)m(ust)f(actually)g
4862
4908
(measure)g(the)150 2528 y(run)m(time)c(of)i(man)m(y)f(p)s(ossible)e
4866
4912
f(computed)h(that)g(the)g(start-up)f(time)h(is)150 2747
4867
4913
y(irrelev)-5 b(an)m(t.)55 b(F)-8 b(or)36 b(short)f(initialization)d
4868
4914
(times)j(but)g(sligh)m(tly)e(slo)m(w)m(er)j(transforms,)g(w)m(e)g(ha)m
4869
 
(v)m(e)g(pro)m(vided)150 2857 y Fo(FFTW_ESTIMATE)p Fu(.)f(The)25
4870
 
b Fo(wisdom)e Fu(mec)m(hanism)i(is)f(a)h(w)m(a)m(y)h(to)g(get)g(the)g
 
4915
(v)m(e)g(pro)m(vided)150 2857 y Fp(FFTW_ESTIMATE)p Fv(.)f(The)25
 
4916
b Fp(wisdom)e Fv(mec)m(hanism)i(is)f(a)h(w)m(a)m(y)h(to)g(get)g(the)g
4871
4917
(b)s(est)e(of)i(b)s(oth)e(w)m(orlds.)38 b(There)24 b(are,)150
4872
4918
2966 y(ho)m(w)m(ev)m(er,)35 b(certain)d(ca)m(v)m(eats)j(that)e(the)g
4873
4919
(user)e(m)m(ust)i(b)s(e)e(a)m(w)m(are)j(of)f(in)e(using)g
4874
 
Fo(wisdom)p Fu(.)45 b(F)-8 b(or)33 b(this)e(reason,)150
4875
 
3076 y Fo(wisdom)e Fu(is)g(an)h(optional)g(feature)h(whic)m(h)e(is)g
 
4920
Fp(wisdom)p Fv(.)45 b(F)-8 b(or)33 b(this)e(reason,)150
 
4921
3076 y Fp(wisdom)e Fv(is)g(an)h(optional)g(feature)h(whic)m(h)e(is)g
4876
4922
(not)i(enabled)e(b)m(y)h(default.)275 3220 y(A)m(t)c(its)g(simplest,)f
4877
 
Fo(wisdom)f Fu(pro)m(vides)h(a)h(w)m(a)m(y)h(of)f(sa)m(ving)g(plans)e
 
4923
Fp(wisdom)f Fv(pro)m(vides)h(a)h(w)m(a)m(y)h(of)f(sa)m(ving)g(plans)e
4878
4924
(to)i(disk)f(so)h(that)g(they)g(can)h(b)s(e)e(reused)150
4879
4925
3329 y(in)38 b(other)i(program)g(runs.)67 b(Y)-8 b(ou)41
4880
 
b(create)g(a)f(plan)e(with)h(the)h(\015ags)g Fo(FFTW_MEASURE)c
4881
 
Fu(and)j Fo(FFTW_USE_)150 3439 y(WISDOM)p Fu(,)29 b(and)h(then)g(sa)m
4882
 
(v)m(e)i(the)e Fo(wisdom)f Fu(using)g Fo(fftw_export_wisdom)p
4883
 
Fu(:)629 3577 y Fo(plan)46 b(=)i(fftw_create_plan\(...,)42
 
4926
b(create)g(a)f(plan)e(with)h(the)h(\015ags)g Fp(FFTW_MEASURE)c
 
4927
Fv(and)j Fp(FFTW_USE_)150 3439 y(WISDOM)p Fv(,)29 b(and)h(then)g(sa)m
 
4928
(v)m(e)i(the)e Fp(wisdom)f Fv(using)g Fp(fftw_export_wisdom)p
 
4929
Fv(:)629 3577 y Fp(plan)46 b(=)i(fftw_create_plan\(...,)42
4884
4930
b(...)47 b(|)g(FFTW_MEASURE)e(|)i(FFTW_USE_WISDOM\);)629
4885
 
3681 y(fftw_export_wisdom\(...\))o(;)275 3824 y Fu(The)37
 
4931
3681 y(fftw_export_wisdom\(...\))o(;)275 3824 y Fv(The)37
4886
4932
b(next)h(time)g(y)m(ou)h(run)d(the)i(program,)i(y)m(ou)f(can)f(restore)
4887
 
h(the)f Fo(wisdom)f Fu(with)f Fo(fftw_import_)150 3934
4888
 
y(wisdom)p Fu(,)28 b(and)g(then)h(recreate)i(the)e(plan)f(using)f(the)i
 
4933
h(the)f Fp(wisdom)f Fv(with)f Fp(fftw_import_)150 3934
 
4934
y(wisdom)p Fv(,)28 b(and)g(then)h(recreate)i(the)e(plan)f(using)f(the)i
4889
4935
(same)h(\015ags)f(as)g(b)s(efore.)40 b(This)28 b(time,)h(ho)m(w)m(ev)m
4890
4936
(er,)i(the)150 4044 y(same)g(optimal)f(plan)g(will)e(b)s(e)j(created)h
4891
4937
(v)m(ery)f(quic)m(kly)f(without)g(measuremen)m(ts.)42
4892
4938
b(\(FFTW)32 b(still)d(needs)150 4153 y(some)i(time)f(to)h(compute)g
4893
4939
(trigonometric)f(tables,)g(ho)m(w)m(ev)m(er.\))43 b(The)30
4894
 
b(basic)f(outline)g(is:)629 4291 y Fo(fftw_import_wisdom\(...\))o(;)629
 
4940
b(basic)f(outline)g(is:)629 4291 y Fp(fftw_import_wisdom\(...\))o(;)629
4895
4941
4395 y(plan)46 b(=)i(fftw_create_plan\(...,)42 b(...)47
4896
 
b(|)g(FFTW_USE_WISDOM\);)275 4539 y Fu(Wisdom)30 b(is)h(more)h(than)f
 
4942
b(|)g(FFTW_USE_WISDOM\);)275 4539 y Fv(Wisdom)30 b(is)h(more)h(than)f
4897
4943
(mere)h(rote)g(memorization,)g(ho)m(w)m(ev)m(er.)46 b(FFTW's)33
4898
 
b Fo(wisdom)d Fu(encompasses)150 4648 y(all)i(of)g(the)h(kno)m(wledge)g
 
4944
b Fp(wisdom)d Fv(encompasses)150 4648 y(all)i(of)g(the)h(kno)m(wledge)g
4899
4945
(and)f(measuremen)m(ts)h(that)g(w)m(ere)g(used)f(to)h(create)i(the)d
4900
4946
(plan)g(for)g(a)h(giv)m(en)g(size.)150 4758 y(Therefore,)d(existing)g
4901
 
Fo(wisdom)e Fu(is)i(also)g(applied)e(to)j(the)g(creation)g(of)f(other)h
 
4947
Fp(wisdom)e Fv(is)i(also)g(applied)e(to)j(the)g(creation)g(of)f(other)h
4902
4948
(plans)e(of)h(di\013eren)m(t)g(sizes.)275 4902 y(Whenev)m(er)c(a)g
4903
 
(plan)f(is)g(created)i(with)e(the)h Fo(FFTW_MEASURE)c
4904
 
Fu(and)k Fo(FFTW_USE_WISDOM)21 b Fu(\015ags,)28 b Fo(wisdom)150
4905
 
5011 y Fu(is)j(generated.)46 b(Thereafter,)33 b(plans)d(for)i(an)m(y)g
 
4949
(plan)f(is)g(created)i(with)e(the)h Fp(FFTW_MEASURE)c
 
4950
Fv(and)k Fp(FFTW_USE_WISDOM)21 b Fv(\015ags,)28 b Fp(wisdom)150
 
4951
5011 y Fv(is)j(generated.)46 b(Thereafter,)33 b(plans)d(for)i(an)m(y)g
4906
4952
(transform)f(with)g(a)h(similar)d(factorization)k(will)c(b)s(e)i(com-)
4907
4953
150 5121 y(puted)f(more)i(quic)m(kly)-8 b(,)30 b(so)i(long)e(as)i(they)
4908
 
f(use)g(the)g Fo(FFTW_USE_WISDOM)c Fu(\015ag.)43 b(In)30
 
4954
f(use)g(the)g Fp(FFTW_USE_WISDOM)c Fv(\015ag.)43 b(In)30
4909
4955
b(fact,)j(for)e(transforms)150 5230 y(with)g(the)h(same)g(factors)h
4910
4956
(and)e(of)h(equal)g(or)g(lesser)f(size,)i(no)e(measuremen)m(ts)i(at)f
4911
4957
(all)f(need)h(to)h(b)s(e)e(made)150 5340 y(and)f(an)g(optimal)f(plan)g
4912
4958
(can)i(b)s(e)f(created)h(with)e(negligible)f(dela)m(y!)p
4913
 
eop
 
4959
eop end
4914
4960
%%Page: 14 16
4915
 
14 15 bop 150 -116 a Fu(14)3232 b(FFTW)275 299 y(F)-8
4916
 
b(or)26 b(example,)h(supp)s(ose)e(that)i(y)m(ou)f(create)i(a)f(plan)e
4917
 
(for)h Fl(N)35 b Fu(=)25 b(2)2427 266 y Fn(16)2498 299
4918
 
y Fu(.)39 b(Then,)26 b(for)g(an)m(y)h(equal)e(or)h(smaller)150
 
4961
TeXDict begin 14 15 bop 150 -116 a Fv(14)3232 b(FFTW)275
 
4962
299 y(F)-8 b(or)26 b(example,)h(supp)s(ose)e(that)i(y)m(ou)f(create)i
 
4963
(a)f(plan)e(for)h Fm(N)35 b Fv(=)25 b(2)2427 266 y Fo(16)2498
 
4964
299 y Fv(.)39 b(Then,)26 b(for)g(an)m(y)h(equal)e(or)h(smaller)150
4919
4965
408 y(p)s(o)m(w)m(er)e(of)f(t)m(w)m(o,)k(FFTW)d(can)g(create)h(a)f
4920
4966
(plan)e(\(with)h(the)g(same)h(direction)e(and)h(\015ags\))h(quic)m(kly)
4921
 
-8 b(,)25 b(using)d(the)150 518 y(precomputed)28 b Fo(wisdom)p
4922
 
Fu(.)39 b(Ev)m(en)29 b(for)f(larger)h(p)s(o)m(w)m(ers)g(of)g(t)m(w)m
 
4967
-8 b(,)25 b(using)d(the)150 518 y(precomputed)28 b Fp(wisdom)p
 
4968
Fv(.)39 b(Ev)m(en)29 b(for)f(larger)h(p)s(o)m(w)m(ers)g(of)g(t)m(w)m
4923
4969
(o,)i(or)e(sizes)f(that)i(are)f(a)g(p)s(o)m(w)m(er)g(of)g(t)m(w)m(o)h
4924
4970
(times)150 628 y(some)h(other)g(prime)f(factors,)i(plans)d(will)f(b)s
4925
4971
(e)i(computed)h(more)g(quic)m(kly)e(than)i(they)g(w)m(ould)f(otherwise)
4926
4972
150 737 y(\(although)g(some)h(measuremen)m(ts)g(still)d(ha)m(v)m(e)j
4927
 
(to)h(b)s(e)d(made\).)275 876 y(The)38 b Fo(wisdom)g
4928
 
Fu(is)g(cum)m(ulativ)m(e,)k(and)c(is)h(stored)g(in)f(a)i(global,)h
4929
 
(priv)-5 b(ate)39 b(data)h(structure)e(managed)150 986
 
4973
(to)h(b)s(e)d(made\).)275 879 y(The)38 b Fp(wisdom)g
 
4974
Fv(is)g(cum)m(ulativ)m(e,)k(and)c(is)h(stored)g(in)f(a)i(global,)h
 
4975
(priv)-5 b(ate)39 b(data)h(structure)e(managed)150 989
4930
4976
y(in)m(ternally)27 b(b)m(y)i(FFTW.)h(The)f(storage)i(space)e(required)f
4931
4977
(is)g(minimal,)f(prop)s(ortional)g(to)j(the)f(logarithm)150
4932
 
1096 y(of)23 b(the)g(sizes)g(the)g Fo(wisdom)e Fu(w)m(as)i(generated)h
4933
 
(from.)38 b(The)22 b Fo(wisdom)f Fu(can)i(b)s(e)f(forgotten)j(\(and)d
4934
 
(its)g(asso)s(ciated)150 1205 y(memory)33 b(freed\))f(b)m(y)h(a)g(call)
4935
 
f(to)h Fo(fftw_forget_wisdom\(\))p Fu(;)c(otherwise,)k(it)f(is)g(remem)
4936
 
m(b)s(ered)g(un)m(til)f(the)150 1315 y(program)f(terminates.)42
 
4978
1099 y(of)23 b(the)g(sizes)g(the)g Fp(wisdom)e Fv(w)m(as)i(generated)h
 
4979
(from.)38 b(The)22 b Fp(wisdom)f Fv(can)i(b)s(e)f(forgotten)j(\(and)d
 
4980
(its)g(asso)s(ciated)150 1208 y(memory)33 b(freed\))f(b)m(y)h(a)g(call)
 
4981
f(to)h Fp(fftw_forget_wisdom\(\))p Fv(;)c(otherwise,)k(it)f(is)g(remem)
 
4982
m(b)s(ered)g(un)m(til)f(the)150 1318 y(program)f(terminates.)42
4937
4983
b(It)31 b(can)g(also)f(b)s(e)g(exp)s(orted)h(to)g(a)g(\014le,)f(a)h
4938
 
(string,)f(or)h(an)m(y)g(other)g(medium)d(using)150 1424
4939
 
y Fo(fftw_export_wisdom)37 b Fu(and)42 b(restored)h(during)d(a)i
 
4984
(string,)f(or)h(an)m(y)g(other)g(medium)d(using)150 1427
 
4985
y Fp(fftw_export_wisdom)37 b Fv(and)42 b(restored)h(during)d(a)i
4940
4986
(subsequen)m(t)g(execution)h(of)f(the)h(program)f(\(or)h(a)150
4941
 
1534 y(di\013eren)m(t)30 b(program\))g(using)f Fo(fftw_import_wisdom)d
4942
 
Fu(\(these)31 b(functions)e(are)h(describ)s(ed)f(b)s(elo)m(w\).)275
4943
 
1673 y(Because)f Fo(wisdom)d Fu(is)h(incorp)s(orated)g(in)m(to)h(FFTW)h
 
4987
1537 y(di\013eren)m(t)30 b(program\))g(using)f Fp(fftw_import_wisdom)d
 
4988
Fv(\(these)31 b(functions)e(are)h(describ)s(ed)f(b)s(elo)m(w\).)275
 
4989
1679 y(Because)f Fp(wisdom)d Fv(is)h(incorp)s(orated)g(in)m(to)h(FFTW)h
4944
4990
(at)f(a)h(v)m(ery)f(lo)m(w)g(lev)m(el,)g(the)h(same)f
4945
 
Fo(wisdom)e Fu(can)j(b)s(e)150 1783 y(used)35 b(for)h(one-dimensional)e
 
4991
Fp(wisdom)e Fv(can)j(b)s(e)150 1789 y(used)35 b(for)h(one-dimensional)e
4946
4992
(transforms,)j(m)m(ulti-dimensional)32 b(transforms,)37
4947
 
b(and)f(ev)m(en)h(the)f(parallel)150 1892 y(extensions)h(to)h(FFTW.)g
4948
 
(Just)e(include)f Fo(FFTW_USE_WISDOM)e Fu(in)j(the)i(\015ags)f(for)g
4949
 
(whatev)m(er)h(plans)e(y)m(ou)150 2002 y(create)c(\(i.e.,)f(alw)m(a)m
4950
 
(ys)g(plan)e(wisely\).)275 2141 y(Plans)22 b(created)i(with)e(the)i
4951
 
Fo(FFTW_ESTIMATE)19 b Fu(plan)j(can)i(use)f Fo(wisdom)p
4952
 
Fu(,)g(but)g(cannot)h(generate)g(it;)i(only)150 2251
4953
 
y Fo(FFTW_MEASURE)j Fu(plans)h(actually)i(pro)s(duce)f
4954
 
Fo(wisdom)p Fu(.)44 b(Also,)33 b(plans)d(can)j(only)e(use)h
4955
 
Fo(wisdom)e Fu(generated)150 2360 y(from)22 b(plans)g(created)h(with)f
 
4993
b(and)f(ev)m(en)h(the)f(parallel)150 1898 y(extensions)h(to)h(FFTW.)g
 
4994
(Just)e(include)f Fp(FFTW_USE_WISDOM)e Fv(in)j(the)i(\015ags)f(for)g
 
4995
(whatev)m(er)h(plans)e(y)m(ou)150 2008 y(create)c(\(i.e.,)f(alw)m(a)m
 
4996
(ys)g(plan)e(wisely\).)275 2150 y(Plans)22 b(created)i(with)e(the)i
 
4997
Fp(FFTW_ESTIMATE)19 b Fv(plan)j(can)i(use)f Fp(wisdom)p
 
4998
Fv(,)g(but)g(cannot)h(generate)g(it;)i(only)150 2260
 
4999
y Fp(FFTW_MEASURE)j Fv(plans)h(actually)i(pro)s(duce)f
 
5000
Fp(wisdom)p Fv(.)44 b(Also,)33 b(plans)d(can)j(only)e(use)h
 
5001
Fp(wisdom)e Fv(generated)150 2369 y(from)22 b(plans)g(created)h(with)f
4956
5002
(the)h(same)g(direction)e(and)i(\015ags.)38 b(F)-8 b(or)23
4957
 
b(example,)i(a)e(size)f Fo(42)h(FFTW_BACKWARD)150 2470
4958
 
y Fu(transform)34 b(will)e(not)j(use)g Fo(wisdom)e Fu(pro)s(duced)g(b)m
4959
 
(y)i(a)g(size)g Fo(42)f(FFTW_FORWARD)d Fu(transform.)54
4960
 
b(The)34 b(only)150 2579 y(exception)40 b(to)h(this)e(rule)g(is)h(that)
4961
 
g Fo(FFTW_ESTIMATE)d Fu(plans)i(can)h(use)g Fo(wisdom)f
4962
 
Fu(from)g Fo(FFTW_MEASURE)150 2689 y Fu(plans.)150 2923
4963
 
y Fi(2.6.1)63 b(Ca)m(v)m(eats)39 b(in)i(Using)h(Wisdom)390
4964
 
3114 y Fu(F)-8 b(or)27 b(in)e(m)m(uc)m(h)h(wisdom)f(is)g(m)m(uc)m(h)i
 
5003
b(example,)i(a)e(size)f Fp(42)h(FFTW_BACKWARD)150 2479
 
5004
y Fv(transform)34 b(will)e(not)j(use)g Fp(wisdom)e Fv(pro)s(duced)g(b)m
 
5005
(y)i(a)g(size)g Fp(42)f(FFTW_FORWARD)d Fv(transform.)54
 
5006
b(The)34 b(only)150 2588 y(exception)40 b(to)h(this)e(rule)g(is)h(that)
 
5007
g Fp(FFTW_ESTIMATE)d Fv(plans)i(can)h(use)g Fp(wisdom)f
 
5008
Fv(from)g Fp(FFTW_MEASURE)150 2698 y Fv(plans.)150 2937
 
5009
y Fh(2.6.1)63 b(Ca)m(v)m(eats)39 b(in)i(Using)h(Wisdom)390
 
5010
3132 y Fv(F)-8 b(or)27 b(in)e(m)m(uc)m(h)h(wisdom)f(is)g(m)m(uc)m(h)i
4965
5011
(grief,)f(and)g(he)g(that)h(increaseth)f(kno)m(wledge)g(increaseth)390
4966
 
3218 y(sorro)m(w.)41 b([Ecclesiastes)31 b(1:18])275 3357
4967
 
y(There)37 b(are)i(pitfalls)d(to)i(using)f Fo(wisdom)p
4968
 
Fu(,)i(in)e(that)h(it)g(can)h(negate)g(FFTW's)g(abilit)m(y)e(to)i
4969
 
(adapt)f(to)150 3467 y(c)m(hanging)g(hardw)m(are)g(and)g(other)h
 
5012
3236 y(sorro)m(w.)41 b([Ecclesiastes)31 b(1:18])275 3378
 
5013
y(There)37 b(are)i(pitfalls)d(to)i(using)f Fp(wisdom)p
 
5014
Fv(,)i(in)e(that)h(it)g(can)h(negate)g(FFTW's)g(abilit)m(y)e(to)i
 
5015
(adapt)f(to)150 3487 y(c)m(hanging)g(hardw)m(are)g(and)g(other)h
4970
5016
(conditions.)63 b(F)-8 b(or)39 b(example,)h(it)e(w)m(ould)f(b)s(e)h(p)s
4971
 
(erfectly)f(p)s(ossible)f(to)150 3576 y(exp)s(ort)41
4972
 
b Fo(wisdom)e Fu(from)i(a)g(program)g(running)d(on)j(one)g(pro)s
 
5017
(erfectly)f(p)s(ossible)f(to)150 3597 y(exp)s(ort)41
 
5018
b Fp(wisdom)e Fv(from)i(a)g(program)g(running)d(on)j(one)g(pro)s
4973
5019
(cessor)g(and)f(imp)s(ort)f(it)i(in)m(to)g(a)g(program)150
4974
 
3686 y(running)35 b(on)i(another)h(pro)s(cessor.)61 b(Doing)38
 
5020
3706 y(running)35 b(on)i(another)h(pro)s(cessor.)61 b(Doing)38
4975
5021
b(so,)h(ho)m(w)m(ev)m(er,)i(w)m(ould)36 b(mean)i(that)g(the)g(second)f
4976
 
(program)150 3795 y(w)m(ould)29 b(use)h(plans)f(optimized)g(for)h(the)h
 
5022
(program)150 3816 y(w)m(ould)29 b(use)h(plans)f(optimized)g(for)h(the)h
4977
5023
(\014rst)f(pro)s(cessor,)g(instead)f(of)i(the)g(one)f(it)g(is)g
4978
 
(running)d(on.)275 3934 y(It)j(should)e(b)s(e)i(safe)g(to)h(reuse)f
4979
 
Fo(wisdom)f Fu(as)h(long)g(as)g(the)h(hardw)m(are)e(and)h(program)g
4980
 
(binaries)e(remain)150 4044 y(unc)m(hanged.)40 b(\(Actually)-8
 
5024
(running)d(on.)275 3958 y(It)j(should)e(b)s(e)i(safe)g(to)h(reuse)f
 
5025
Fp(wisdom)f Fv(as)h(long)g(as)g(the)h(hardw)m(are)e(and)h(program)g
 
5026
(binaries)e(remain)150 4068 y(unc)m(hanged.)40 b(\(Actually)-8
4981
5027
b(,)31 b(the)f(optimal)f(plan)g(ma)m(y)i(c)m(hange)g(ev)m(en)g(b)s(et)m
4982
 
(w)m(een)g(runs)d(of)j(the)f(same)h(binary)150 4154 y(on)22
 
5028
(w)m(een)g(runs)d(of)j(the)f(same)h(binary)150 4177 y(on)22
4983
5029
b(iden)m(tical)e(hardw)m(are,)k(due)d(to)i(di\013erences)e(in)f(the)i
4984
5030
(virtual)f(memory)g(en)m(vironmen)m(t,)j(etcetera.)40
4985
 
b(Users)150 4263 y(seriously)26 b(in)m(terested)i(in)e(p)s(erformance)h
 
5031
b(Users)150 4287 y(seriously)26 b(in)m(terested)i(in)e(p)s(erformance)h
4986
5032
(should)f(w)m(orry)i(ab)s(out)f(this)g(problem,)g(to)s(o.\))41
4987
 
b(It)28 b(is)f(lik)m(ely)f(that,)150 4373 y(if)35 b(the)i(same)g
4988
 
Fo(wisdom)e Fu(is)g(used)h(for)g(t)m(w)m(o)i(di\013eren)m(t)e(program)g
 
5033
b(It)28 b(is)f(lik)m(ely)f(that,)150 4397 y(if)35 b(the)i(same)g
 
5034
Fp(wisdom)e Fv(is)g(used)h(for)g(t)m(w)m(o)i(di\013eren)m(t)e(program)g
4989
5035
(binaries,)g(ev)m(en)h(running)d(on)i(the)h(same)150
4990
 
4482 y(mac)m(hine,)28 b(the)f(plans)f(ma)m(y)i(b)s(e)f(sub-optimal)e(b)
 
5036
4506 y(mac)m(hine,)28 b(the)f(plans)f(ma)m(y)i(b)s(e)f(sub-optimal)e(b)
4991
5037
s(ecause)j(of)f(di\013ering)e(co)s(de)j(alignmen)m(ts.)39
4992
 
b(It)27 b(is)f(therefore)150 4592 y(wise)k(to)i(recreate)g
4993
 
Fo(wisdom)d Fu(ev)m(ery)j(time)f(an)f(application)g(is)g(recompiled.)40
4994
 
b(The)31 b(more)g(the)g(underlying)150 4702 y(hardw)m(are)40
 
5038
b(It)27 b(is)f(therefore)150 4616 y(wise)k(to)i(recreate)g
 
5039
Fp(wisdom)d Fv(ev)m(ery)j(time)f(an)f(application)g(is)g(recompiled.)40
 
5040
b(The)31 b(more)g(the)g(underlying)150 4725 y(hardw)m(are)40
4995
5041
b(and)g(soft)m(w)m(are)i(c)m(hanges)g(b)s(et)m(w)m(een)f(the)g
4996
 
(creation)g(of)g Fo(wisdom)e Fu(and)h(its)g(use,)j(the)e(greater)150
4997
 
4811 y(gro)m(ws)31 b(the)f(risk)f(of)i(sub-optimal)d(plans.)150
4998
 
5045 y Fi(2.6.2)63 b(Imp)s(orting)41 b(and)g(Exp)s(orting)h(Wisdom)390
4999
 
5236 y Fo(void)47 b(fftw_export_wisdom_to_f)o(ile\()o(FIL)o(E)42
 
5042
(creation)g(of)g Fp(wisdom)e Fv(and)h(its)g(use,)j(the)e(greater)150
 
5043
4835 y(gro)m(ws)31 b(the)f(risk)f(of)i(sub-optimal)d(plans.)150
 
5044
5074 y Fh(2.6.2)63 b(Imp)s(orting)41 b(and)g(Exp)s(orting)h(Wisdom)390
 
5045
5236 y Fp(void)47 b(fftw_export_wisdom_to_f)o(ile\()o(FIL)o(E)42
5000
5046
b(*output_file\);)390 5340 y(fftw_status)j(fftw_import_wisdom_from)o
5001
 
(_fil)o(e\(FI)o(LE)c(*input_file\);)p eop
 
5047
(_fil)o(e\(FI)o(LE)c(*input_file\);)p eop end
5002
5048
%%Page: 15 17
5003
 
15 16 bop 150 -116 a Fu(Chapter)30 b(2:)41 b(T)-8 b(utorial)2722
5004
 
b(15)275 299 y Fo(fftw_export_wisdom_to_f)o(ile)21 b
5005
 
Fu(writes)26 b(the)i Fo(wisdom)d Fu(to)j Fo(output_file)p
5006
 
Fu(,)d(whic)m(h)h(m)m(ust)h(b)s(e)g(a)h(\014le)150 408
5007
 
y(op)s(en)39 b(for)h(writing.)68 b Fo(fftw_import_wisdom_from_f)o(ile)
5008
 
33 b Fu(reads)40 b(the)h Fo(wisdom)d Fu(from)h Fo(input_file)p
5009
 
Fu(,)150 518 y(whic)m(h)33 b(m)m(ust)h(b)s(e)g(a)g(\014le)g(op)s(en)f
5010
 
(for)h(reading,)h(and)f(returns)f Fo(FFTW_SUCCESS)d Fu(if)k(successful)
5011
 
f(and)g Fo(FFTW_)150 628 y(FAILURE)c Fu(otherwise.)43
 
5049
TeXDict begin 15 16 bop 150 -116 a Fv(Chapter)30 b(2:)41
 
5050
b(T)-8 b(utorial)2722 b(15)275 299 y Fp(fftw_export_wisdom_to_f)o(ile)
 
5051
21 b Fv(writes)26 b(the)i Fp(wisdom)d Fv(to)j Fp(output_file)p
 
5052
Fv(,)d(whic)m(h)h(m)m(ust)h(b)s(e)g(a)h(\014le)150 408
 
5053
y(op)s(en)39 b(for)h(writing.)68 b Fp(fftw_import_wisdom_from_f)o(ile)
 
5054
33 b Fv(reads)40 b(the)h Fp(wisdom)d Fv(from)h Fp(input_file)p
 
5055
Fv(,)150 518 y(whic)m(h)33 b(m)m(ust)h(b)s(e)g(a)g(\014le)g(op)s(en)f
 
5056
(for)h(reading,)h(and)f(returns)f Fp(FFTW_SUCCESS)d Fv(if)k(successful)
 
5057
f(and)g Fp(FFTW_)150 628 y(FAILURE)c Fv(otherwise.)43
5012
5058
b(In)30 b(b)s(oth)h(cases,)h(the)g(\014le)e(is)g(left)h(op)s(en)g(and)f
5013
5059
(m)m(ust)h(b)s(e)g(closed)g(b)m(y)g(the)g(caller.)43
5014
5060
b(It)150 737 y(is)30 b(p)s(erfectly)h(\014ne)g(if)f(other)i(data)g(lie)
5015
 
e(b)s(efore)h(or)g(after)h(the)g Fo(wisdom)d Fu(in)h(the)i(\014le,)f
 
5061
e(b)s(efore)h(or)g(after)h(the)g Fp(wisdom)d Fv(in)h(the)i(\014le,)f
5016
5062
(as)g(long)g(as)h(the)f(\014le)g(is)150 847 y(p)s(ositioned)d(at)j(the)
5017
 
g(b)s(eginning)d(of)i(the)h Fo(wisdom)d Fu(data)k(b)s(efore)e(imp)s
5018
 
(ort.)390 975 y Fo(char)47 b(*fftw_export_wisdom_to_)o(stri)o(ng\()o
 
5063
g(b)s(eginning)d(of)i(the)h Fp(wisdom)d Fv(data)k(b)s(efore)e(imp)s
 
5064
(ort.)390 975 y Fp(char)47 b(*fftw_export_wisdom_to_)o(stri)o(ng\()o
5019
5065
(void)o(\);)390 1079 y(fftw_status)e(fftw_import_wisdom_from)o(_str)o
5020
5066
(ing\()o(con)o(st)d(char)k(*input_string\))275 1212 y
5021
 
(fftw_export_wisdom_to_s)o(tri)o(ng)29 b Fu(allo)s(cates)36
5022
 
b(a)g(string,)f(exp)s(orts)g(the)h Fo(wisdom)e Fu(to)i(it)f(in)f
5023
 
Fo(NULL)p Fu(-)150 1322 y(terminated)j(format,)i(and)e(returns)e(a)j(p)
 
5067
(fftw_export_wisdom_to_s)o(tri)o(ng)29 b Fv(allo)s(cates)36
 
5068
b(a)g(string,)f(exp)s(orts)g(the)h Fp(wisdom)e Fv(to)i(it)f(in)f
 
5069
Fp(NULL)p Fv(-)150 1322 y(terminated)j(format,)i(and)e(returns)e(a)j(p)
5024
5070
s(oin)m(ter)e(to)i(the)f(string.)60 b(If)37 b(there)g(is)f(an)h(error)g
5025
5071
(in)f(allo)s(cating)150 1432 y(or)h(writing)e(the)i(data,)i(it)d
5026
 
(returns)g Fo(NULL)p Fu(.)58 b(The)37 b(caller)f(is)g(resp)s(onsible)d
 
5072
(returns)g Fp(NULL)p Fv(.)58 b(The)37 b(caller)f(is)g(resp)s(onsible)d
5027
5073
(for)k(deallo)s(cating)f(the)h(string)150 1541 y(\(with)e
5028
 
Fo(fftw_free)p Fu(\))e(when)i(she)g(is)g(done)g(with)f(it.)56
5029
 
b Fo(fftw_import_wisdom_from_st)o(rin)o(g)30 b Fu(imp)s(orts)150
5030
 
1651 y(the)k Fo(wisdom)f Fu(from)h Fo(input_string)p
5031
 
Fu(,)e(returning)g Fo(FFTW_SUCCESS)f Fu(if)i(successful)g(and)g
5032
 
Fo(FFTW_FAILURE)150 1760 y Fu(otherwise.)275 1894 y(Exp)s(orting)h
5033
 
Fo(wisdom)h Fu(do)s(es)h(not)g(a\013ect)i(the)f(store)g(of)f
5034
 
Fo(wisdom)p Fu(.)57 b(Imp)s(orted)35 b Fo(wisdom)g Fu(supplemen)m(ts)
 
5074
Fp(fftw_free)p Fv(\))e(when)i(she)g(is)g(done)g(with)f(it.)56
 
5075
b Fp(fftw_import_wisdom_from_st)o(rin)o(g)30 b Fv(imp)s(orts)150
 
5076
1651 y(the)k Fp(wisdom)f Fv(from)h Fp(input_string)p
 
5077
Fv(,)e(returning)g Fp(FFTW_SUCCESS)f Fv(if)i(successful)g(and)g
 
5078
Fp(FFTW_FAILURE)150 1760 y Fv(otherwise.)275 1894 y(Exp)s(orting)h
 
5079
Fp(wisdom)h Fv(do)s(es)h(not)g(a\013ect)i(the)f(store)g(of)f
 
5080
Fp(wisdom)p Fv(.)57 b(Imp)s(orted)35 b Fp(wisdom)g Fv(supplemen)m(ts)
5035
5081
150 2004 y(the)26 b(curren)m(t)g(store)h(rather)e(than)h(replacing)f
5036
5082
(it)g(\(except)j(when)d(there)h(is)f(con\015icting)g
5037
 
Fo(wisdom)p Fu(,)h(in)e(whic)m(h)150 2113 y(case)30 b(the)e(older)g
5038
 
Fo(wisdom)f Fu(is)h(discarded\).)39 b(The)28 b(format)h(of)g(the)f(exp)
5039
 
s(orted)h Fo(wisdom)e Fu(is)g(\\nerd-readable")150 2223
 
5083
Fp(wisdom)p Fv(,)h(in)e(whic)m(h)150 2113 y(case)30 b(the)e(older)g
 
5084
Fp(wisdom)f Fv(is)h(discarded\).)39 b(The)28 b(format)h(of)g(the)f(exp)
 
5085
s(orted)h Fp(wisdom)e Fv(is)g(\\nerd-readable")150 2223
5040
5086
y(LISP-lik)m(e)k(ASCI)s(I)f(text;)35 b(w)m(e)d(will)e(not)i(do)s(cumen)
5041
5087
m(t)g(it)g(here)g(except)h(to)g(note)g(that)g(it)f(is)f(insensitiv)m(e)
5042
5088
f(to)150 2333 y(white)f(space)i(\(in)m(terested)g(users)f(can)g(con)m
5043
5089
(tact)j(us)d(for)g(more)g(details\).)275 2466 y(See)h(Chapter)g(3)h
5044
5090
([FFTW)h(Reference],)g(page)g(17,)g(for)e(more)h(information,)e(and)h
5045
5091
(for)h(a)g(description)150 2576 y(of)e(ho)m(w)g(y)m(ou)h(can)f
5046
 
(implemen)m(t)f Fo(wisdom)f Fu(imp)s(ort/exp)s(ort)h(for)h(other)g
 
5092
(implemen)m(t)f Fp(wisdom)f Fv(imp)s(ort/exp)s(ort)h(for)h(other)g
5047
5093
(media)g(b)s(esides)e(\014les)h(and)h(strings.)275 2710
5048
5094
y(The)25 b(follo)m(wing)f(is)h(a)h(brief)e(example)i(in)e(whic)m(h)h
5049
 
(the)h Fo(wisdom)e Fu(is)h(read)h(from)f(a)h(\014le,)g(a)h(plan)d(is)h
 
5095
(the)h Fp(wisdom)e Fv(is)h(read)h(from)f(a)h(\014le,)g(a)h(plan)d(is)h
5050
5096
(created)150 2819 y(\(p)s(ossibly)e(generating)i(more)h
5051
 
Fo(wisdom)p Fu(\),)f(and)g(then)g(the)h Fo(wisdom)d Fu(is)i(exp)s
 
5097
Fp(wisdom)p Fv(\),)f(and)g(then)g(the)h Fp(wisdom)d Fv(is)i(exp)s
5052
5098
(orted)g(to)h(a)f(string)g(and)f(prin)m(ted)150 2929
5053
 
y(to)31 b Fo(stdout)p Fu(.)390 3057 y Fo({)629 3161 y(fftw_plan)45
 
5099
y(to)31 b Fp(stdout)p Fv(.)390 3057 y Fp({)629 3161 y(fftw_plan)45
5054
5100
b(plan;)629 3264 y(char)h(*wisdom_string;)629 3368 y(FILE)g
5055
5101
(*input_file;)629 3576 y(/*)h(open)f(file)h(to)g(read)g(wisdom)f(from)h
5056
5102
(*/)629 3680 y(input_file)e(=)i(fopen\("sample.wisdom",)42
5067
5113
4925 y(fftw_destroy_plan\(plan\))o(;)629 5132 y(/*)h(write)f(the)h
5068
5114
(wisdom)f(to)h(a)h(string)e(*/)629 5236 y(wisdom_string)e(=)j
5069
5115
(fftw_export_wisdom_to_str)o(ing\()o(\);)629 5340 y(if)g
5070
 
(\(wisdom_string)d(!=)j(NULL\))f({)p eop
 
5116
(\(wisdom_string)d(!=)j(NULL\))f({)p eop end
5071
5117
%%Page: 16 18
5072
 
16 17 bop 150 -116 a Fu(16)3232 b(FFTW)867 299 y Fo
5073
 
(printf\("Accumulated)43 b(wisdom:)j(\045s\\n",wisdom_string\);)867
5074
 
506 y(/*)h(Just)g(for)g(fun,)g(destroy)e(and)i(restore)f(the)h(wisdom)f
5075
 
(*/)867 610 y(fftw_forget_wisdom\(\);)c(/*)47 b(all)g(gone!)g(*/)867
5076
 
714 y(fftw_import_wisdom_from_s)o(trin)o(g\(wi)o(sdo)o(m_st)o(ring)o
5077
 
(\);)867 818 y(/*)g(wisdom)g(is)g(back!)f(*/)867 1025
5078
 
y(fftw_free\(wisdom_string\);)41 b(/*)47 b(deallocate)e(it)j(since)e
5079
 
(we're)g(done)h(*/)629 1129 y(})390 1233 y(})p eop
 
5118
TeXDict begin 16 17 bop 150 -116 a Fv(16)3232 b(FFTW)867
 
5119
299 y Fp(printf\("Accumulated)43 b(wisdom:)j
 
5120
(\045s\\n",wisdom_string\);)867 506 y(/*)h(Just)g(for)g(fun,)g(destroy)
 
5121
e(and)i(restore)f(the)h(wisdom)f(*/)867 610 y(fftw_forget_wisdom\(\);)c
 
5122
(/*)47 b(all)g(gone!)g(*/)867 714 y(fftw_import_wisdom_from_s)o(trin)o
 
5123
(g\(wi)o(sdo)o(m_st)o(ring)o(\);)867 818 y(/*)g(wisdom)g(is)g(back!)f
 
5124
(*/)867 1025 y(fftw_free\(wisdom_string\);)41 b(/*)47
 
5125
b(deallocate)e(it)j(since)e(we're)g(done)h(*/)629 1129
 
5126
y(})390 1233 y(})p eop end
5080
5127
%%Page: 17 19
5081
 
17 18 bop 150 -116 a Fu(Chapter)30 b(3:)41 b(FFTW)31
5082
 
b(Reference)2354 b(17)150 299 y Fr(3)80 b(FFTW)55 b(Reference)275
5083
 
526 y Fu(This)25 b(c)m(hapter)j(pro)m(vides)e(a)i(complete)f(reference)
5084
 
h(for)f(all)f(sequen)m(tial)h(\(i.e.,)h(one-pro)s(cessor\))g(FFTW)150
5085
 
636 y(functions.)52 b(W)-8 b(e)35 b(\014rst)f(de\014ne)g(the)g(data)i
5086
 
(t)m(yp)s(es)e(up)s(on)f(whic)m(h)g(FFTW)i(op)s(erates,)i(that)e(is,)g
5087
 
(real,)g(com-)150 746 y(plex,)24 b(and)f(\\halfcomplex")g(n)m(um)m(b)s
5088
 
(ers)f(\(see)i(Section)g(3.1)g([Data)i(T)m(yp)s(es],)e(page)g(17\).)40
 
5128
TeXDict begin 17 18 bop 150 -116 a Fv(Chapter)30 b(3:)41
 
5129
b(FFTW)31 b(Reference)2354 b(17)150 299 y Fs(3)80 b(FFTW)55
 
5130
b(Reference)275 526 y Fv(This)25 b(c)m(hapter)j(pro)m(vides)e(a)i
 
5131
(complete)f(reference)h(for)f(all)f(sequen)m(tial)h(\(i.e.,)h(one-pro)s
 
5132
(cessor\))g(FFTW)150 636 y(functions.)52 b(W)-8 b(e)35
 
5133
b(\014rst)f(de\014ne)g(the)g(data)i(t)m(yp)s(es)e(up)s(on)f(whic)m(h)g
 
5134
(FFTW)i(op)s(erates,)i(that)e(is,)g(real,)g(com-)150
 
5135
746 y(plex,)24 b(and)f(\\halfcomplex")g(n)m(um)m(b)s(ers)f(\(see)i
 
5136
(Section)g(3.1)g([Data)i(T)m(yp)s(es],)e(page)g(17\).)40
5089
5137
b(Then,)24 b(in)e(four)h(sec-)150 855 y(tions,)i(w)m(e)g(explain)e(the)
5090
5138
i(FFTW)g(program)f(in)m(terface)h(for)g(complex)f(one-dimensional)e
5091
5139
(transforms)i(\(see)150 965 y(Section)32 b(3.2)i([One-dimensional)c(T)
5098
5146
(ulti-dimensional)c(transforms)k(\(see)h(Section)e(3.5)j([Real)e
5099
5147
(Multi-dimensional)c(T)-8 b(rans-)150 1403 y(forms)30
5100
5148
b(Reference],)j(page)e(29\).)44 b(Section)30 b(3.6)i([Wisdom)f
5101
 
(Reference],)h(page)g(34)f(describ)s(es)f(the)h Fo(wisdom)150
5102
 
1513 y Fu(mec)m(hanism)c(for)g(exp)s(orting)g(and)g(imp)s(orting)e
 
5149
(Reference],)h(page)g(34)f(describ)s(es)f(the)h Fp(wisdom)150
 
5150
1513 y Fv(mec)m(hanism)c(for)g(exp)s(orting)g(and)g(imp)s(orting)e
5103
5151
(plans.)38 b(Finally)-8 b(,)27 b(Section)g(3.7)i([Memory)f(Allo)s
5104
 
(cator)g(Ref-)150 1622 y(erence],)36 b(page)f(36)g(describ)s(es)e(ho)m
 
5152
(cator)g(Ref-)150 1622 y(erence],)36 b(page)f(35)g(describ)s(es)e(ho)m
5105
5153
(w)h(to)h(c)m(hange)h(FFTW's)f(default)e(memory)h(allo)s(cator.)53
5106
5154
b(F)-8 b(or)35 b(parallel)150 1732 y(transforms,)30 b(See)g(Chapter)g
5107
 
(4)h([P)m(arallel)f(FFTW],)h(page)g(37.)150 1985 y Ft(3.1)68
5108
 
b(Data)46 b(T)l(yp)t(es)275 2177 y Fu(The)e(routines)f(in)g(the)i(FFTW)
 
5155
(4)h([P)m(arallel)f(FFTW],)h(page)g(37.)150 1985 y Fu(3.1)68
 
5156
b(Data)46 b(T)l(yp)t(es)275 2177 y Fv(The)e(routines)f(in)g(the)i(FFTW)
5109
5157
h(pac)m(k)-5 b(age)46 b(use)f(three)f(main)g(kinds)f(of)h(data)i(t)m
5110
 
(yp)s(es.)83 b Fp(Real)49 b Fu(and)150 2286 y Fp(complex)31
5111
 
b Fu(n)m(um)m(b)s(ers)23 b(should)f(b)s(e)j(already)f(kno)m(wn)g(to)h
 
5158
(yp)s(es.)83 b Fq(Real)49 b Fv(and)150 2286 y Fq(complex)31
 
5159
b Fv(n)m(um)m(b)s(ers)23 b(should)f(b)s(e)j(already)f(kno)m(wn)g(to)h
5112
5160
(the)g(reader.)39 b(W)-8 b(e)26 b(also)e(use)h(the)g(term)f
5113
 
Fp(halfcomplex)150 2396 y Fu(to)41 b(describ)s(e)e(complex)h(arra)m(ys)
 
5161
Fq(halfcomplex)150 2396 y Fv(to)41 b(describ)s(e)e(complex)h(arra)m(ys)
5114
5162
g(in)f(a)i(sp)s(ecial)e(pac)m(k)m(ed)i(format)g(used)f(b)m(y)g(the)g
5115
5163
(one-dimensional)e(real)150 2505 y(transforms)30 b(\(taking)g(adv)-5
5116
 
b(an)m(tage)32 b(of)f(the)g Fp(hermitian)d Fu(symmetry)i(that)h(arises)
5117
 
f(in)f(those)i(cases\).)275 2638 y(By)f(including)d Fo(<fftw.h>)h
5118
 
Fu(or)j Fo(<rfftw.h>)p Fu(,)d(y)m(ou)i(will)e(ha)m(v)m(e)k(access)f(to)
5119
 
h(the)e(follo)m(wing)f(de\014nitions:)390 2766 y Fo(typedef)46
 
5164
b(an)m(tage)32 b(of)f(the)g Fq(hermitian)d Fv(symmetry)i(that)h(arises)
 
5165
f(in)f(those)i(cases\).)275 2638 y(By)f(including)d Fp(<fftw.h>)h
 
5166
Fv(or)j Fp(<rfftw.h>)p Fv(,)d(y)m(ou)i(will)e(ha)m(v)m(e)k(access)f(to)
 
5167
h(the)e(follo)m(wing)f(de\014nitions:)390 2766 y Fp(typedef)46
5120
5168
b(double)g(fftw_real;)390 2973 y(typedef)g(struct)g({)629
5121
5169
3077 y(fftw_real)f(re,)i(im;)390 3181 y(})g(fftw_complex;)390
5122
5170
3388 y(#define)f(c_re\(c\))93 b(\(\(c\).re\))390 3492
5123
5171
y(#define)46 b(c_im\(c\))93 b(\(\(c\).im\))275 3625 y
5124
 
Fu(All)31 b(FFTW)i(op)s(erations)f(are)h(p)s(erformed)e(on)h(the)h
5125
 
Fo(fftw_real)d Fu(and)i Fo(fftw_complex)d Fu(data)34
5126
 
b(t)m(yp)s(es.)150 3735 y(F)-8 b(or)32 b Fo(fftw_complex)c
5127
 
Fu(n)m(um)m(b)s(ers,)j(the)h(t)m(w)m(o)h(macros)f Fo(c_re)e
5128
 
Fu(and)h Fo(c_im)f Fu(retriev)m(e,)j(resp)s(ectiv)m(ely)-8
 
5172
Fv(All)31 b(FFTW)i(op)s(erations)f(are)h(p)s(erformed)e(on)h(the)h
 
5173
Fp(fftw_real)d Fv(and)i Fp(fftw_complex)d Fv(data)34
 
5174
b(t)m(yp)s(es.)150 3735 y(F)-8 b(or)32 b Fp(fftw_complex)c
 
5175
Fv(n)m(um)m(b)s(ers,)j(the)h(t)m(w)m(o)h(macros)f Fp(c_re)e
 
5176
Fv(and)h Fp(c_im)f Fv(retriev)m(e,)j(resp)s(ectiv)m(ely)-8
5129
5177
b(,)32 b(the)f(real)150 3845 y(and)f(imaginary)f(parts)h(of)g(the)h(n)m
5130
 
(um)m(b)s(er.)275 3978 y(A)43 b Fp(real)g(arra)m(y)51
5131
 
b Fu(is)42 b(an)h(arra)m(y)h(of)g(real)e(n)m(um)m(b)s(ers.)78
5132
 
b(A)43 b Fp(complex)g(arra)m(y)52 b Fu(is)42 b(an)h(arra)m(y)h(of)f
 
5178
(um)m(b)s(er.)275 3978 y(A)43 b Fq(real)g(arra)m(y)51
 
5179
b Fv(is)42 b(an)h(arra)m(y)h(of)g(real)e(n)m(um)m(b)s(ers.)78
 
5180
b(A)43 b Fq(complex)g(arra)m(y)52 b Fv(is)42 b(an)h(arra)m(y)h(of)f
5133
5181
(complex)150 4087 y(n)m(um)m(b)s(ers.)53 b(A)36 b(one-dimensional)d
5134
 
(arra)m(y)i Fl(X)43 b Fu(of)35 b Fl(n)f Fu(complex)h(n)m(um)m(b)s(ers)f
5135
 
(is)g Fp(hermitian)f Fu(if)h(the)h(follo)m(wing)150 4197
 
5182
(arra)m(y)i Fm(X)43 b Fv(of)35 b Fm(n)f Fv(complex)h(n)m(um)m(b)s(ers)f
 
5183
(is)g Fq(hermitian)f Fv(if)h(the)h(follo)m(wing)150 4197
5136
5184
y(prop)s(ert)m(y)40 b(holds:)60 b(for)41 b(all)f(0)j
5137
 
Fs(\024)g Fl(i)g(<)f(n)p Fu(,)h(w)m(e)f(ha)m(v)m(e)g
5138
 
Fl(X)2065 4211 y Fk(i)2136 4197 y Fu(=)g Fl(X)2331 4164
5139
 
y Fj(\003)2324 4219 y Fk(n)p Fj(\000)p Fk(i)2445 4197
5140
 
y Fu(,)i(where)c Fl(x)2839 4164 y Fj(\003)2918 4197 y
5141
 
Fu(denotes)h(the)g(complex)150 4307 y(conjugate)30 b(of)e
5142
 
Fl(x)p Fu(.)40 b(Hermitian)27 b(arra)m(ys)i(are)f(relev)-5
 
5185
Ft(\024)g Fm(i)g(<)f(n)p Fv(,)h(w)m(e)f(ha)m(v)m(e)g
 
5186
Fm(X)2065 4211 y Fl(i)2136 4197 y Fv(=)g Fm(X)2331 4164
 
5187
y Fk(\003)2324 4219 y Fl(n)p Fk(\000)p Fl(i)2445 4197
 
5188
y Fv(,)i(where)c Fm(x)2839 4164 y Fk(\003)2918 4197 y
 
5189
Fv(denotes)h(the)g(complex)150 4307 y(conjugate)30 b(of)e
 
5190
Fm(x)p Fv(.)40 b(Hermitian)27 b(arra)m(ys)i(are)f(relev)-5
5143
5191
b(an)m(t)29 b(to)g(FFTW)g(b)s(ecause)f(the)h(F)-8 b(ourier)27
5144
5192
b(transform)h(of)h(a)150 4416 y(real)h(arra)m(y)h(is)e(hermitian.)275
5145
5193
4549 y(Because)h(of)e(its)g(symmetry)-8 b(,)30 b(a)f(hermitian)d(arra)m
5146
5194
(y)k(can)f(b)s(e)f(stored)g(in)g(half)f(the)i(space)g(of)g(a)g(complex)
5147
5195
150 4659 y(arra)m(y)i(of)g(the)h(same)f(size.)42 b(FFTW's)32
5148
5196
b(one-dimensional)c(real)j(transforms)f(store)i(hermitian)c(arra)m(ys)k
5149
 
(as)150 4768 y Fp(halfcomplex)40 b Fu(arra)m(ys.)56 b(A)35
5150
 
b(halfcomplex)f(arra)m(y)i(of)g(size)f Fl(n)f Fu(is)h(a)g
5151
 
(one-dimensional)e(arra)m(y)j(of)g Fl(n)e Fo(fftw_)150
5152
 
4878 y(real)e Fu(n)m(um)m(b)s(ers.)47 b(A)33 b(hermitian)e(arra)m(y)i
5153
 
Fl(X)41 b Fu(in)31 b(stored)i(in)m(to)g(a)h(halfcomplex)d(arra)m(y)j
5154
 
Fl(Y)52 b Fu(as)34 b(follo)m(ws.)47 b(F)-8 b(or)150 4988
5155
 
y(all)36 b(in)m(tegers)h Fl(i)h Fu(suc)m(h)f(that)g(0)g
5156
 
Fs(\024)f Fl(i)h Fs(\024)f Fl(n=)p Fu(2,)k(w)m(e)d(ha)m(v)m(e)i
5157
 
Fl(Y)2094 5002 y Fk(i)2158 4988 y Fu(:=)d(Re\()p Fl(X)2507
5158
 
5002 y Fk(i)2536 4988 y Fu(\).)61 b(F)-8 b(or)38 b(all)e(in)m(tegers)h
5159
 
Fl(i)h Fu(suc)m(h)e(that)150 5097 y(0)26 b Fl(<)f(i)g(<)g(n=)p
5160
 
Fu(2,)31 b(w)m(e)g(ha)m(v)m(e)h Fl(Y)1067 5111 y Fk(n)p
5161
 
Fj(\000)p Fk(i)1212 5097 y Fu(:=)25 b(Im\()p Fl(X)1552
5162
 
5111 y Fk(i)1580 5097 y Fu(\).)275 5230 y(W)-8 b(e)27
 
5197
(as)150 4768 y Fq(halfcomplex)40 b Fv(arra)m(ys.)56 b(A)35
 
5198
b(halfcomplex)f(arra)m(y)i(of)g(size)f Fm(n)f Fv(is)h(a)g
 
5199
(one-dimensional)e(arra)m(y)j(of)g Fm(n)e Fp(fftw_)150
 
5200
4878 y(real)e Fv(n)m(um)m(b)s(ers.)47 b(A)33 b(hermitian)e(arra)m(y)i
 
5201
Fm(X)41 b Fv(in)31 b(stored)i(in)m(to)g(a)h(halfcomplex)d(arra)m(y)j
 
5202
Fm(Y)52 b Fv(as)34 b(follo)m(ws.)47 b(F)-8 b(or)150 4988
 
5203
y(all)36 b(in)m(tegers)h Fm(i)h Fv(suc)m(h)f(that)g(0)g
 
5204
Ft(\024)f Fm(i)h Ft(\024)f Fm(n=)p Fv(2,)k(w)m(e)d(ha)m(v)m(e)i
 
5205
Fm(Y)2094 5002 y Fl(i)2158 4988 y Fv(:=)d(Re\()p Fm(X)2507
 
5206
5002 y Fl(i)2536 4988 y Fv(\).)61 b(F)-8 b(or)38 b(all)e(in)m(tegers)h
 
5207
Fm(i)h Fv(suc)m(h)e(that)150 5097 y(0)26 b Fm(<)f(i)g(<)g(n=)p
 
5208
Fv(2,)31 b(w)m(e)g(ha)m(v)m(e)h Fm(Y)1067 5111 y Fl(n)p
 
5209
Fk(\000)p Fl(i)1212 5097 y Fv(:=)25 b(Im\()p Fm(X)1552
 
5210
5111 y Fl(i)1580 5097 y Fv(\).)275 5230 y(W)-8 b(e)27
5163
5211
b(no)m(w)f(illustrate)e(halfcomplex)h(storage)i(for)f
5164
 
Fl(n)f Fu(=)g(4)h(and)f Fl(n)g Fu(=)g(5,)j(since)d(the)h(sc)m(heme)h
5165
 
(dep)s(ends)d(on)150 5340 y(the)g(parit)m(y)g(of)h Fl(n)p
5166
 
Fu(.)38 b(Let)25 b Fl(n)f Fu(=)h(4.)39 b(In)24 b(this)f(case,)k(w)m(e)e
5167
 
(ha)m(v)m(e)g Fl(Y)2095 5354 y Fn(0)2157 5340 y Fu(:=)h(Re\()p
5168
 
Fl(X)2496 5354 y Fn(0)2534 5340 y Fu(\),)g Fl(Y)2673
5169
 
5354 y Fn(1)2735 5340 y Fu(:=)f(Re)q(\()p Fl(X)3074 5354
5170
 
y Fn(1)3112 5340 y Fu(\),)h Fl(Y)3251 5354 y Fn(2)3313
5171
 
5340 y Fu(:=)f(Re\()p Fl(X)3651 5354 y Fn(2)3689 5340
5172
 
y Fu(\),)p eop
 
5212
Fm(n)f Fv(=)g(4)h(and)f Fm(n)g Fv(=)g(5,)j(since)d(the)h(sc)m(heme)h
 
5213
(dep)s(ends)d(on)150 5340 y(the)g(parit)m(y)g(of)h Fm(n)p
 
5214
Fv(.)38 b(Let)25 b Fm(n)f Fv(=)h(4.)39 b(In)24 b(this)f(case,)k(w)m(e)e
 
5215
(ha)m(v)m(e)g Fm(Y)2095 5354 y Fo(0)2157 5340 y Fv(:=)h(Re\()p
 
5216
Fm(X)2496 5354 y Fo(0)2534 5340 y Fv(\),)g Fm(Y)2673
 
5217
5354 y Fo(1)2735 5340 y Fv(:=)f(Re)q(\()p Fm(X)3074 5354
 
5218
y Fo(1)3112 5340 y Fv(\),)h Fm(Y)3251 5354 y Fo(2)3313
 
5219
5340 y Fv(:=)f(Re\()p Fm(X)3651 5354 y Fo(2)3689 5340
 
5220
y Fv(\),)p eop end
5173
5221
%%Page: 18 20
5174
 
18 19 bop 150 -116 a Fu(18)3232 b(FFTW)150 299 y(and)37
5175
 
b Fl(Y)387 313 y Fn(3)461 299 y Fu(:=)g(Im)o(\()p Fl(X)812
5176
 
313 y Fn(1)850 299 y Fu(\).)62 b(Let)38 b(no)m(w)g Fl(n)e
5177
 
Fu(=)h(5.)62 b(In)37 b(this)f(case,)41 b(w)m(e)d(ha)m(v)m(e)g
5178
 
Fl(Y)2612 313 y Fn(0)2686 299 y Fu(:=)f(Re)q(\()p Fl(X)3037
5179
 
313 y Fn(0)3074 299 y Fu(\),)j Fl(Y)3227 313 y Fn(1)3301
5180
 
299 y Fu(:=)d(Re\()p Fl(X)3651 313 y Fn(1)3689 299 y
5181
 
Fu(\),)150 408 y Fl(Y)203 422 y Fn(2)265 408 y Fu(:=)25
5182
 
b(Re)q(\()p Fl(X)604 422 y Fn(2)642 408 y Fu(\),)31 b
5183
 
Fl(Y)786 422 y Fn(3)848 408 y Fu(:=)25 b(Im\()p Fl(X)1188
5184
 
422 y Fn(2)1226 408 y Fu(\),)31 b(and)e Fl(Y)1546 422
5185
 
y Fn(4)1609 408 y Fu(:=)c(Im)o(\()p Fl(X)1948 422 y Fn(1)1986
5186
 
408 y Fu(\).)275 546 y(By)32 b(default,)g(the)h(t)m(yp)s(e)f
5187
 
Fo(fftw_real)e Fu(equals)i(the)g(C)g(t)m(yp)s(e)h Fo(double)p
5188
 
Fu(.)45 b(T)-8 b(o)32 b(w)m(ork)h(in)e(single)g(precision)150
5189
 
655 y(rather)c(than)g(double)f(precision,)g Fo(#define)g
5190
 
Fu(the)h(sym)m(b)s(ol)f Fo(FFTW_ENABLE_FLOAT)d Fu(in)j
5191
 
Fo(fftw.h)g Fu(and)g(then)150 765 y(recompile)32 b(the)i(library)-8
 
5222
TeXDict begin 18 19 bop 150 -116 a Fv(18)3232 b(FFTW)150
 
5223
299 y(and)37 b Fm(Y)387 313 y Fo(3)461 299 y Fv(:=)g(Im)o(\()p
 
5224
Fm(X)812 313 y Fo(1)850 299 y Fv(\).)62 b(Let)38 b(no)m(w)g
 
5225
Fm(n)e Fv(=)h(5.)62 b(In)37 b(this)f(case,)41 b(w)m(e)d(ha)m(v)m(e)g
 
5226
Fm(Y)2612 313 y Fo(0)2686 299 y Fv(:=)f(Re)q(\()p Fm(X)3037
 
5227
313 y Fo(0)3074 299 y Fv(\),)j Fm(Y)3227 313 y Fo(1)3301
 
5228
299 y Fv(:=)d(Re\()p Fm(X)3651 313 y Fo(1)3689 299 y
 
5229
Fv(\),)150 408 y Fm(Y)203 422 y Fo(2)265 408 y Fv(:=)25
 
5230
b(Re)q(\()p Fm(X)604 422 y Fo(2)642 408 y Fv(\),)31 b
 
5231
Fm(Y)786 422 y Fo(3)848 408 y Fv(:=)25 b(Im\()p Fm(X)1188
 
5232
422 y Fo(2)1226 408 y Fv(\),)31 b(and)e Fm(Y)1546 422
 
5233
y Fo(4)1609 408 y Fv(:=)c(Im)o(\()p Fm(X)1948 422 y Fo(1)1986
 
5234
408 y Fv(\).)275 542 y(By)32 b(default,)g(the)h(t)m(yp)s(e)f
 
5235
Fp(fftw_real)e Fv(equals)i(the)g(C)g(t)m(yp)s(e)h Fp(double)p
 
5236
Fv(.)45 b(T)-8 b(o)32 b(w)m(ork)h(in)e(single)g(precision)150
 
5237
652 y(rather)c(than)g(double)f(precision,)g Fp(#define)g
 
5238
Fv(the)h(sym)m(b)s(ol)f Fp(FFTW_ENABLE_FLOAT)d Fv(in)j
 
5239
Fp(fftw.h)g Fv(and)g(then)150 761 y(recompile)32 b(the)i(library)-8
5192
5240
b(.)48 b(On)32 b(Unix)g(systems,)j(y)m(ou)f(can)f(instead)g(use)g
5193
 
Fo(configure)28 b(--enable-float)150 875 y Fu(at)j(installation)d(time)
 
5241
Fp(configure)28 b(--enable-float)150 871 y Fv(at)j(installation)d(time)
5194
5242
i(\(see)i(Chapter)e(6)g([Installation)f(and)h(Customization],)g(page)h
5195
 
(55\).)275 1012 y(In)j(v)m(ersion)g(1)h(of)g(FFTW,)h(the)f(data)h(t)m
5196
 
(yp)s(es)f(w)m(ere)g(called)f Fo(FFTW_REAL)f Fu(and)h
5197
 
Fo(FFTW_COMPLEX)p Fu(.)51 b(W)-8 b(e)150 1122 y(c)m(hanged)35
 
5243
(55\).)275 1004 y(In)j(v)m(ersion)g(1)h(of)g(FFTW,)h(the)f(data)h(t)m
 
5244
(yp)s(es)f(w)m(ere)g(called)f Fp(FFTW_REAL)f Fv(and)h
 
5245
Fp(FFTW_COMPLEX)p Fv(.)51 b(W)-8 b(e)150 1114 y(c)m(hanged)35
5198
5246
b(the)g(capitalization)f(for)h(consistency)g(with)e(the)i(rest)g(of)g
5199
5247
(FFTW's)h(con)m(v)m(en)m(tions.)55 b(The)34 b(old)150
5200
 
1231 y(names)c(are)h(still)d(supp)s(orted,)h(but)h(their)f(use)h(is)g
5201
 
(deprecated.)150 1496 y Ft(3.2)68 b(One-dimensional)46
5202
 
b(T)-11 b(ransforms)45 b(Reference)275 1692 y Fu(The)d(one-dimensional)
 
5248
1223 y(names)c(are)h(still)d(supp)s(orted,)h(but)h(their)f(use)h(is)g
 
5249
(deprecated.)150 1478 y Fu(3.2)68 b(One-dimensional)46
 
5250
b(T)-11 b(ransforms)45 b(Reference)275 1669 y Fv(The)d(one-dimensional)
5203
5251
f(complex)i(routines)e(are)j(generally)e(pre\014xed)g(with)g
5204
 
Fo(fftw_)p Fu(.)77 b(Programs)150 1801 y(using)31 b(FFTW)j(should)d(b)s
5205
 
(e)h(link)m(ed)g(with)f Fo(-lfftw)e(-lm)j Fu(on)h(Unix)e(systems,)j(or)
5206
 
f(with)f(the)h(FFTW)h(and)150 1911 y(standard)c(math)g(libraries)d(in)i
5207
 
(general.)150 2141 y Fi(3.2.1)63 b(Plan)40 b(Creation)h(for)g
5208
 
(One-dimensional)h(T)-10 b(ransforms)390 2330 y Fo(#include)46
5209
 
b(<fftw.h>)390 2538 y(fftw_plan)f(fftw_create_plan\(int)e(n,)k
5210
 
(fftw_direction)d(dir,)1679 2642 y(int)j(flags\);)390
5211
 
2849 y(fftw_plan)e(fftw_create_plan_specific)o(\(int)c(n,)47
5212
 
b(fftw_direction)d(dir,)2108 2953 y(int)j(flags,)2108
5213
 
3057 y(fftw_complex)e(*in,)h(int)h(istride,)2108 3161
5214
 
y(fftw_complex)e(*out,)h(int)h(ostride\);)275 3298 y
5215
 
Fu(The)27 b(function)f Fo(fftw_create_plan)e Fu(creates)29
 
5252
Fp(fftw_)p Fv(.)77 b(Programs)150 1779 y(using)31 b(FFTW)j(should)d(b)s
 
5253
(e)h(link)m(ed)g(with)f Fp(-lfftw)e(-lm)j Fv(on)h(Unix)e(systems,)j(or)
 
5254
f(with)f(the)h(FFTW)h(and)150 1889 y(standard)c(math)g(libraries)d(in)i
 
5255
(general.)150 2110 y Fh(3.2.1)63 b(Plan)40 b(Creation)h(for)g
 
5256
(One-dimensional)h(T)-10 b(ransforms)390 2272 y Fp(#include)46
 
5257
b(<fftw.h>)390 2479 y(fftw_plan)f(fftw_create_plan\(int)e(n,)k
 
5258
(fftw_direction)d(dir,)1679 2583 y(int)j(flags\);)390
 
5259
2791 y(fftw_plan)e(fftw_create_plan_specific)o(\(int)c(n,)47
 
5260
b(fftw_direction)d(dir,)2108 2894 y(int)j(flags,)2108
 
5261
2998 y(fftw_complex)e(*in,)h(int)h(istride,)2108 3102
 
5262
y(fftw_complex)e(*out,)h(int)h(ostride\);)275 3235 y
 
5263
Fv(The)27 b(function)f Fp(fftw_create_plan)e Fv(creates)29
5216
5264
b(a)f(plan,)f(whic)m(h)g(is)g(a)h(data)g(structure)g(con)m(taining)f
5217
 
(all)150 3408 y(the)35 b(information)e(that)i Fo(fftw)e
5218
 
Fu(needs)i(in)e(order)h(to)h(compute)g(the)g(1D)h(F)-8
 
5265
(all)150 3345 y(the)35 b(information)e(that)i Fp(fftw)e
 
5266
Fv(needs)i(in)e(order)h(to)h(compute)g(the)g(1D)h(F)-8
5219
5267
b(ourier)34 b(transform.)53 b(Y)-8 b(ou)35 b(can)150
5220
 
3517 y(create)i(as)e(man)m(y)h(plans)d(as)j(y)m(ou)f(need,)i(but)d
 
5268
3455 y(create)i(as)e(man)m(y)h(plans)d(as)j(y)m(ou)f(need,)i(but)d
5221
5269
(only)h(one)g(plan)f(for)h(a)h(giv)m(en)f(arra)m(y)g(size)h(is)e
5222
 
(required)f(\(a)150 3627 y(plan)c(can)i(b)s(e)e(reused)h(man)m(y)g
5223
 
(times\).)275 3764 y Fo(fftw_create_plan)f Fu(returns)j(a)i(v)-5
5224
 
b(alid)32 b(plan,)i(or)f Fo(NULL)g Fu(if,)g(for)h(some)g(reason,)h(the)
5225
 
f(plan)e(can't)j(b)s(e)150 3874 y(created.)78 b(In)42
 
5270
(required)f(\(a)150 3564 y(plan)c(can)i(b)s(e)e(reused)h(man)m(y)g
 
5271
(times\).)275 3698 y Fp(fftw_create_plan)f Fv(returns)j(a)i(v)-5
 
5272
b(alid)32 b(plan,)i(or)f Fp(NULL)g Fv(if,)g(for)h(some)g(reason,)h(the)
 
5273
f(plan)e(can't)j(b)s(e)150 3807 y(created.)78 b(In)42
5226
5274
b(the)h(default)e(installation,)j(this)d(cannot)i(happ)s(en,)i(but)c
5227
 
(it)h(is)g(p)s(ossible)e(to)j(con\014gure)150 3983 y(FFTW)27
 
5275
(it)h(is)g(p)s(ossible)e(to)j(con\014gure)150 3917 y(FFTW)27
5228
5276
b(in)e(suc)m(h)h(a)h(w)m(a)m(y)h(that)f(some)g(input)d(sizes)i(are)h
5229
5277
(forbidden,)e(and)h(FFTW)h(cannot)g(create)h(a)f(plan.)275
5230
 
4121 y(The)44 b Fo(fftw_create_plan_specific)38 b Fu(v)-5
 
5278
4050 y(The)44 b Fp(fftw_create_plan_specific)38 b Fv(v)-5
5231
5279
b(arian)m(t)46 b(tak)m(es)g(as)f(additional)f(argumen)m(ts)h(sp)s
5232
 
(eci\014c)f(in-)150 4230 y(put/output)f(arra)m(ys)g(and)f(their)h
 
5280
(eci\014c)f(in-)150 4160 y(put/output)f(arra)m(ys)g(and)f(their)h
5233
5281
(strides.)77 b(F)-8 b(or)44 b(the)f(last)g(four)g(argumen)m(ts,)k(y)m
5234
 
(ou)c(should)e(pass)i(the)150 4340 y(arra)m(ys)31 b(and)g(strides)f
 
5282
(ou)c(should)e(pass)i(the)150 4270 y(arra)m(ys)31 b(and)g(strides)f
5235
5283
(that)h(y)m(ou)g(will)e(ev)m(en)m(tually)i(b)s(e)f(passing)g(to)i
5236
 
Fo(fftw)p Fu(.)41 b(The)31 b(resulting)e(plans)g(will)g(b)s(e)150
5237
 
4450 y(optimized)g(for)h(those)h(arra)m(ys)g(and)e(strides,)h(although)
 
5284
Fp(fftw)p Fv(.)41 b(The)31 b(resulting)e(plans)g(will)g(b)s(e)150
 
5285
4379 y(optimized)g(for)h(those)h(arra)m(ys)g(and)e(strides,)h(although)
5238
5286
f(they)i(ma)m(y)g(b)s(e)e(used)h(on)g(other)h(arra)m(ys)f(as)h(w)m
5239
 
(ell.)150 4559 y(Note:)40 b(the)27 b(con)m(ten)m(ts)i(of)e(the)g(in)e
5240
 
(and)h(out)h(arra)m(ys)g(are)h Fq(destr)-5 b(oye)g(d)38
5241
 
b Fu(b)m(y)27 b(the)g(sp)s(eci\014c)e(planner)g(\(the)j(initial)150
5242
 
4669 y(con)m(ten)m(ts)k(are)f(ignored,)f(so)g(the)h(arra)m(ys)g(need)f
5243
 
(not)g(ha)m(v)m(e)i(b)s(een)d(initialized\).)150 4899
5244
 
y Fi(Argumen)m(ts)225 5094 y Fs(\017)60 b Fo(n)30 b Fu(is)f(the)i(size)
 
5287
(ell.)150 4489 y(Note:)40 b(the)27 b(con)m(ten)m(ts)i(of)e(the)g(in)e
 
5288
(and)h(out)h(arra)m(ys)g(are)h Fr(destr)-5 b(oye)g(d)38
 
5289
b Fv(b)m(y)27 b(the)g(sp)s(eci\014c)e(planner)g(\(the)j(initial)150
 
5290
4598 y(con)m(ten)m(ts)k(are)f(ignored,)f(so)g(the)h(arra)m(ys)g(need)f
 
5291
(not)g(ha)m(v)m(e)i(b)s(een)d(initialized\).)150 4820
 
5292
y Fh(Argumen)m(ts)225 4987 y Ft(\017)60 b Fp(n)30 b Fv(is)f(the)i(size)
5245
5293
f(of)h(the)f(transform.)40 b(It)31 b(can)g(b)s(e)e(an)m(y)i(p)s(ositiv)
5246
 
m(e)e(in)m(teger.)379 5230 y Fs(\000)60 b Fu(FFTW)22
 
5294
m(e)e(in)m(teger.)379 5121 y Ft(\000)60 b Fv(FFTW)22
5247
5295
b(is)f(b)s(est)g(at)h(handling)c(sizes)k(of)f(the)h(form)f(2)2237
5248
 
5197 y Fk(a)2277 5230 y Fu(3)2322 5197 y Fk(b)2356 5230
5249
 
y Fu(5)2401 5197 y Fk(c)2435 5230 y Fu(7)2480 5197 y
5250
 
Fk(d)2519 5230 y Fu(11)2609 5197 y Fk(e)2646 5230 y Fu(13)2736
5251
 
5197 y Fk(f)2780 5230 y Fu(,)i(where)e Fl(e)p Fu(+)p
5252
 
Fl(f)31 b Fu(is)20 b(either)h(0)h(or)510 5340 y(1,)30
 
5296
5088 y Fl(a)2277 5121 y Fv(3)2322 5088 y Fl(b)2356 5121
 
5297
y Fv(5)2401 5088 y Fl(c)2435 5121 y Fv(7)2480 5088 y
 
5298
Fl(d)2519 5121 y Fv(11)2609 5088 y Fl(e)2646 5121 y Fv(13)2736
 
5299
5088 y Fl(f)2780 5121 y Fv(,)i(where)e Fm(e)p Fv(+)p
 
5300
Fm(f)31 b Fv(is)20 b(either)h(0)h(or)510 5230 y(1,)30
5253
5301
b(and)e(the)h(other)g(exp)s(onen)m(ts)f(are)i(arbitrary)-8
5254
 
b(.)39 b(Other)28 b(sizes)h(are)g(computed)f(b)m(y)h(means)g(of)g(a)p
5255
 
eop
 
5302
b(.)39 b(Other)28 b(sizes)h(are)g(computed)f(b)m(y)h(means)g(of)g(a)510
 
5303
5340 y(slo)m(w,)h(general-purp)s(ose)f(routine)h(\(whic)m(h)f(nev)m
 
5304
(ertheless)i(retains)f Fm(O)s Fv(\()p Fm(n)15 b Fv(log)h
 
5305
Fm(n)p Fv(\))30 b(p)s(erformance,)p eop end
5256
5306
%%Page: 19 21
5257
 
19 20 bop 150 -116 a Fu(Chapter)30 b(3:)41 b(FFTW)31
5258
 
b(Reference)2354 b(19)510 299 y(slo)m(w,)30 b(general-purp)s(ose)f
5259
 
(routine)h(\(whic)m(h)f(nev)m(ertheless)i(retains)f Fl(O)s
5260
 
Fu(\()p Fl(n)15 b Fu(log)h Fl(n)p Fu(\))30 b(p)s(erformance,)510
5261
 
408 y(ev)m(en)i(for)f(prime)e(sizes\).)44 b(\(It)31 b(is)g(p)s(ossible)
5262
 
d(to)k(customize)g(FFTW)g(for)f(di\013eren)m(t)f(arra)m(y)i(sizes.)510
5263
 
518 y(See)k(Chapter)g(6)g([Installation)f(and)g(Customization],)i(page)
5264
 
g(55,)h(for)e(more)g(information.\))510 628 y(T)-8 b(ransforms)29
 
5307
TeXDict begin 19 20 bop 150 -116 a Fv(Chapter)30 b(3:)41
 
5308
b(FFTW)31 b(Reference)2354 b(19)510 299 y(ev)m(en)32
 
5309
b(for)f(prime)e(sizes\).)44 b(\(It)31 b(is)g(p)s(ossible)d(to)k
 
5310
(customize)g(FFTW)g(for)f(di\013eren)m(t)f(arra)m(y)i(sizes.)510
 
5311
408 y(See)k(Chapter)g(6)g([Installation)f(and)g(Customization],)i(page)
 
5312
g(55,)h(for)e(more)g(information.\))510 518 y(T)-8 b(ransforms)29
5265
5313
b(whose)h(sizes)h(are)f(p)s(o)m(w)m(ers)h(of)f(2)h(are)g(esp)s(ecially)
5266
 
d(fast.)225 761 y Fs(\017)60 b Fo(dir)34 b Fu(is)g(the)h(sign)f(of)h
 
5314
d(fast.)225 646 y Ft(\017)60 b Fp(dir)34 b Fv(is)g(the)h(sign)f(of)h
5267
5315
(the)g(exp)s(onen)m(t)f(in)g(the)h(form)m(ula)f(that)h(de\014nes)f(the)
5268
 
h(F)-8 b(ourier)34 b(transform.)54 b(It)330 871 y(can)31
5269
 
b(b)s(e)f Fs(\000)p Fu(1)h(or)g(+)o(1.)43 b(The)30 b(aliases)g
5270
 
Fo(FFTW_FORWARD)d Fu(and)j Fo(FFTW_BACKWARD)e Fu(are)j(pro)m(vided,)e
5271
 
(where)330 981 y Fo(FFTW_FORWARD)e Fu(stands)j(for)g
5272
 
Fs(\000)p Fu(1.)225 1114 y Fs(\017)60 b Fo(flags)29 b
5273
 
Fu(is)g(a)i(b)s(o)s(olean)e(OR)h(\(`)p Fo(|)p Fu('\))i(of)e(zero)h(or)g
5274
 
(more)f(of)h(the)f(follo)m(wing:)379 1248 y Fs(\000)60
5275
 
b Fo(FFTW_MEASURE)p Fu(:)33 b(this)19 b(\015ag)j(tells)e(FFTW)h(to)h
5276
 
(\014nd)d(the)i(optimal)f(plan)g(b)m(y)g(actually)h Fq(c)-5
5277
 
b(omputing)510 1358 y Fu(sev)m(eral)31 b(FFTs)g(and)f(measuring)f
 
5316
h(F)-8 b(ourier)34 b(transform.)54 b(It)330 755 y(can)31
 
5317
b(b)s(e)f Ft(\000)p Fv(1)h(or)g(+)o(1.)43 b(The)30 b(aliases)g
 
5318
Fp(FFTW_FORWARD)d Fv(and)j Fp(FFTW_BACKWARD)e Fv(are)j(pro)m(vided,)e
 
5319
(where)330 865 y Fp(FFTW_FORWARD)e Fv(stands)j(for)g
 
5320
Ft(\000)p Fv(1.)225 992 y Ft(\017)60 b Fp(flags)29 b
 
5321
Fv(is)g(a)i(b)s(o)s(olean)e(OR)h(\(`)p Fp(|)p Fv('\))i(of)e(zero)h(or)g
 
5322
(more)f(of)h(the)f(follo)m(wing:)379 1120 y Ft(\000)60
 
5323
b Fp(FFTW_MEASURE)p Fv(:)33 b(this)19 b(\015ag)j(tells)e(FFTW)h(to)h
 
5324
(\014nd)d(the)i(optimal)f(plan)g(b)m(y)g(actually)h Fr(c)-5
 
5325
b(omputing)510 1230 y Fv(sev)m(eral)31 b(FFTs)g(and)f(measuring)f
5278
5326
(their)h(execution)h(time.)41 b(Dep)s(ending)29 b(on)i(the)g
5279
 
(installation,)510 1467 y(this)e(can)i(tak)m(e)h(some)f(time.)1481
5280
 
1434 y Fn(1)379 1601 y Fs(\000)60 b Fo(FFTW_ESTIMATE)p
5281
 
Fu(:)34 b(do)25 b(not)g(run)f(an)m(y)h(FFT)h(and)e(pro)m(vide)g(a)h
5282
 
(\\reasonable")h(plan)e(\(for)h(a)g(RISC)510 1711 y(pro)s(cessor)37
 
5327
(installation,)510 1339 y(this)e(can)i(tak)m(e)h(some)f(time.)1481
 
5328
1306 y Fo(1)379 1467 y Ft(\000)60 b Fp(FFTW_ESTIMATE)p
 
5329
Fv(:)34 b(do)25 b(not)g(run)f(an)m(y)h(FFT)h(and)e(pro)m(vide)g(a)h
 
5330
(\\reasonable")h(plan)e(\(for)h(a)g(RISC)510 1576 y(pro)s(cessor)37
5283
5331
b(with)g(man)m(y)h(registers\).)62 b(If)38 b(neither)e
5284
 
Fo(FFTW_ESTIMATE)e Fu(nor)k Fo(FFTW_MEASURE)c Fu(is)510
5285
 
1820 y(pro)m(vided,)29 b(the)i(default)e(is)h Fo(FFTW_ESTIMATE)p
5286
 
Fu(.)379 1954 y Fs(\000)60 b Fo(FFTW_OUT_OF_PLACE)p Fu(:)41
 
5332
Fp(FFTW_ESTIMATE)e Fv(nor)k Fp(FFTW_MEASURE)c Fv(is)510
 
5333
1686 y(pro)m(vided,)29 b(the)i(default)e(is)h Fp(FFTW_ESTIMATE)p
 
5334
Fv(.)379 1813 y Ft(\000)60 b Fp(FFTW_OUT_OF_PLACE)p Fv(:)41
5287
5335
b(pro)s(duce)32 b(a)i(plan)e(assuming)f(that)j(the)f(input)e(and)i
5288
 
(output)g(arra)m(ys)510 2063 y(will)28 b(b)s(e)h(distinct)g(\(this)h
5289
 
(is)f(the)i(default\).)379 2197 y Fs(\000)60 b Fo(FFTW_IN_PLACE)p
5290
 
Fu(:)39 b(pro)s(duce)31 b(a)h(plan)e(assuming)g(that)i(y)m(ou)g(w)m(an)
5291
 
m(t)g(the)g(output)f(in)f(the)i(input)510 2307 y(arra)m(y)-8
 
5336
(output)g(arra)m(ys)510 1923 y(will)28 b(b)s(e)h(distinct)g(\(this)h
 
5337
(is)f(the)i(default\).)379 2051 y Ft(\000)60 b Fp(FFTW_IN_PLACE)p
 
5338
Fv(:)39 b(pro)s(duce)31 b(a)h(plan)e(assuming)g(that)i(y)m(ou)g(w)m(an)
 
5339
m(t)g(the)g(output)f(in)f(the)i(input)510 2160 y(arra)m(y)-8
5292
5340
b(.)50 b(The)33 b(algorithm)f(used)g(is)g(not)i(necessarily)e(in)g
5293
 
(place:)46 b(FFTW)34 b(is)e(able)h(to)h(compute)510 2416
 
5341
(place:)46 b(FFTW)34 b(is)e(able)h(to)h(compute)510 2270
5294
5342
y(true)23 b(in-place)f(transforms)h(only)f(for)h(small)f(v)-5
5295
 
b(alues)23 b(of)g Fo(n)p Fu(.)38 b(If)23 b(FFTW)h(is)e(not)i(able)f(to)
5296
 
h(compute)510 2526 y(the)31 b(transform)g(in-place,)f(it)h(will)e(allo)
 
5343
b(alues)23 b(of)g Fp(n)p Fv(.)38 b(If)23 b(FFTW)h(is)e(not)i(able)f(to)
 
5344
h(compute)510 2379 y(the)31 b(transform)g(in-place,)f(it)h(will)e(allo)
5297
5345
s(cate)i(a)h(temp)s(orary)e(arra)m(y)i(\(unless)e(y)m(ou)h(pro)m(vide)g
5298
 
(one)510 2636 y(y)m(ourself)7 b(\),)27 b(compute)e(the)h(transform)f
 
5346
(one)510 2489 y(y)m(ourself)7 b(\),)27 b(compute)e(the)h(transform)f
5299
5347
(out)h(of)g(place,)h(and)e(cop)m(y)h(the)g(result)f(bac)m(k.)40
5300
 
b Fq(Warning:)510 2745 y(This)f(option)i(changes)e(the)h(me)-5
 
5348
b Fr(Warning:)510 2599 y(This)f(option)i(changes)e(the)h(me)-5
5301
5349
b(aning)40 b(of)f(some)g(p)-5 b(ar)g(ameters)42 b(of)e
5302
 
Fo(fftw)c Fu(\(see)i(Section)f(3.2.3)510 2855 y([Computing)29
 
5350
Fp(fftw)c Fv(\(see)i(Section)f(3.2.3)510 2708 y([Computing)29
5303
5351
b(the)i(One-dimensional)c(T)-8 b(ransform],)30 b(page)h(20\).)510
5304
 
2988 y(The)d(in-place)f(option)h(is)f(mainly)g(pro)m(vided)f(for)j(p)s
 
5352
2836 y(The)d(in-place)f(option)h(is)f(mainly)g(pro)m(vided)f(for)j(p)s
5305
5353
(eople)e(who)h(w)m(an)m(t)h(to)g(write)e(their)h(o)m(wn)g(in-)510
5306
 
3098 y(place)k(m)m(ulti-dimensional)d(F)-8 b(ourier)32
 
5354
2945 y(place)k(m)m(ulti-dimensional)d(F)-8 b(ourier)32
5307
5355
b(transform,)g(using)f(FFTW)i(as)f(a)h(base.)47 b(F)-8
5308
 
b(or)33 b(example,)510 3208 y(consider)f(a)i(three-dimensional)e
5309
 
Fo(n)e(*)g(n)g(*)g(n)j Fu(transform.)49 b(An)33 b(out-of-place)i
5310
 
(algorithm)e(will)510 3317 y(need)45 b(another)g(arra)m(y)g(\(whic)m(h)
 
5356
b(or)33 b(example,)510 3055 y(consider)f(a)i(three-dimensional)e
 
5357
Fp(n)e(*)g(n)g(*)g(n)j Fv(transform.)49 b(An)33 b(out-of-place)i
 
5358
(algorithm)e(will)510 3164 y(need)45 b(another)g(arra)m(y)g(\(whic)m(h)
5311
5359
f(ma)m(y)i(b)s(e)e(h)m(uge\).)86 b(Ho)m(w)m(ev)m(er,)50
5312
 
b(FFTW)c(can)f(compute)h(the)510 3427 y(in-place)35 b(transform)f
 
5360
b(FFTW)c(can)f(compute)h(the)510 3274 y(in-place)35 b(transform)f
5313
5361
(along)i(eac)m(h)h(dimension)c(using)h(only)h(a)h(temp)s(orary)f(arra)m
5314
 
(y)h(of)g(size)f Fo(n)p Fu(.)510 3536 y(Moreo)m(v)m(er,)h(if)31
 
5362
(y)h(of)g(size)f Fp(n)p Fv(.)510 3384 y(Moreo)m(v)m(er,)h(if)31
5315
5363
b(FFTW)j(happ)s(ens)c(to)k(b)s(e)e(able)g(to)i(compute)f(the)f
5316
 
(transform)g(truly)g(in-place,)510 3646 y(no)k(temp)s(orary)g(arra)m(y)
 
5364
(transform)g(truly)g(in-place,)510 3493 y(no)k(temp)s(orary)g(arra)m(y)
5317
5365
h(and)e(no)i(cop)m(ying)f(are)g(needed.)58 b(As)37 b(distributed,)e
5318
 
(FFTW)i(`kno)m(ws')510 3756 y(ho)m(w)32 b(to)h(compute)f(in-place)g
 
5366
(FFTW)i(`kno)m(ws')510 3603 y(ho)m(w)32 b(to)h(compute)f(in-place)g
5319
5367
(transforms)f(of)h(size)g(1,)h(2,)h(3,)f(4,)g(5,)g(6,)g(7,)g(8,)h(9,)f
5320
 
(10,)h(11,)f(12,)h(13,)510 3865 y(14,)d(15,)h(16,)f(32)h(and)d(64.)510
5321
 
3999 y(The)h(default)f(mo)s(de)h(of)h(op)s(eration)f(is)f
5322
 
Fo(FFTW_OUT_OF_PLACE)p Fu(.)379 4133 y Fs(\000)60 b Fo(FFTW_USE_WISDOM)
5323
 
p Fu(:)37 b(use)30 b(an)m(y)h Fo(wisdom)e Fu(that)i(is)e(a)m(v)-5
 
5368
(10,)h(11,)f(12,)h(13,)510 3712 y(14,)d(15,)h(16,)f(32)h(and)d(64.)510
 
5369
3840 y(The)h(default)f(mo)s(de)h(of)h(op)s(eration)f(is)f
 
5370
Fp(FFTW_OUT_OF_PLACE)p Fv(.)379 3968 y Ft(\000)60 b Fp(FFTW_USE_WISDOM)
 
5371
p Fv(:)37 b(use)30 b(an)m(y)h Fp(wisdom)e Fv(that)i(is)e(a)m(v)-5
5324
5372
b(ailable)30 b(to)h(help)e(in)h(the)g(creation)h(of)g(the)510
5325
 
4242 y(plan.)50 b(\(See)34 b(Section)g(2.6)h([W)-8 b(ords)34
 
5373
4077 y(plan.)50 b(\(See)34 b(Section)g(2.6)h([W)-8 b(ords)34
5326
5374
b(of)h(Wisdom],)f(page)h(13.\))52 b(This)32 b(can)i(greatly)h(sp)s(eed)
5327
 
e(the)510 4352 y(creation)27 b(of)g(plans,)g(esp)s(ecially)e(with)g
5328
 
(the)j Fo(FFTW_MEASURE)23 b Fu(option.)39 b Fo(FFTW_ESTIMATE)23
5329
 
b Fu(plans)510 4461 y(can)28 b(also)g(tak)m(e)i(adv)-5
5330
 
b(an)m(tage)30 b(of)e Fo(wisdom)e Fu(to)j(pro)s(duce)e(a)h(more)g
5331
 
(optimal)f(plan)g(\(based)h(on)g(past)510 4571 y(measuremen)m(ts\))i
 
5375
e(the)510 4187 y(creation)27 b(of)g(plans,)g(esp)s(ecially)e(with)g
 
5376
(the)j Fp(FFTW_MEASURE)23 b Fv(option.)39 b Fp(FFTW_ESTIMATE)23
 
5377
b Fv(plans)510 4296 y(can)28 b(also)g(tak)m(e)i(adv)-5
 
5378
b(an)m(tage)30 b(of)e Fp(wisdom)e Fv(to)j(pro)s(duce)e(a)h(more)g
 
5379
(optimal)f(plan)g(\(based)h(on)g(past)510 4406 y(measuremen)m(ts\))i
5332
5380
(than)g(the)g(estimation)f(heuristic)f(w)m(ould)g(normally)g(generate.)
5333
 
42 b(When)30 b(the)510 4681 y Fo(FFTW_MEASURE)36 b Fu(option)j(is)f
5334
 
(used,)j(new)d Fo(wisdom)g Fu(will)f(also)i(b)s(e)f(generated)j(if)d
5335
 
(the)h(curren)m(t)510 4790 y(transform)30 b(size)g(is)f(not)i
5336
 
(completely)f(understo)s(o)s(d)f(b)m(y)h(existing)f Fo(wisdom)p
5337
 
Fu(.)225 4924 y Fs(\017)60 b Fo(in)p Fu(,)35 b Fo(out)p
5338
 
Fu(,)f Fo(istride)p Fu(,)g Fo(ostride)e Fu(\(only)i(for)g
5339
 
Fo(fftw_create_plan_specifi)o(c)p Fu(\):)43 b(see)35
5340
 
b(corresp)s(ond-)330 5033 y(ing)41 b(argumen)m(ts)h(in)e(the)i
5341
 
(description)d(of)j Fo(fftw)p Fu(.)73 b(\(See)42 b(Section)g(3.2.3)h
5342
 
([Computing)d(the)i(One-)p 150 5131 1200 4 v 192 5197
5343
 
a Fn(1)275 5230 y Fu(The)28 b(basic)h(problem)f(is)g(the)i(resolution)e
5344
 
(of)i(the)f(clo)s(c)m(k:)41 b(FFTW)30 b(needs)f(to)h(run)e(for)h(a)h
5345
 
(certain)f(time)275 5340 y(for)h(the)g(clo)s(c)m(k)h(to)g(b)s(e)f
5346
 
(reliable.)p eop
 
5381
42 b(When)30 b(the)510 4515 y Fp(FFTW_MEASURE)36 b Fv(option)j(is)f
 
5382
(used,)j(new)d Fp(wisdom)g Fv(will)f(also)i(b)s(e)f(generated)j(if)d
 
5383
(the)h(curren)m(t)510 4625 y(transform)30 b(size)g(is)f(not)i
 
5384
(completely)f(understo)s(o)s(d)f(b)m(y)h(existing)f Fp(wisdom)p
 
5385
Fv(.)225 4753 y Ft(\017)60 b Fp(in)p Fv(,)35 b Fp(out)p
 
5386
Fv(,)f Fp(istride)p Fv(,)g Fp(ostride)e Fv(\(only)i(for)g
 
5387
Fp(fftw_create_plan_specifi)o(c)p Fv(\):)43 b(see)35
 
5388
b(corresp)s(ond-)330 4862 y(ing)41 b(argumen)m(ts)h(in)e(the)i
 
5389
(description)d(of)j Fp(fftw)p Fv(.)73 b(\(See)42 b(Section)g(3.2.3)h
 
5390
([Computing)d(the)i(One-)330 4972 y(dimensional)e(T)-8
 
5391
b(ransform],)45 b(page)f(20.\))79 b(In)42 b(particular,)i(the)f
 
5392
Fp(out)f Fv(and)g Fp(ostride)f Fv(parameters)330 5081
 
5393
y(ha)m(v)m(e)32 b(the)e(same)h(sp)s(ecial)e(meaning)h(for)g
 
5394
Fp(FFTW_IN_PLACE)c Fv(transforms)k(as)h(they)f(ha)m(v)m(e)i(for)e
 
5395
Fp(fftw)p Fv(.)p 150 5149 1200 4 v 199 5217 a Fo(1)275
 
5396
5249 y Fj(The)25 b(basic)h(problem)e(is)i(the)e(resolution)j(of)e(the)g
 
5397
(clo)r(c)n(k:)35 b(FFTW)26 b(needs)e(to)i(run)e(for)i(a)g(certain)f
 
5398
(time)g(for)g(the)g(clo)r(c)n(k)h(to)275 5340 y(b)r(e)f(reliable.)p
 
5399
eop end
5347
5400
%%Page: 20 22
5348
 
20 21 bop 150 -116 a Fu(20)3232 b(FFTW)330 299 y(dimensional)40
5349
 
b(T)-8 b(ransform],)45 b(page)f(20.\))79 b(In)42 b(particular,)i(the)f
5350
 
Fo(out)f Fu(and)g Fo(ostride)f Fu(parameters)330 408
5351
 
y(ha)m(v)m(e)32 b(the)e(same)h(sp)s(ecial)e(meaning)h(for)g
5352
 
Fo(FFTW_IN_PLACE)c Fu(transforms)k(as)h(they)f(ha)m(v)m(e)i(for)e
5353
 
Fo(fftw)p Fu(.)150 636 y Fi(3.2.2)63 b(Discussion)42
5354
 
b(on)g(Sp)s(eci\014c)f(Plans)275 830 y Fu(W)-8 b(e)22
5355
 
b(recommend)f(the)g(use)g(of)h(the)f(sp)s(eci\014c)f(planners,)i(ev)m
5356
 
(en)g(in)e(cases)i(where)e(y)m(ou)i(will)d(b)s(e)h(transform-)150
5357
 
939 y(ing)30 b(arra)m(ys)h(di\013eren)m(t)f(from)g(those)h(passed)f(to)
5358
 
i(the)f(sp)s(eci\014c)e(planners,)g(as)i(they)g(confer)g(the)f(follo)m
5359
 
(wing)150 1049 y(adv)-5 b(an)m(tages:)225 1185 y Fs(\017)60
5360
 
b Fu(The)36 b(resulting)f(plans)g(will)f(b)s(e)i(optimized)g(for)g(y)m
5361
 
(our)h(sp)s(eci\014c)e(arra)m(ys)i(and)f(strides.)58
5362
 
b(This)35 b(ma)m(y)330 1294 y(or)d(ma)m(y)h(not)g(mak)m(e)g(a)f
5363
 
(signi\014can)m(t)g(di\013erence,)g(but)g(it)f(certainly)h(do)s(esn't)g
5364
 
(h)m(urt.)46 b(\(The)32 b(ordinary)330 1404 y(planner)d(do)s(es)h(its)f
5365
 
(planning)f(based)i(up)s(on)f(a)i(stride-one)f(temp)s(orary)g(arra)m(y)
5366
 
g(that)h(it)f(allo)s(cates.\))225 1539 y Fs(\017)60 b
5367
 
Fu(Less)27 b(in)m(termediate)g(storage)i(is)e(required)e(during)g(the)j
5368
 
(planning)d(pro)s(cess.)39 b(\(The)27 b(ordinary)f(plan-)330
5369
 
1649 y(ner)32 b(uses)g(O\()p Fo(N)p Fu(\))h(temp)s(orary)f(storage,)j
5370
 
(where)d Fo(N)g Fu(is)g(the)h(maxim)m(um)e(dimension,)g(while)g(it)h
5371
 
(is)f(cre-)330 1758 y(ating)f(the)h(plan.\))225 1894
5372
 
y Fs(\017)60 b Fu(F)-8 b(or)32 b(m)m(ulti-dimensional)27
5373
 
b(transforms,)j(new)h(parameters)g(b)s(ecome)h(accessible)e(for)h
5374
 
(optimization)330 2003 y(b)m(y)36 b(the)h(planner.)58
5375
 
b(\(Since)36 b(m)m(ulti-dimensional)c(arra)m(ys)37 b(can)g(b)s(e)f(v)m
5376
 
(ery)h(large,)h(w)m(e)f(don't)g(dare)f(to)330 2113 y(allo)s(cate)30
5377
 
b(one)g(in)f(the)g(ordinary)f(planner)g(for)i(exp)s(erimen)m(tation.)39
5378
 
b(This)28 b(prev)m(en)m(ts)j(us)e(from)g(doing)330 2222
 
5401
TeXDict begin 20 21 bop 150 -116 a Fv(20)3232 b(FFTW)150
 
5402
299 y Fh(3.2.2)63 b(Discussion)42 b(on)g(Sp)s(eci\014c)f(Plans)275
 
5403
499 y Fv(W)-8 b(e)22 b(recommend)f(the)g(use)g(of)h(the)f(sp)s
 
5404
(eci\014c)f(planners,)i(ev)m(en)g(in)e(cases)i(where)e(y)m(ou)i(will)d
 
5405
(b)s(e)h(transform-)150 608 y(ing)30 b(arra)m(ys)h(di\013eren)m(t)f
 
5406
(from)g(those)h(passed)f(to)i(the)f(sp)s(eci\014c)e(planners,)g(as)i
 
5407
(they)g(confer)g(the)f(follo)m(wing)150 718 y(adv)-5
 
5408
b(an)m(tages:)225 860 y Ft(\017)60 b Fv(The)36 b(resulting)f(plans)g
 
5409
(will)f(b)s(e)i(optimized)g(for)g(y)m(our)h(sp)s(eci\014c)e(arra)m(ys)i
 
5410
(and)f(strides.)58 b(This)35 b(ma)m(y)330 969 y(or)d(ma)m(y)h(not)g
 
5411
(mak)m(e)g(a)f(signi\014can)m(t)g(di\013erence,)g(but)g(it)f(certainly)
 
5412
h(do)s(esn't)g(h)m(urt.)46 b(\(The)32 b(ordinary)330
 
5413
1079 y(planner)d(do)s(es)h(its)f(planning)f(based)i(up)s(on)f(a)i
 
5414
(stride-one)f(temp)s(orary)g(arra)m(y)g(that)h(it)f(allo)s(cates.\))225
 
5415
1217 y Ft(\017)60 b Fv(Less)27 b(in)m(termediate)g(storage)i(is)e
 
5416
(required)e(during)g(the)j(planning)d(pro)s(cess.)39
 
5417
b(\(The)27 b(ordinary)f(plan-)330 1327 y(ner)32 b(uses)g(O\()p
 
5418
Fp(N)p Fv(\))h(temp)s(orary)f(storage,)j(where)d Fp(N)g
 
5419
Fv(is)g(the)h(maxim)m(um)e(dimension,)g(while)g(it)h(is)f(cre-)330
 
5420
1436 y(ating)f(the)h(plan.\))225 1574 y Ft(\017)60 b
 
5421
Fv(F)-8 b(or)32 b(m)m(ulti-dimensional)27 b(transforms,)j(new)h
 
5422
(parameters)g(b)s(ecome)h(accessible)e(for)h(optimization)330
 
5423
1684 y(b)m(y)36 b(the)h(planner.)58 b(\(Since)36 b(m)m
 
5424
(ulti-dimensional)c(arra)m(ys)37 b(can)g(b)s(e)f(v)m(ery)h(large,)h(w)m
 
5425
(e)f(don't)g(dare)f(to)330 1794 y(allo)s(cate)30 b(one)g(in)f(the)g
 
5426
(ordinary)f(planner)g(for)i(exp)s(erimen)m(tation.)39
 
5427
b(This)28 b(prev)m(en)m(ts)j(us)e(from)g(doing)330 1903
5379
5428
y(certain)h(optimizations)f(that)i(can)g(yield)e(dramatic)h(impro)m(v)m
5380
 
(emen)m(ts)g(in)f(some)i(cases.\))275 2384 y(On)36 b(the)h(other)g
5381
 
(hand,)h(note)g(that)f Fq(the)i(sp)-5 b(e)g(ci\014c)40
 
5429
(emen)m(ts)g(in)f(some)i(cases.\))275 2073 y(On)36 b(the)h(other)g
 
5430
(hand,)h(note)g(that)f Fr(the)i(sp)-5 b(e)g(ci\014c)40
5382
5431
b(planner)g(destr)-5 b(oys)41 b(the)e(c)-5 b(ontents)40
5383
 
b(of)f(the)g Fo(in)f Fq(and)150 2494 y Fo(out)32 b Fq(arr)-5
5384
 
b(ays)p Fu(.)150 2721 y Fi(3.2.3)63 b(Computing)41 b(the)f
5385
 
(One-dimensional)i(T)-10 b(ransform)390 2909 y Fo(#include)46
5386
 
b(<fftw.h>)390 3117 y(void)h(fftw\(fftw_plan)d(plan,)i(int)h(howmany,)
5387
 
867 3220 y(fftw_complex)e(*in,)h(int)h(istride,)f(int)h(idist,)867
5388
 
3324 y(fftw_complex)e(*out,)h(int)h(ostride,)f(int)g(odist\);)390
5389
 
3532 y(void)h(fftw_one\(fftw_plan)42 b(plan,)47 b(fftw_complex)d(*in,)
5390
 
867 3636 y(fftw_complex)h(*out\);)275 3772 y Fu(The)22
5391
 
b(function)g Fo(fftw)g Fu(computes)h(the)g(one-dimensional)e(F)-8
 
5432
b(of)f(the)g Fp(in)f Fr(and)150 2183 y Fp(out)32 b Fr(arr)-5
 
5433
b(ays)p Fv(.)150 2422 y Fh(3.2.3)63 b(Computing)41 b(the)f
 
5434
(One-dimensional)i(T)-10 b(ransform)390 2584 y Fp(#include)46
 
5435
b(<fftw.h>)390 2791 y(void)h(fftw\(fftw_plan)d(plan,)i(int)h(howmany,)
 
5436
867 2895 y(fftw_complex)e(*in,)h(int)h(istride,)f(int)h(idist,)867
 
5437
2999 y(fftw_complex)e(*out,)h(int)h(ostride,)f(int)g(odist\);)390
 
5438
3206 y(void)h(fftw_one\(fftw_plan)42 b(plan,)47 b(fftw_complex)d(*in,)
 
5439
867 3310 y(fftw_complex)h(*out\);)275 3452 y Fv(The)22
 
5440
b(function)g Fp(fftw)g Fv(computes)h(the)g(one-dimensional)e(F)-8
5392
5441
b(ourier)23 b(transform,)h(using)e(a)h(plan)f(created)150
5393
 
3881 y(b)m(y)31 b Fo(fftw_create_plan)c Fu(\(See)k(Section)g(3.2.1)i
 
5442
3561 y(b)m(y)31 b Fp(fftw_create_plan)c Fv(\(See)k(Section)g(3.2.1)i
5394
5443
([Plan)e(Creation)f(for)h(One-dimensional)e(T)-8 b(ransforms],)150
5395
 
3991 y(page)38 b(18.\))62 b(The)37 b(function)f Fo(fftw_one)f
5396
 
Fu(pro)m(vides)h(a)i(simpli\014ed)33 b(in)m(terface)38
5397
 
b(for)f(the)g(common)h(case)g(of)150 4100 y(single)29
5398
 
b(input)f(arra)m(y)j(of)g(stride)e(1.)150 4327 y Fi(Argumen)m(ts)225
5399
 
4522 y Fs(\017)60 b Fo(plan)35 b Fu(is)g(the)i(plan)e(created)i(b)m(y)f
5400
 
Fo(fftw_create_plan)c Fu(\(see)38 b(Section)e(3.2.1)i([Plan)d(Creation)
5401
 
h(for)330 4631 y(One-dimensional)27 b(T)-8 b(ransforms],)30
5402
 
b(page)h(18\).)225 4766 y Fs(\017)60 b Fo(howmany)30
5403
 
b Fu(is)h(the)h(n)m(um)m(b)s(er)e(of)i(transforms)f Fo(fftw)g
5404
 
Fu(will)e(compute.)46 b(It)32 b(is)f(faster)h(to)h(tell)e(FFTW)h(to)330
5405
 
4876 y(compute)f(man)m(y)f(transforms,)g(instead)g(of)g(simply)e
5406
 
(calling)h Fo(fftw)g Fu(man)m(y)i(times.)225 5011 y Fs(\017)60
5407
 
b Fo(in)p Fu(,)33 b Fo(istride)e Fu(and)i Fo(idist)e
5408
 
Fu(describ)s(e)h(the)h(input)e(arra)m(y\(s\).)50 b(There)33
5409
 
b(are)g Fo(howmany)e Fu(input)g(arra)m(ys;)330 5121 y(the)26
5410
 
b(\014rst)f(one)h(is)f(p)s(oin)m(ted)g(to)h(b)m(y)g Fo(in)p
5411
 
Fu(,)g(the)g(second)g(one)g(is)f(p)s(oin)m(ted)g(to)h(b)m(y)g
5412
 
Fo(in)k(+)g(idist)p Fu(,)c(and)f(so)h(on,)330 5230 y(up)f(to)h
5413
 
Fo(in)k(+)g(\(howmany)e(-)i(1\))g(*)g(idist)p Fu(.)37
 
5444
3671 y(page)38 b(18.\))62 b(The)37 b(function)f Fp(fftw_one)f
 
5445
Fv(pro)m(vides)h(a)i(simpli\014ed)33 b(in)m(terface)38
 
5446
b(for)f(the)g(common)h(case)g(of)150 3781 y(single)29
 
5447
b(input)f(arra)m(y)j(of)g(stride)e(1.)150 4019 y Fh(Argumen)m(ts)225
 
5448
4187 y Ft(\017)60 b Fp(plan)35 b Fv(is)g(the)i(plan)e(created)i(b)m(y)f
 
5449
Fp(fftw_create_plan)c Fv(\(see)38 b(Section)e(3.2.1)i([Plan)d(Creation)
 
5450
h(for)330 4297 y(One-dimensional)27 b(T)-8 b(ransforms],)30
 
5451
b(page)h(18\).)225 4435 y Ft(\017)60 b Fp(howmany)30
 
5452
b Fv(is)h(the)h(n)m(um)m(b)s(er)e(of)i(transforms)f Fp(fftw)g
 
5453
Fv(will)e(compute.)46 b(It)32 b(is)f(faster)h(to)h(tell)e(FFTW)h(to)330
 
5454
4544 y(compute)f(man)m(y)f(transforms,)g(instead)g(of)g(simply)e
 
5455
(calling)h Fp(fftw)g Fv(man)m(y)i(times.)225 4682 y Ft(\017)60
 
5456
b Fp(in)p Fv(,)33 b Fp(istride)e Fv(and)i Fp(idist)e
 
5457
Fv(describ)s(e)h(the)h(input)e(arra)m(y\(s\).)50 b(There)33
 
5458
b(are)g Fp(howmany)e Fv(input)g(arra)m(ys;)330 4792 y(the)26
 
5459
b(\014rst)f(one)h(is)f(p)s(oin)m(ted)g(to)h(b)m(y)g Fp(in)p
 
5460
Fv(,)g(the)g(second)g(one)g(is)f(p)s(oin)m(ted)g(to)h(b)m(y)g
 
5461
Fp(in)k(+)g(idist)p Fv(,)c(and)f(so)h(on,)330 4902 y(up)f(to)h
 
5462
Fp(in)k(+)g(\(howmany)e(-)i(1\))g(*)g(idist)p Fv(.)37
5414
5463
b(Eac)m(h)27 b(input)c(arra)m(y)k(consists)e(of)g(complex)h(n)m(um)m(b)
5415
 
s(ers)e(\(see)330 5340 y(Section)35 b(3.1)i([Data)g(T)m(yp)s(es],)g
 
5464
s(ers)e(\(see)330 5011 y(Section)35 b(3.1)i([Data)g(T)m(yp)s(es],)g
5416
5465
(page)g(17\),)h(whic)m(h)c(are)i(not)g(necessarily)e(con)m(tiguous)i
5417
 
(in)e(memory)-8 b(.)p eop
 
5466
(in)e(memory)-8 b(.)330 5121 y(Sp)s(eci\014cally)g(,)39
 
5467
b Fp(in[0])f Fv(is)g(the)h(\014rst)g(elemen)m(t)g(of)g(the)h(\014rst)e
 
5468
(arra)m(y)-8 b(,)42 b Fp(in[istride])36 b Fv(is)j(the)g(second)330
 
5469
5230 y(elemen)m(t)f(of)f(the)g(\014rst)g(arra)m(y)-8
 
5470
b(,)40 b(and)c(so)i(on.)61 b(In)36 b(general,)j(the)f
 
5471
Fp(i)p Fv(-th)f(elemen)m(t)g(of)h(the)f Fp(j)p Fv(-th)g(input)330
 
5472
5340 y(arra)m(y)31 b(will)d(b)s(e)h(in)g(p)s(osition)g
 
5473
Fp(in[i)g(*)h(istride)e(+)j(j)f(*)g(idist])p Fv(.)p eop
 
5474
end
5418
5475
%%Page: 21 23
5419
 
21 22 bop 150 -116 a Fu(Chapter)30 b(3:)41 b(FFTW)31
5420
 
b(Reference)2354 b(21)330 299 y(Sp)s(eci\014cally)-8
5421
 
b(,)39 b Fo(in[0])f Fu(is)g(the)h(\014rst)g(elemen)m(t)g(of)g(the)h
5422
 
(\014rst)e(arra)m(y)-8 b(,)42 b Fo(in[istride])36 b Fu(is)j(the)g
5423
 
(second)330 408 y(elemen)m(t)f(of)f(the)g(\014rst)g(arra)m(y)-8
5424
 
b(,)40 b(and)c(so)i(on.)61 b(In)36 b(general,)j(the)f
5425
 
Fo(i)p Fu(-th)f(elemen)m(t)g(of)h(the)f Fo(j)p Fu(-th)g(input)330
5426
 
518 y(arra)m(y)31 b(will)d(b)s(e)h(in)g(p)s(osition)g
5427
 
Fo(in[i)g(*)h(istride)e(+)j(j)f(*)g(idist])p Fu(.)225
5428
 
652 y Fs(\017)60 b Fo(out)p Fu(,)33 b Fo(ostride)f Fu(and)g
5429
 
Fo(odist)g Fu(describ)s(e)g(the)h(output)g(arra)m(y\(s\).)50
5430
 
b(The)33 b(format)g(is)g(the)g(same)h(as)f(for)330 762
5431
 
y(the)e(input)d(arra)m(y)-8 b(.)379 895 y Fs(\000)60
5432
 
b Fq(In-plac)-5 b(e)46 b(tr)-5 b(ansforms)7 b Fu(:)73
5433
 
b(If)44 b(the)h Fo(plan)e Fu(sp)s(eci\014es)g(an)i(in-place)e
5434
 
(transform,)48 b Fo(ostride)43 b Fu(and)510 1005 y Fo(odist)33
5435
 
b Fu(are)h(alw)m(a)m(ys)h(ignored.)51 b(If)33 b Fo(out)g
5436
 
Fu(is)h Fo(NULL)p Fu(,)g Fo(out)f Fu(is)g(ignored,)h(to)s(o.)53
5437
 
b(Otherwise,)34 b Fo(out)f Fu(is)510 1115 y(in)m(terpreted)h(as)h(a)g
5438
 
(p)s(oin)m(ter)f(to)i(an)e(arra)m(y)i(of)f Fo(n)f Fu(complex)h(n)m(um)m
5439
 
(b)s(ers,)f(that)i(FFTW)f(will)d(use)510 1224 y(as)j(temp)s(orary)f
5440
 
(space)h(to)h(p)s(erform)d(the)i(in-place)e(computation.)54
5441
 
b Fo(out)34 b Fu(is)f(used)h(as)h(scratc)m(h)510 1334
5442
 
y(space)41 b(and)g(its)f(con)m(ten)m(ts)j(destro)m(y)m(ed.)73
5443
 
b(In)40 b(this)g(case,)k Fo(out)c Fu(m)m(ust)h(b)s(e)f(an)h(ordinary)e
5444
 
(arra)m(y)510 1443 y(whose)30 b(elemen)m(ts)h(are)g(con)m(tiguous)f(in)
5445
 
f(memory)i(\(no)f(striding\).)275 1602 y(The)45 b(function)f
5446
 
Fo(fftw_one)f Fu(transforms)i(a)h(single,)i(con)m(tiguous)e(input)d
5447
 
(arra)m(y)k(to)f(a)g(con)m(tiguous)150 1711 y(output)30
5448
 
b(arra)m(y)-8 b(.)42 b(By)30 b(de\014nition,)f(the)h(call)390
5449
 
1839 y Fo(fftw_one\(plan,)44 b(in,)j(out\))275 1973 y
5450
 
Fu(is)29 b(equiv)-5 b(alen)m(t)30 b(to)390 2101 y Fo(fftw\(plan,)45
5451
 
b(1,)i(in,)g(1,)g(1,)g(out,)g(1,)g(1\))150 2324 y Fi(3.2.4)63
5452
 
b(Destro)m(ying)41 b(a)g(One-dimensional)g(Plan)390 2510
5453
 
y Fo(#include)46 b(<fftw.h>)390 2718 y(void)h(fftw_destroy_plan\(fftw_)
5454
 
o(plan)41 b(plan\);)275 2852 y Fu(The)27 b(function)f
5455
 
Fo(fftw_destroy_plan)d Fu(frees)28 b(the)f(plan)g Fo(plan)f
5456
 
Fu(and)h(releases)h(all)f(the)h(memory)f(asso-)150 2961
5457
 
y(ciated)k(with)e(it.)40 b(After)31 b(destruction,)e(a)i(plan)e(is)h
5458
 
(no)g(longer)g(v)-5 b(alid.)150 3184 y Fi(3.2.5)63 b(What)40
5459
 
b(FFTW)h(Really)f(Computes)275 3376 y Fu(In)26 b(this)g(section,)i(w)m
5460
 
(e)g(de\014ne)e(precisely)g(what)h(FFTW)h(computes.)40
5461
 
b(Please)27 b(b)s(e)f(w)m(arned)h(that)h(di\013er-)150
5462
 
3485 y(en)m(t)j(authors)f(and)g(soft)m(w)m(are)i(pac)m(k)-5
 
5476
TeXDict begin 21 22 bop 150 -116 a Fv(Chapter)30 b(3:)41
 
5477
b(FFTW)31 b(Reference)2354 b(21)225 299 y Ft(\017)60
 
5478
b Fp(out)p Fv(,)33 b Fp(ostride)f Fv(and)g Fp(odist)g
 
5479
Fv(describ)s(e)g(the)h(output)g(arra)m(y\(s\).)50 b(The)33
 
5480
b(format)g(is)g(the)g(same)h(as)f(for)330 408 y(the)e(input)d(arra)m(y)
 
5481
-8 b(.)379 540 y Ft(\000)60 b Fr(In-plac)-5 b(e)46 b(tr)-5
 
5482
b(ansforms)7 b Fv(:)73 b(If)44 b(the)h Fp(plan)e Fv(sp)s(eci\014es)g
 
5483
(an)i(in-place)e(transform,)48 b Fp(ostride)43 b Fv(and)510
 
5484
649 y Fp(odist)33 b Fv(are)h(alw)m(a)m(ys)h(ignored.)51
 
5485
b(If)33 b Fp(out)g Fv(is)h Fp(NULL)p Fv(,)g Fp(out)f
 
5486
Fv(is)g(ignored,)h(to)s(o.)53 b(Otherwise,)34 b Fp(out)f
 
5487
Fv(is)510 759 y(in)m(terpreted)h(as)h(a)g(p)s(oin)m(ter)f(to)i(an)e
 
5488
(arra)m(y)i(of)f Fp(n)f Fv(complex)h(n)m(um)m(b)s(ers,)f(that)i(FFTW)f
 
5489
(will)d(use)510 868 y(as)j(temp)s(orary)f(space)h(to)h(p)s(erform)d
 
5490
(the)i(in-place)e(computation.)54 b Fp(out)34 b Fv(is)f(used)h(as)h
 
5491
(scratc)m(h)510 978 y(space)41 b(and)g(its)f(con)m(ten)m(ts)j(destro)m
 
5492
(y)m(ed.)73 b(In)40 b(this)g(case,)k Fp(out)c Fv(m)m(ust)h(b)s(e)f(an)h
 
5493
(ordinary)e(arra)m(y)510 1088 y(whose)30 b(elemen)m(ts)h(are)g(con)m
 
5494
(tiguous)f(in)f(memory)i(\(no)f(striding\).)275 1240
 
5495
y(The)45 b(function)f Fp(fftw_one)f Fv(transforms)i(a)h(single,)i(con)m
 
5496
(tiguous)e(input)d(arra)m(y)k(to)f(a)g(con)m(tiguous)150
 
5497
1350 y(output)30 b(arra)m(y)-8 b(.)42 b(By)30 b(de\014nition,)f(the)h
 
5498
(call)390 1475 y Fp(fftw_one\(plan,)44 b(in,)j(out\))275
 
5499
1607 y Fv(is)29 b(equiv)-5 b(alen)m(t)30 b(to)390 1732
 
5500
y Fp(fftw\(plan,)45 b(1,)i(in,)g(1,)g(0,)g(out,)g(1,)g(0\))150
 
5501
1946 y Fh(3.2.4)63 b(Destro)m(ying)41 b(a)g(One-dimensional)g(Plan)390
 
5502
2108 y Fp(#include)46 b(<fftw.h>)390 2316 y(void)h
 
5503
(fftw_destroy_plan\(fftw_)o(plan)41 b(plan\);)275 2447
 
5504
y Fv(The)27 b(function)f Fp(fftw_destroy_plan)d Fv(frees)28
 
5505
b(the)f(plan)g Fp(plan)f Fv(and)h(releases)h(all)f(the)h(memory)f
 
5506
(asso-)150 2557 y(ciated)k(with)e(it.)40 b(After)31 b(destruction,)e(a)
 
5507
i(plan)e(is)h(no)g(longer)g(v)-5 b(alid.)150 2771 y Fh(3.2.5)63
 
5508
b(What)40 b(FFTW)h(Really)f(Computes)275 2960 y Fv(In)26
 
5509
b(this)g(section,)i(w)m(e)g(de\014ne)e(precisely)g(what)h(FFTW)h
 
5510
(computes.)40 b(Please)27 b(b)s(e)f(w)m(arned)h(that)h(di\013er-)150
 
5511
3070 y(en)m(t)j(authors)f(and)g(soft)m(w)m(are)i(pac)m(k)-5
5463
5512
b(ages)32 b(migh)m(t)e(emplo)m(y)g(di\013eren)m(t)g(con)m(v)m(en)m
5464
 
(tions)h(than)g(FFTW)g(do)s(es.)275 3619 y(The)e(forw)m(ard)h
5465
 
(transform)g(of)g(a)h(complex)f(arra)m(y)h Fl(X)38 b
5466
 
Fu(of)30 b(size)h Fl(n)f Fu(computes)g(an)g(arra)m(y)h
5467
 
Fl(Y)20 b Fu(,)31 b(where)1475 3892 y Fl(Y)1528 3906
5468
 
y Fk(i)1581 3892 y Fu(=)1677 3786 y Fk(n)p Fj(\000)p
5469
 
Fn(1)1680 3811 y Fh(X)1682 3988 y Fk(j)s Fn(=0)1818 3892
5470
 
y Fl(X)1893 3906 y Fk(j)1928 3892 y Fl(e)1970 3854 y
5471
 
Fj(\000)p Fn(2)p Fk(\031)r(ij)2149 3809 y Fj(p)p 2205
5472
 
3809 85 3 v 2205 3854 a(\000)p Fn(1)p Fk(=n)2400 3892
5473
 
y Fl(:)275 4198 y Fu(The)e(bac)m(kw)m(ard)i(transform)f(computes)1501
5474
 
4471 y Fl(Y)1554 4485 y Fk(i)1607 4471 y Fu(=)1703 4365
5475
 
y Fk(n)p Fj(\000)p Fn(1)1706 4390 y Fh(X)1708 4567 y
5476
 
Fk(j)s Fn(=0)1844 4471 y Fl(X)1919 4485 y Fk(j)1954 4471
5477
 
y Fl(e)1996 4434 y Fn(2)p Fk(\031)r(ij)2123 4389 y Fj(p)p
5478
 
2179 4389 V 2179 4434 a(\000)p Fn(1)p Fk(=n)2374 4471
5479
 
y Fl(:)275 4877 y Fu(FFTW)h(computes)g(an)f(unnormalized)f(transform,)h
5480
 
(that)h(is,)f(the)h(equation)g Fl(I)7 b(F)13 b(F)g(T)g
5481
 
Fu(\()p Fl(F)g(F)g(T)g Fu(\()p Fl(X)7 b Fu(\)\))27 b(=)150
5482
 
4987 y Fl(nX)47 b Fu(holds.)70 b(In)40 b(other)g(w)m(ords,)j(applying)c
 
5513
(tions)h(than)g(FFTW)g(do)s(es.)275 3201 y(The)e(forw)m(ard)h
 
5514
(transform)g(of)g(a)h(complex)f(arra)m(y)h Fm(X)38 b
 
5515
Fv(of)30 b(size)h Fm(n)f Fv(computes)g(an)g(arra)m(y)h
 
5516
Fm(Y)20 b Fv(,)31 b(where)1475 3449 y Fm(Y)1528 3463
 
5517
y Fl(i)1581 3449 y Fv(=)1677 3344 y Fl(n)p Fk(\000)p
 
5518
Fo(1)1680 3369 y Fg(X)1682 3545 y Fl(j)s Fo(=0)1818 3449
 
5519
y Fm(X)1893 3463 y Fl(j)1928 3449 y Fm(e)1970 3412 y
 
5520
Fk(\000)p Fo(2)p Fl(\031)r(ij)2149 3367 y Fk(p)p 2205
 
5521
3367 85 3 v 2205 3412 a(\000)p Fo(1)p Fl(=n)2400 3449
 
5522
y Fm(:)275 3729 y Fv(The)e(bac)m(kw)m(ard)i(transform)f(computes)1501
 
5523
3977 y Fm(Y)1554 3991 y Fl(i)1607 3977 y Fv(=)1703 3872
 
5524
y Fl(n)p Fk(\000)p Fo(1)1706 3897 y Fg(X)1708 4073 y
 
5525
Fl(j)s Fo(=0)1844 3977 y Fm(X)1919 3991 y Fl(j)1954 3977
 
5526
y Fm(e)1996 3940 y Fo(2)p Fl(\031)r(ij)2123 3895 y Fk(p)p
 
5527
2179 3895 V 2179 3940 a(\000)p Fo(1)p Fl(=n)2374 3977
 
5528
y Fm(:)275 4332 y Fv(FFTW)h(computes)g(an)f(unnormalized)f(transform,)h
 
5529
(that)h(is,)f(the)h(equation)g Fm(I)7 b(F)13 b(F)g(T)g
 
5530
Fv(\()p Fm(F)g(F)g(T)g Fv(\()p Fm(X)7 b Fv(\)\))27 b(=)150
 
5531
4442 y Fm(nX)47 b Fv(holds.)70 b(In)40 b(other)g(w)m(ords,)j(applying)c
5483
5532
(the)h(forw)m(ard)g(and)g(then)g(the)h(bac)m(kw)m(ard)g(transform)f
5484
 
(will)150 5096 y(m)m(ultiply)28 b(the)i(input)f(b)m(y)h
5485
 
Fl(n)p Fu(.)275 5230 y(An)j Fo(FFTW_FORWARD)e Fu(transform)i(corresp)s
5486
 
(onds)f(to)j(a)f(sign)f(of)h Fs(\000)p Fu(1)g(in)e(the)i(exp)s(onen)m
5487
 
(t)g(of)g(the)g(DFT.)150 5340 y(Note)23 b(also)g(that)f(w)m(e)h(use)f
 
5533
(will)150 4551 y(m)m(ultiply)28 b(the)i(input)f(b)m(y)h
 
5534
Fm(n)p Fv(.)275 4682 y(An)j Fp(FFTW_FORWARD)e Fv(transform)i(corresp)s
 
5535
(onds)f(to)j(a)f(sign)f(of)h Ft(\000)p Fv(1)g(in)e(the)i(exp)s(onen)m
 
5536
(t)g(of)g(the)g(DFT.)150 4792 y(Note)23 b(also)g(that)f(w)m(e)h(use)f
5488
5537
(the)g(standard)g(\\in-order")f(output)h(ordering|the)f
5489
 
Fl(k)s Fu(-th)i(output)e(corresp)s(onds)p eop
5490
 
%%Page: 22 24
5491
 
22 23 bop 150 -116 a Fu(22)3232 b(FFTW)150 299 y(to)26
5492
 
b(the)g(frequency)f Fl(k)s(=n)h Fu(\(or)g Fl(k)s(=T)13
5493
 
b Fu(,)27 b(where)e Fl(T)38 b Fu(is)25 b(y)m(our)h(total)g(sampling)e
 
5538
Fm(k)s Fv(-th)i(output)e(corresp)s(onds)150 4902 y(to)26
 
5539
b(the)g(frequency)f Fm(k)s(=n)h Fv(\(or)g Fm(k)s(=T)13
 
5540
b Fv(,)27 b(where)e Fm(T)38 b Fv(is)25 b(y)m(our)h(total)g(sampling)e
5494
5541
(p)s(erio)s(d\).)37 b(F)-8 b(or)26 b(those)g(who)f(lik)m(e)g(to)150
5495
 
408 y(think)g(in)h(terms)h(of)f(p)s(ositiv)m(e)g(and)g(negativ)m(e)i
 
5542
5011 y(think)g(in)h(terms)h(of)f(p)s(ositiv)m(e)g(and)g(negativ)m(e)i
5496
5543
(frequencies,)f(this)f(means)h(that)g(the)g(p)s(ositiv)m(e)f
5497
 
(frequencies)150 518 y(are)e(stored)f(in)f(the)h(\014rst)g(half)f(of)i
 
5544
(frequencies)150 5121 y(are)e(stored)f(in)f(the)h(\014rst)g(half)f(of)i
5498
5545
(the)f(output)g(and)f(the)i(negativ)m(e)g(frequencies)f(are)g(stored)h
5499
 
(in)e(bac)m(kw)m(ards)150 628 y(order)31 b(in)f(the)i(second)g(half)e
5500
 
(of)i(the)g(output.)44 b(\(The)32 b(frequency)f Fs(\000)p
5501
 
Fl(k)s(=n)g Fu(is)g(the)h(same)g(as)g(the)f(frequency)150
5502
 
737 y(\()p Fl(n)20 b Fs(\000)g Fl(k)s Fu(\))p Fl(=n)p
5503
 
Fu(.\))150 1042 y Ft(3.3)68 b(Multi-dimensional)46 b(T)-11
5504
 
b(ransforms)45 b(Reference)275 1253 y Fu(The)26 b(m)m(ulti-dimensional)
5505
 
d(complex)j(routines)g(are)h(generally)f(pre\014xed)g(with)f
5506
 
Fo(fftwnd_)p Fu(.)38 b(Programs)150 1363 y(using)d(FFTWND)i(should)d(b)
5507
 
s(e)h(link)m(ed)g(with)g Fo(-lfftw)28 b(-lm)35 b Fu(on)h(Unix)f
5508
 
(systems,)j(or)e(with)f(the)h(FFTW)150 1473 y(and)30
5509
 
b(standard)f(math)i(libraries)c(in)i(general.)150 1735
5510
 
y Fi(3.3.1)63 b(Plan)40 b(Creation)h(for)g(Multi-dimensional)h(T)-10
5511
 
b(ransforms)390 1940 y Fo(#include)46 b(<fftw.h>)390
5512
 
2148 y(fftwnd_plan)f(fftwnd_create_plan\(int)c(rank,)47
5513
 
b(const)f(int)h(*n,)1870 2252 y(fftw_direction)c(dir,)k(int)g(flags\);)
5514
 
390 2459 y(fftwnd_plan)e(fftw2d_create_plan\(int)c(nx,)47
5515
 
b(int)g(ny,)1870 2563 y(fftw_direction)c(dir,)k(int)g(flags\);)390
5516
 
2771 y(fftwnd_plan)e(fftw3d_create_plan\(int)c(nx,)47
5517
 
b(int)g(ny,)g(int)g(nz,)1870 2874 y(fftw_direction)c(dir,)k(int)g
5518
 
(flags\);)390 3082 y(fftwnd_plan)e(fftwnd_create_plan_spec)o(ific)o
5519
 
(\(int)c(rank,)46 b(const)h(int)g(*n,)2299 3186 y(fftw_direction)d
5520
 
(dir,)2299 3289 y(int)j(flags,)2299 3393 y(fftw_complex)e(*in,)h(int)h
5521
 
(istride,)2299 3497 y(fftw_complex)e(*out,)h(int)h(ostride\);)390
5522
 
3705 y(fftwnd_plan)e(fftw2d_create_plan_spec)o(ific)o(\(int)c(nx,)47
5523
 
b(int)g(ny,)2299 3808 y(fftw_direction)d(dir,)2299 3912
5524
 
y(int)j(flags,)2299 4016 y(fftw_complex)e(*in,)h(int)h(istride,)2299
5525
 
4120 y(fftw_complex)e(*out,)h(int)h(ostride\);)390 4327
 
5546
(in)e(bac)m(kw)m(ards)150 5230 y(order)31 b(in)f(the)i(second)g(half)e
 
5547
(of)i(the)g(output.)44 b(\(The)32 b(frequency)f Ft(\000)p
 
5548
Fm(k)s(=n)g Fv(is)g(the)h(same)g(as)g(the)f(frequency)150
 
5549
5340 y(\()p Fm(n)20 b Ft(\000)g Fm(k)s Fv(\))p Fm(=n)p
 
5550
Fv(.\))p eop end
 
5551
%%Page: 22 24
 
5552
TeXDict begin 22 23 bop 150 -116 a Fv(22)3232 b(FFTW)150
 
5553
299 y Fu(3.3)68 b(Multi-dimensional)46 b(T)-11 b(ransforms)45
 
5554
b(Reference)275 499 y Fv(The)26 b(m)m(ulti-dimensional)d(complex)j
 
5555
(routines)g(are)h(generally)f(pre\014xed)g(with)f Fp(fftwnd_)p
 
5556
Fv(.)38 b(Programs)150 609 y(using)d(FFTWND)i(should)d(b)s(e)h(link)m
 
5557
(ed)g(with)g Fp(-lfftw)28 b(-lm)35 b Fv(on)h(Unix)f(systems,)j(or)e
 
5558
(with)f(the)h(FFTW)150 718 y(and)30 b(standard)f(math)i(libraries)c(in)
 
5559
i(general.)150 958 y Fh(3.3.1)63 b(Plan)40 b(Creation)h(for)g
 
5560
(Multi-dimensional)h(T)-10 b(ransforms)390 1120 y Fp(#include)46
 
5561
b(<fftw.h>)390 1327 y(fftwnd_plan)f(fftwnd_create_plan\(int)c(rank,)47
 
5562
b(const)f(int)h(*n,)1870 1431 y(fftw_direction)c(dir,)k(int)g(flags\);)
 
5563
390 1638 y(fftwnd_plan)e(fftw2d_create_plan\(int)c(nx,)47
 
5564
b(int)g(ny,)1870 1742 y(fftw_direction)c(dir,)k(int)g(flags\);)390
 
5565
1950 y(fftwnd_plan)e(fftw3d_create_plan\(int)c(nx,)47
 
5566
b(int)g(ny,)g(int)g(nz,)1870 2054 y(fftw_direction)c(dir,)k(int)g
 
5567
(flags\);)390 2261 y(fftwnd_plan)e(fftwnd_create_plan_spec)o(ific)o
 
5568
(\(int)c(rank,)46 b(const)h(int)g(*n,)2299 2365 y(fftw_direction)d
 
5569
(dir,)2299 2469 y(int)j(flags,)2299 2572 y(fftw_complex)e(*in,)h(int)h
 
5570
(istride,)2299 2676 y(fftw_complex)e(*out,)h(int)h(ostride\);)390
 
5571
2884 y(fftwnd_plan)e(fftw2d_create_plan_spec)o(ific)o(\(int)c(nx,)47
 
5572
b(int)g(ny,)2299 2988 y(fftw_direction)d(dir,)2299 3091
 
5573
y(int)j(flags,)2299 3195 y(fftw_complex)e(*in,)h(int)h(istride,)2299
 
5574
3299 y(fftw_complex)e(*out,)h(int)h(ostride\);)390 3506
5526
5575
y(fftwnd_plan)e(fftw3d_create_plan_spec)o(ific)o(\(int)c(nx,)47
5527
 
b(int)g(ny,)g(int)g(nz,)2299 4431 y(fftw_direction)d(dir,)j(int)g
5528
 
(flags,)2299 4535 y(fftw_complex)e(*in,)h(int)h(istride,)2299
5529
 
4639 y(fftw_complex)e(*out,)h(int)h(ostride\);)275 4792
5530
 
y Fu(The)29 b(function)h Fo(fftwnd_create_plan)25 b Fu(creates)32
 
5576
b(int)g(ny,)g(int)g(nz,)2299 3610 y(fftw_direction)d(dir,)j(int)g
 
5577
(flags,)2299 3714 y(fftw_complex)e(*in,)h(int)h(istride,)2299
 
5578
3818 y(fftw_complex)e(*out,)h(int)h(ostride\);)275 3960
 
5579
y Fv(The)29 b(function)h Fp(fftwnd_create_plan)25 b Fv(creates)32
5531
5580
b(a)f(plan,)e(whic)m(h)g(is)h(a)g(data)h(structure)f(con)m(taining)150
5532
 
4902 y(all)41 b(the)h(information)e(that)j Fo(fftwnd)d
5533
 
Fu(needs)h(in)g(order)g(to)i(compute)f(a)g(m)m(ulti-dimensional)c(F)-8
5534
 
b(ourier)150 5011 y(transform.)42 b(Y)-8 b(ou)32 b(can)f(create)i(as)e
 
5581
4069 y(all)41 b(the)h(information)e(that)j Fp(fftwnd)d
 
5582
Fv(needs)h(in)g(order)g(to)i(compute)f(a)g(m)m(ulti-dimensional)c(F)-8
 
5583
b(ourier)150 4179 y(transform.)42 b(Y)-8 b(ou)32 b(can)f(create)i(as)e
5535
5584
(man)m(y)g(plans)f(as)h(y)m(ou)g(need,)h(but)e(only)g(one)h(plan)f(for)
5536
 
h(a)g(giv)m(en)g(arra)m(y)150 5121 y(size)37 b(is)f(required)f(\(a)j
 
5585
h(a)g(giv)m(en)g(arra)m(y)150 4289 y(size)37 b(is)f(required)f(\(a)j
5537
5586
(plan)e(can)h(b)s(e)g(reused)f(man)m(y)h(times\).)61
5538
 
b(The)36 b(functions)g Fo(fftw2d_create_plan)150 5230
5539
 
y Fu(and)e Fo(fftw3d_create_plan)29 b Fu(are)35 b(optional,)g
5540
 
(alternativ)m(e)g(in)m(terfaces)g(to)h Fo(fftwnd_create_plan)29
5541
 
b Fu(for)150 5340 y(t)m(w)m(o)j(and)d(three)i(dimensions,)d(resp)s
5542
 
(ectiv)m(ely)-8 b(.)p eop
5543
 
%%Page: 23 25
5544
 
23 24 bop 150 -116 a Fu(Chapter)30 b(3:)41 b(FFTW)31
5545
 
b(Reference)2354 b(23)275 299 y Fo(fftwnd_create_plan)21
5546
 
b Fu(returns)26 b(a)h(v)-5 b(alid)25 b(plan,)i(or)g Fo(NULL)f
5547
 
Fu(if,)g(for)h(some)g(reason,)h(the)f(plan)f(can't)i(b)s(e)150
5548
 
408 y(created.)46 b(This)30 b(can)i(happ)s(en)e(if)h(memory)h(runs)e
 
5587
b(The)36 b(functions)g Fp(fftw2d_create_plan)150 4398
 
5588
y Fv(and)e Fp(fftw3d_create_plan)29 b Fv(are)35 b(optional,)g
 
5589
(alternativ)m(e)g(in)m(terfaces)g(to)h Fp(fftwnd_create_plan)29
 
5590
b Fv(for)150 4508 y(t)m(w)m(o)j(and)d(three)i(dimensions,)d(resp)s
 
5591
(ectiv)m(ely)-8 b(.)275 4650 y Fp(fftwnd_create_plan)21
 
5592
b Fv(returns)26 b(a)h(v)-5 b(alid)25 b(plan,)i(or)g Fp(NULL)f
 
5593
Fv(if,)g(for)h(some)g(reason,)h(the)f(plan)f(can't)i(b)s(e)150
 
5594
4760 y(created.)46 b(This)30 b(can)i(happ)s(en)e(if)h(memory)h(runs)e
5549
5595
(out)i(or)g(if)f(the)h(argumen)m(ts)g(are)g(in)m(v)-5
5550
 
b(alid)29 b(in)i(some)h(w)m(a)m(y)150 518 y(\(e.g.)42
5551
 
b(if)29 b Fo(rank)h(<)g Fu(0\).)275 654 y(The)22 b Fo
5552
 
(create_plan_specific)17 b Fu(v)-5 b(arian)m(ts)22 b(tak)m(e)i(as)f
 
5596
b(alid)29 b(in)i(some)h(w)m(a)m(y)150 4869 y(\(e.g.)42
 
5597
b(if)29 b Fp(rank)h(<)g Fv(0\).)275 5011 y(The)22 b Fp
 
5598
(create_plan_specific)17 b Fv(v)-5 b(arian)m(ts)22 b(tak)m(e)i(as)f
5553
5599
(additional)e(argumen)m(ts)h(sp)s(eci\014c)g(input/output)150
5554
 
763 y(arra)m(ys)k(and)f(their)g(strides.)38 b(F)-8 b(or)26
 
5600
5121 y(arra)m(ys)k(and)f(their)g(strides.)38 b(F)-8 b(or)26
5555
5601
b(the)g(last)g(four)f(argumen)m(ts,)i(y)m(ou)f(should)e(pass)h(the)h
5556
 
(arra)m(ys)g(and)f(strides)150 873 y(that)37 b(y)m(ou)g(will)c(ev)m(en)
5557
 
m(tually)k(b)s(e)f(passing)f(to)i Fo(fftwnd)p Fu(.)57
 
5602
(arra)m(ys)g(and)f(strides)150 5230 y(that)37 b(y)m(ou)g(will)c(ev)m
 
5603
(en)m(tually)k(b)s(e)f(passing)f(to)i Fp(fftwnd)p Fv(.)57
5558
5604
b(The)36 b(resulting)f(plans)f(will)g(b)s(e)i(optimized)f(for)150
5559
 
982 y(those)j(arra)m(ys)g(and)g(strides,)g(although)g(they)f(ma)m(y)i
 
5605
5340 y(those)j(arra)m(ys)g(and)g(strides,)g(although)g(they)f(ma)m(y)i
5560
5606
(b)s(e)e(used)g(on)h(other)g(arra)m(ys)g(as)g(w)m(ell.)62
5561
 
b(Note:)57 b(the)150 1092 y(con)m(ten)m(ts)32 b(of)e(the)h(in)d(and)i
5562
 
(out)g(arra)m(ys)h(are)f Fq(destr)-5 b(oye)g(d)42 b Fu(b)m(y)30
5563
 
b(the)h(sp)s(eci\014c)e(planner)f(\(the)j(initial)c(con)m(ten)m(ts)150
5564
 
1202 y(are)k(ignored,)f(so)h(the)g(arra)m(ys)h(need)e(not)h(ha)m(v)m(e)
5565
 
h(b)s(een)e(initialized\).)39 b(See)31 b(Section)g(3.2.2)i([Discussion)
5566
 
c(on)150 1311 y(Sp)s(eci\014c)g(Plans],)h(page)h(20,)g(for)f(a)h
5567
 
(discussion)d(on)i(sp)s(eci\014c)f(plans.)150 1537 y
5568
 
Fi(Argumen)m(ts)225 1731 y Fs(\017)60 b Fo(rank)27 b
5569
 
Fu(is)g(the)h(dimensionalit)m(y)d(of)j(the)g(arra)m(ys)g(to)g(b)s(e)f
5570
 
(transformed.)40 b(It)27 b(can)i(b)s(e)e(an)m(y)h(non-negativ)m(e)330
5571
 
1841 y(in)m(teger.)225 1976 y Fs(\017)60 b Fo(n)38 b
5572
 
Fu(is)f(a)h(p)s(oin)m(ter)f(to)i(an)f(arra)m(y)h(of)f
5573
 
Fo(rank)f Fu(in)m(tegers,)j(giving)d(the)i(size)f(of)g(eac)m(h)h
5574
 
(dimension)c(of)k(the)330 2085 y(arra)m(ys)h(to)h(b)s(e)f(transformed.)
5575
 
69 b(These)40 b(sizes,)i(whic)m(h)d(m)m(ust)h(b)s(e)f(p)s(ositiv)m(e)g
5576
 
(in)m(tegers,)k(corresp)s(ond)330 2195 y(to)37 b(the)f(dimensions)d(of)
5577
 
j(ro)m(w-ma)5 b(jor)36 b(arra)m(ys|i.e.)58 b Fo(n[0])35
5578
 
b Fu(is)g(the)h(size)g(of)g(the)g(dimension)d(whose)330
5579
 
2304 y(indices)21 b(v)-5 b(ary)23 b(most)g(slo)m(wly)-8
 
5607
b(Note:)57 b(the)p eop end
 
5608
%%Page: 23 25
 
5609
TeXDict begin 23 24 bop 150 -116 a Fv(Chapter)30 b(3:)41
 
5610
b(FFTW)31 b(Reference)2354 b(23)150 299 y(con)m(ten)m(ts)32
 
5611
b(of)e(the)h(in)d(and)i(out)g(arra)m(ys)h(are)f Fr(destr)-5
 
5612
b(oye)g(d)42 b Fv(b)m(y)30 b(the)h(sp)s(eci\014c)e(planner)f(\(the)j
 
5613
(initial)c(con)m(ten)m(ts)150 408 y(are)k(ignored,)f(so)h(the)g(arra)m
 
5614
(ys)h(need)e(not)h(ha)m(v)m(e)h(b)s(een)e(initialized\).)39
 
5615
b(See)31 b(Section)g(3.2.2)i([Discussion)c(on)150 518
 
5616
y(Sp)s(eci\014c)g(Plans],)h(page)h(20,)g(for)f(a)h(discussion)d(on)i
 
5617
(sp)s(eci\014c)f(plans.)150 736 y Fh(Argumen)m(ts)225
 
5618
903 y Ft(\017)60 b Fp(rank)27 b Fv(is)g(the)h(dimensionalit)m(y)d(of)j
 
5619
(the)g(arra)m(ys)g(to)g(b)s(e)f(transformed.)40 b(It)27
 
5620
b(can)i(b)s(e)e(an)m(y)h(non-negativ)m(e)330 1013 y(in)m(teger.)225
 
5621
1145 y Ft(\017)60 b Fp(n)38 b Fv(is)f(a)h(p)s(oin)m(ter)f(to)i(an)f
 
5622
(arra)m(y)h(of)f Fp(rank)f Fv(in)m(tegers,)j(giving)d(the)i(size)f(of)g
 
5623
(eac)m(h)h(dimension)c(of)k(the)330 1255 y(arra)m(ys)h(to)h(b)s(e)f
 
5624
(transformed.)69 b(These)40 b(sizes,)i(whic)m(h)d(m)m(ust)h(b)s(e)f(p)s
 
5625
(ositiv)m(e)g(in)m(tegers,)k(corresp)s(ond)330 1365 y(to)37
 
5626
b(the)f(dimensions)d(of)j(ro)m(w-ma)5 b(jor)36 b(arra)m(ys|i.e.)58
 
5627
b Fp(n[0])35 b Fv(is)g(the)h(size)g(of)g(the)g(dimension)d(whose)330
 
5628
1474 y(indices)21 b(v)-5 b(ary)23 b(most)g(slo)m(wly)-8
5580
5629
b(,)24 b(and)f(so)g(on.)38 b(\(See)24 b(Section)e(2.5)j
5581
5630
([Multi-dimensional)19 b(Arra)m(y)k(F)-8 b(ormat],)330
5582
 
2414 y(page)29 b(11,)g(for)f(more)g(information)e(on)i(ro)m(w-ma)5
 
5631
1584 y(page)29 b(11,)g(for)f(more)g(information)e(on)i(ro)m(w-ma)5
5583
5632
b(jor)28 b(storage.\))42 b(See)28 b(Section)g(3.2.1)h([Plan)e(Creation)
5584
 
330 2524 y(for)44 b(One-dimensional)e(T)-8 b(ransforms],)48
 
5633
330 1693 y(for)44 b(One-dimensional)e(T)-8 b(ransforms],)48
5585
5634
b(page)d(18,)50 b(for)44 b(more)h(information)e(regarding)h(optimal)330
5586
 
2633 y(arra)m(y)31 b(sizes.)225 2768 y Fs(\017)60 b Fo(nx)35
5587
 
b Fu(and)f Fo(ny)h Fu(in)f Fo(fftw2d_create_plan)c Fu(are)36
 
5635
1803 y(arra)m(y)31 b(sizes.)225 1935 y Ft(\017)60 b Fp(nx)35
 
5636
b Fv(and)f Fp(ny)h Fv(in)f Fp(fftw2d_create_plan)c Fv(are)36
5588
5637
b(p)s(ositiv)m(e)e(in)m(tegers)i(sp)s(ecifying)d(the)i(dimensions)e(of)
5589
 
330 2878 y(the)h(rank)f(2)h(arra)m(y)g(to)h(b)s(e)e(transformed.)50
 
5638
330 2045 y(the)h(rank)f(2)h(arra)m(y)g(to)h(b)s(e)e(transformed.)50
5590
5639
b(i.e.)h(they)34 b(sp)s(ecify)e(that)i(the)g(transform)f(will)e(op)s
5591
 
(erate)330 2987 y(on)k Fo(nx)30 b(x)g(ny)35 b Fu(arra)m(ys)g(in)f(ro)m
5592
 
(w-ma)5 b(jor)36 b(order,)g(where)f Fo(nx)g Fu(is)f(the)h(n)m(um)m(b)s
5593
 
(er)f(of)i(ro)m(ws)f(and)g Fo(ny)f Fu(is)h(the)330 3097
5594
 
y(n)m(um)m(b)s(er)29 b(of)i(columns.)225 3232 y Fs(\017)60
5595
 
b Fo(nx)p Fu(,)31 b Fo(ny)f Fu(and)g Fo(nz)h Fu(in)e
5596
 
Fo(fftw3d_create_plan)d Fu(are)32 b(p)s(ositiv)m(e)e(in)m(tegers)h(sp)s
5597
 
(ecifying)e(the)i(dimensions)330 3342 y(of)c(the)h(rank)e(3)i(arra)m(y)
 
5640
(erate)330 2154 y(on)k Fp(nx)30 b(x)g(ny)35 b Fv(arra)m(ys)g(in)f(ro)m
 
5641
(w-ma)5 b(jor)36 b(order,)g(where)f Fp(nx)g Fv(is)f(the)h(n)m(um)m(b)s
 
5642
(er)f(of)i(ro)m(ws)f(and)g Fp(ny)f Fv(is)h(the)330 2264
 
5643
y(n)m(um)m(b)s(er)29 b(of)i(columns.)225 2396 y Ft(\017)60
 
5644
b Fp(nx)p Fv(,)31 b Fp(ny)f Fv(and)g Fp(nz)h Fv(in)e
 
5645
Fp(fftw3d_create_plan)d Fv(are)32 b(p)s(ositiv)m(e)e(in)m(tegers)h(sp)s
 
5646
(ecifying)e(the)i(dimensions)330 2506 y(of)c(the)h(rank)e(3)i(arra)m(y)
5598
5647
g(to)g(b)s(e)e(transformed.)39 b(i.e.)h(they)27 b(sp)s(ecify)f(that)i
5599
 
(the)f(transform)g(will)d(op)s(erate)330 3451 y(on)30
5600
 
b Fo(nx)g(x)g(ny)g(x)g(nz)g Fu(arra)m(ys)g(in)f(ro)m(w-ma)5
5601
 
b(jor)31 b(order.)225 3586 y Fs(\017)60 b Fo(dir)34 b
5602
 
Fu(is)g(the)h(sign)f(of)h(the)g(exp)s(onen)m(t)f(in)g(the)h(form)m(ula)
 
5648
(the)f(transform)g(will)d(op)s(erate)330 2616 y(on)30
 
5649
b Fp(nx)g(x)g(ny)g(x)g(nz)g Fv(arra)m(ys)g(in)f(ro)m(w-ma)5
 
5650
b(jor)31 b(order.)225 2748 y Ft(\017)60 b Fp(dir)34 b
 
5651
Fv(is)g(the)h(sign)f(of)h(the)g(exp)s(onen)m(t)f(in)g(the)h(form)m(ula)
5603
5652
f(that)h(de\014nes)f(the)h(F)-8 b(ourier)34 b(transform.)54
5604
 
b(It)330 3696 y(can)31 b(b)s(e)f Fs(\000)p Fu(1)h(or)g(+)o(1.)43
5605
 
b(The)30 b(aliases)g Fo(FFTW_FORWARD)d Fu(and)j Fo(FFTW_BACKWARD)e
5606
 
Fu(are)j(pro)m(vided,)e(where)330 3805 y Fo(FFTW_FORWARD)e
5607
 
Fu(stands)j(for)g Fs(\000)p Fu(1.)225 3940 y Fs(\017)60
5608
 
b Fo(flags)29 b Fu(is)g(a)i(b)s(o)s(olean)e(OR)h(\(`)p
5609
 
Fo(|)p Fu('\))i(of)e(zero)h(or)g(more)f(of)h(the)f(follo)m(wing:)379
5610
 
4075 y Fs(\000)60 b Fo(FFTW_MEASURE)p Fu(:)33 b(this)19
 
5653
b(It)330 2857 y(can)31 b(b)s(e)f Ft(\000)p Fv(1)h(or)g(+)o(1.)43
 
5654
b(The)30 b(aliases)g Fp(FFTW_FORWARD)d Fv(and)j Fp(FFTW_BACKWARD)e
 
5655
Fv(are)j(pro)m(vided,)e(where)330 2967 y Fp(FFTW_FORWARD)e
 
5656
Fv(stands)j(for)g Ft(\000)p Fv(1.)225 3099 y Ft(\017)60
 
5657
b Fp(flags)29 b Fv(is)g(a)i(b)s(o)s(olean)e(OR)h(\(`)p
 
5658
Fp(|)p Fv('\))i(of)e(zero)h(or)g(more)f(of)h(the)f(follo)m(wing:)379
 
5659
3232 y Ft(\000)60 b Fp(FFTW_MEASURE)p Fv(:)33 b(this)19
5611
5660
b(\015ag)j(tells)e(FFTW)h(to)h(\014nd)d(the)i(optimal)f(plan)g(b)m(y)g
5612
 
(actually)h Fq(c)-5 b(omputing)510 4185 y Fu(sev)m(eral)31
 
5661
(actually)h Fr(c)-5 b(omputing)510 3341 y Fv(sev)m(eral)31
5613
5662
b(FFTs)f(and)g(measuring)f(their)h(execution)g(time.)379
5614
 
4320 y Fs(\000)60 b Fo(FFTW_ESTIMATE)p Fu(:)34 b(do)25
 
5663
3474 y Ft(\000)60 b Fp(FFTW_ESTIMATE)p Fv(:)34 b(do)25
5615
5664
b(not)g(run)f(an)m(y)h(FFT)h(and)e(pro)m(vide)g(a)h(\\reasonable")h
5616
 
(plan)e(\(for)h(a)g(RISC)510 4430 y(pro)s(cessor)37 b(with)g(man)m(y)h
5617
 
(registers\).)62 b(If)38 b(neither)e Fo(FFTW_ESTIMATE)e
5618
 
Fu(nor)k Fo(FFTW_MEASURE)c Fu(is)510 4539 y(pro)m(vided,)29
5619
 
b(the)i(default)e(is)h Fo(FFTW_ESTIMATE)p Fu(.)379 4674
5620
 
y Fs(\000)60 b Fo(FFTW_OUT_OF_PLACE)p Fu(:)41 b(pro)s(duce)32
 
5665
(plan)e(\(for)h(a)g(RISC)510 3583 y(pro)s(cessor)37 b(with)g(man)m(y)h
 
5666
(registers\).)62 b(If)38 b(neither)e Fp(FFTW_ESTIMATE)e
 
5667
Fv(nor)k Fp(FFTW_MEASURE)c Fv(is)510 3693 y(pro)m(vided,)29
 
5668
b(the)i(default)e(is)h Fp(FFTW_ESTIMATE)p Fv(.)379 3825
 
5669
y Ft(\000)60 b Fp(FFTW_OUT_OF_PLACE)p Fv(:)41 b(pro)s(duce)32
5621
5670
b(a)i(plan)e(assuming)f(that)j(the)f(input)e(and)i(output)g(arra)m(ys)
5622
 
510 4784 y(will)28 b(b)s(e)h(distinct)g(\(this)h(is)f(the)i(default\).)
5623
 
379 4919 y Fs(\000)60 b Fo(FFTW_IN_PLACE)p Fu(:)54 b(pro)s(duce)37
 
5671
510 3935 y(will)28 b(b)s(e)h(distinct)g(\(this)h(is)f(the)i(default\).)
 
5672
379 4067 y Ft(\000)60 b Fp(FFTW_IN_PLACE)p Fv(:)54 b(pro)s(duce)37
5624
5673
b(a)i(plan)e(assuming)h(that)h(y)m(ou)g(w)m(an)m(t)g(to)h(p)s(erform)d
5625
 
(the)i(trans-)510 5028 y(form)30 b(in-place.)40 b(\(Unlik)m(e)30
 
5674
(the)i(trans-)510 4177 y(form)30 b(in-place.)40 b(\(Unlik)m(e)30
5626
5675
b(the)h(one-dimensional)d(transform,)j(this)e(\\really")3176
5627
 
4995 y Fn(2)3244 5028 y Fu(p)s(erforms)g(the)p 150 5131
5628
 
1200 4 v 192 5197 a Fn(2)275 5230 y Fo(fftwnd)e Fu(actually)i(ma)m(y)h
5629
 
(use)f(some)h(temp)s(orary)e(storage)j(\(hidden)c(in)h(the)i(plan\),)f
5630
 
(but)f(this)g(storage)275 5340 y(space)40 b(is)e(only)h(the)h(size)f
5631
 
(of)h(the)g(largest)g(dimension)d(of)j(the)g(arra)m(y)-8
5632
 
b(,)43 b(rather)c(than)h(b)s(eing)e(as)i(big)p eop
 
5676
4144 y Fo(2)3244 4177 y Fv(p)s(erforms)g(the)510 4286
 
5677
y(transform)36 b(in-place.\))58 b(Note)37 b(that,)i(if)c(y)m(ou)i(w)m
 
5678
(an)m(t)g(to)g(p)s(erform)e(in-place)g(transforms,)j(y)m(ou)510
 
5679
4396 y Fr(must)h Fv(use)31 b(a)f(plan)f(created)j(with)d(this)g
 
5680
(option.)510 4528 y(The)h(default)f(mo)s(de)h(of)h(op)s(eration)f(is)f
 
5681
Fp(FFTW_OUT_OF_PLACE)p Fv(.)379 4661 y Ft(\000)60 b Fp(FFTW_USE_WISDOM)
 
5682
p Fv(:)h(use)42 b(an)m(y)h Fp(wisdom)e Fv(that)i(is)f(a)m(v)-5
 
5683
b(ailable)41 b(to)j(help)d(in)g(the)i(creation)g(of)510
 
5684
4770 y(the)34 b(plan.)50 b(\(See)35 b(Section)e(2.6)i([W)-8
 
5685
b(ords)35 b(of)f(Wisdom],)g(page)h(13.\))52 b(This)32
 
5686
b(can)i(greatly)h(sp)s(eed)510 4880 y(the)g(creation)h(of)f(plans,)g
 
5687
(esp)s(ecially)e(with)h(the)h Fp(FFTW_MEASURE)d Fv(option.)55
 
5688
b Fp(FFTW_ESTIMATE)p 150 4967 1200 4 v 199 5034 a Fo(2)275
 
5689
5066 y Fi(fftwnd)32 b Fj(actually)f(ma)n(y)f(use)g(some)h(temp)r(orary)
 
5690
f(storage)i(\(hidden)e(in)h(the)f(plan\),)i(but)e(this)h(storage)h
 
5691
(space)g(is)f(only)275 5157 y(the)26 b(size)j(of)f(the)f(largest)h
 
5692
(dimension)f(of)h(the)f(arra)n(y)-6 b(,)28 b(rather)f(than)g(b)r(eing)g
 
5693
(as)h(big)g(as)g(the)f(en)n(tire)g(arra)n(y)-6 b(.)39
 
5694
b(\(Unless)28 b(y)n(ou)275 5249 y(use)k Fi(fftwnd)i Fj(to)f(p)r(erform)
 
5695
f(one-dimensional)h(transforms,)i(in)d(whic)n(h)h(case)g(the)f(temp)r
 
5696
(orary)g(storage)i(required)e(for)275 5340 y(in-place)25
 
5697
b(transforms)h Ff(is)32 b Fj(as)26 b(big)g(as)h(the)e(en)n(tire)h(arra)
 
5698
n(y)-6 b(.\))p eop end
5633
5699
%%Page: 24 26
5634
 
24 25 bop 150 -116 a Fu(24)3232 b(FFTW)510 299 y(transform)36
5635
 
b(in-place.\))58 b(Note)37 b(that,)i(if)c(y)m(ou)i(w)m(an)m(t)g(to)g(p)
5636
 
s(erform)e(in-place)g(transforms,)j(y)m(ou)510 408 y
5637
 
Fq(must)h Fu(use)31 b(a)f(plan)f(created)j(with)d(this)g(option.)510
5638
 
546 y(The)h(default)f(mo)s(de)h(of)h(op)s(eration)f(is)f
5639
 
Fo(FFTW_OUT_OF_PLACE)p Fu(.)379 684 y Fs(\000)60 b Fo(FFTW_USE_WISDOM)p
5640
 
Fu(:)h(use)42 b(an)m(y)h Fo(wisdom)e Fu(that)i(is)f(a)m(v)-5
5641
 
b(ailable)41 b(to)j(help)d(in)g(the)i(creation)g(of)510
5642
 
793 y(the)34 b(plan.)50 b(\(See)35 b(Section)e(2.6)i([W)-8
5643
 
b(ords)35 b(of)f(Wisdom],)g(page)h(13.\))52 b(This)32
5644
 
b(can)i(greatly)h(sp)s(eed)510 903 y(the)g(creation)h(of)f(plans,)g
5645
 
(esp)s(ecially)e(with)h(the)h Fo(FFTW_MEASURE)d Fu(option.)55
5646
 
b Fo(FFTW_ESTIMATE)510 1012 y Fu(plans)33 b(can)i(also)f(tak)m(e)i(adv)
5647
 
-5 b(an)m(tage)36 b(of)f Fo(wisdom)e Fu(to)i(pro)s(duce)e(a)i(more)f
5648
 
(optimal)g(plan)f(\(based)510 1122 y(on)42 b(past)g(measuremen)m(ts\))g
5649
 
(than)f(the)h(estimation)g(heuristic)e(w)m(ould)g(normally)g(generate.)
5650
 
510 1232 y(When)30 b(the)h Fo(FFTW_MEASURE)c Fu(option)j(is)g(used,)g
5651
 
(new)g Fo(wisdom)f Fu(will)f(also)j(b)s(e)f(generated)h(if)f(the)510
5652
 
1341 y(curren)m(t)42 b(transform)g(size)g(is)g(not)g(completely)g
5653
 
(understo)s(o)s(d)f(b)m(y)h(existing)g Fo(wisdom)p Fu(.)75
5654
 
b(Note)510 1451 y(that)37 b(the)g(same)g Fo(wisdom)e
5655
 
Fu(is)h(shared)g(b)s(et)m(w)m(een)h(one-dimensional)e(and)h(m)m
5656
 
(ulti-dimensional)510 1560 y(transforms.)225 1698 y Fs(\017)60
5657
 
b Fo(in)p Fu(,)34 b Fo(out)p Fu(,)g Fo(istride)p Fu(,)f
5658
 
Fo(ostride)f Fu(\(only)h(for)g(the)h Fo(_create_plan_specific)28
5659
 
b Fu(v)-5 b(arian)m(ts\):)48 b(see)34 b(cor-)330 1808
5660
 
y(resp)s(onding)25 b(argumen)m(ts)j(in)f(the)g(description)f(of)i
5661
 
Fo(fftwnd)p Fu(.)38 b(\(See)28 b(Section)g(3.3.2)h([Computing)e(the)330
5662
 
1917 y(Multi-dimensional)g(T)-8 b(ransform],)29 b(page)j(24.\))150
5663
 
2154 y Fi(3.3.2)63 b(Computing)41 b(the)f(Multi-dimensional)i(T)-10
5664
 
b(ransform)390 2347 y Fo(#include)46 b(<fftw.h>)390 2554
5665
 
y(void)h(fftwnd\(fftwnd_plan)42 b(plan,)47 b(int)g(howmany,)963
5666
 
2658 y(fftw_complex)d(*in,)j(int)g(istride,)e(int)i(idist,)963
5667
 
2762 y(fftw_complex)d(*out,)j(int)f(ostride,)g(int)h(odist\);)390
5668
 
2969 y(void)g(fftwnd_one\(fftwnd_plan)41 b(p,)48 b(fftw_complex)c(*in,)
5669
 
1154 3073 y(fftw_complex)g(*out\);)275 3214 y Fu(The)33
5670
 
b(function)f Fo(fftwnd)g Fu(computes)h(one)h(or)g(more)f(m)m
5671
 
(ulti-dimensional)d(F)-8 b(ourier)33 b(T)-8 b(ransforms,)33
5672
 
b(us-)150 3324 y(ing)39 b(a)i(plan)e(created)i(b)m(y)f
5673
 
Fo(fftwnd_create_plan)35 b Fu(\(see)41 b(Section)f(3.3.1)i([Plan)d
5674
 
(Creation)h(for)g(Multi-)150 3433 y(dimensional)20 b(T)-8
 
5700
TeXDict begin 24 25 bop 150 -116 a Fv(24)3232 b(FFTW)510
 
5701
299 y(plans)33 b(can)i(also)f(tak)m(e)i(adv)-5 b(an)m(tage)36
 
5702
b(of)f Fp(wisdom)e Fv(to)i(pro)s(duce)e(a)i(more)f(optimal)g(plan)f
 
5703
(\(based)510 408 y(on)42 b(past)g(measuremen)m(ts\))g(than)f(the)h
 
5704
(estimation)g(heuristic)e(w)m(ould)g(normally)g(generate.)510
 
5705
518 y(When)30 b(the)h Fp(FFTW_MEASURE)c Fv(option)j(is)g(used,)g(new)g
 
5706
Fp(wisdom)f Fv(will)f(also)j(b)s(e)f(generated)h(if)f(the)510
 
5707
628 y(curren)m(t)42 b(transform)g(size)g(is)g(not)g(completely)g
 
5708
(understo)s(o)s(d)f(b)m(y)h(existing)g Fp(wisdom)p Fv(.)75
 
5709
b(Note)510 737 y(that)37 b(the)g(same)g Fp(wisdom)e Fv(is)h(shared)g(b)
 
5710
s(et)m(w)m(een)h(one-dimensional)e(and)h(m)m(ulti-dimensional)510
 
5711
847 y(transforms.)225 982 y Ft(\017)60 b Fp(in)p Fv(,)34
 
5712
b Fp(out)p Fv(,)g Fp(istride)p Fv(,)f Fp(ostride)f Fv(\(only)h(for)g
 
5713
(the)h Fp(_create_plan_specific)28 b Fv(v)-5 b(arian)m(ts\):)48
 
5714
b(see)34 b(cor-)330 1092 y(resp)s(onding)25 b(argumen)m(ts)j(in)f(the)g
 
5715
(description)f(of)i Fp(fftwnd)p Fv(.)38 b(\(See)28 b(Section)g(3.3.2)h
 
5716
([Computing)e(the)330 1202 y(Multi-dimensional)g(T)-8
 
5717
b(ransform],)29 b(page)j(24.\))150 1430 y Fh(3.3.2)63
 
5718
b(Computing)41 b(the)f(Multi-dimensional)i(T)-10 b(ransform)390
 
5719
1592 y Fp(#include)46 b(<fftw.h>)390 1799 y(void)h(fftwnd\(fftwnd_plan)
 
5720
42 b(plan,)47 b(int)g(howmany,)963 1903 y(fftw_complex)d(*in,)j(int)g
 
5721
(istride,)e(int)i(idist,)963 2007 y(fftw_complex)d(*out,)j(int)f
 
5722
(ostride,)g(int)h(odist\);)390 2215 y(void)g(fftwnd_one\(fftwnd_plan)41
 
5723
b(p,)48 b(fftw_complex)c(*in,)1154 2318 y(fftw_complex)g(*out\);)275
 
5724
2455 y Fv(The)33 b(function)f Fp(fftwnd)g Fv(computes)h(one)h(or)g
 
5725
(more)f(m)m(ulti-dimensional)d(F)-8 b(ourier)33 b(T)-8
 
5726
b(ransforms,)33 b(us-)150 2565 y(ing)39 b(a)i(plan)e(created)i(b)m(y)f
 
5727
Fp(fftwnd_create_plan)35 b Fv(\(see)41 b(Section)f(3.3.1)i([Plan)d
 
5728
(Creation)h(for)g(Multi-)150 2674 y(dimensional)20 b(T)-8
5675
5729
b(ransforms],)24 b(page)g(22\).)40 b(\(Note)24 b(that)g(the)f(plan)f
5676
 
(determines)g(the)i(rank)e(and)h(dimensions)150 3543
 
5730
(determines)g(the)i(rank)e(and)h(dimensions)150 2784
5677
5731
y(of)34 b(the)h(arra)m(y)g(to)g(b)s(e)e(transformed.\))52
5678
 
b(The)34 b(function)f Fo(fftwnd_one)e Fu(pro)m(vides)i(a)i
5679
 
(simpli\014ed)30 b(in)m(terface)150 3652 y(for)g(the)h(common)f(case)i
 
5732
b(The)34 b(function)f Fp(fftwnd_one)e Fv(pro)m(vides)i(a)i
 
5733
(simpli\014ed)30 b(in)m(terface)150 2893 y(for)g(the)h(common)f(case)i
5680
5734
(of)e(single)f(input)g(arra)m(y)i(of)f(stride)f(1.)150
5681
 
3889 y Fi(Argumen)m(ts)225 4088 y Fs(\017)60 b Fo(plan)35
5682
 
b Fu(is)h(the)h(plan)e(created)i(b)m(y)g Fo(fftwnd_create_plan)p
5683
 
Fu(.)54 b(\(see)37 b(Section)f(3.3.1)j([Plan)c(Creation)330
5684
 
4197 y(for)28 b(Multi-dimensional)d(T)-8 b(ransforms],)28
5685
 
b(page)i(22\).)41 b(In)28 b(the)h(case)g(of)g(t)m(w)m(o)h(and)e
5686
 
(three-dimensional)330 4307 y(transforms,)57 b(it)51
5687
 
b(could)g(also)h(ha)m(v)m(e)i(b)s(een)d(created)i(b)m(y)f
5688
 
Fo(fftw2d_create_plan)47 b Fu(or)52 b Fo(fftw3d_)330
5689
 
4417 y(create_plan)p Fu(,)28 b(resp)s(ectiv)m(ely)-8
5690
 
b(.)225 4554 y Fs(\017)60 b Fo(howmany)28 b Fu(is)i(the)g(n)m(um)m(b)s
5691
 
(er)f(of)i(m)m(ulti-dimensional)26 b(transforms)k Fo(fftwnd)f
5692
 
Fu(will)e(compute.)225 4692 y Fs(\017)60 b Fo(in)p Fu(,)f
5693
 
Fo(istride)52 b Fu(and)h Fo(idist)f Fu(describ)s(e)g(the)i(input)e
5694
 
(arra)m(y\(s\).)112 b(There)53 b(are)h Fo(howmany)e Fu(m)m(ulti-)330
5695
 
4801 y(dimensional)24 b(input)g(arra)m(ys;)29 b(the)e(\014rst)f(one)h
5696
 
(is)e(p)s(oin)m(ted)h(to)h(b)m(y)g Fo(in)p Fu(,)g(the)f(second)h(one)g
5697
 
(is)f(p)s(oin)m(ted)f(to)330 4911 y(b)m(y)i Fo(in)j(+)g(idist)p
5698
 
Fu(,)d(and)f(so)i(on,)g(up)e(to)i Fo(in)i(+)g(\(howmany)e(-)i(1\))g(*)g
5699
 
(idist)p Fu(.)38 b(Eac)m(h)28 b(m)m(ulti-dimensional)p
5700
 
150 5022 1200 4 v 275 5121 a(as)33 b(the)h(en)m(tire)f(arra)m(y)-8
5701
 
b(.)51 b(\(Unless)32 b(y)m(ou)i(use)f Fo(fftwnd)f Fu(to)i(p)s(erform)e
5702
 
(one-dimensional)f(transforms,)i(in)275 5230 y(whic)m(h)26
5703
 
b(case)i(the)g(temp)s(orary)f(storage)i(required)d(for)i(in-place)e
5704
 
(transforms)h Fq(is)35 b Fu(as)28 b(big)e(as)i(the)g(en)m(tire)275
5705
 
5340 y(arra)m(y)-8 b(.\))p eop
 
5735
3122 y Fh(Argumen)m(ts)225 3289 y Ft(\017)60 b Fp(plan)67
 
5736
b Fv(is)g(the)i(plan)d(created)k(b)m(y)e Fp(fftwnd_create_plan)p
 
5737
Fv(.)149 b(\(see)69 b(Section)f(3.3.1)i([Plan)330 3399
 
5738
y(Creation)56 b(for)g(Multi-dimensional)c(T)-8 b(ransforms],)62
 
5739
b(page)57 b(22\).)119 b(In)55 b(the)i(case)g(of)f(t)m(w)m(o)i(and)330
 
5740
3509 y(three-dimensional)24 b(transforms,)j(it)e(could)h(also)g(ha)m(v)
 
5741
m(e)i(b)s(een)d(created)j(b)m(y)e Fp(fftw2d_create_plan)330
 
5742
3618 y Fv(or)k Fp(fftw3d_create_plan)p Fv(,)c(resp)s(ectiv)m(ely)-8
 
5743
b(.)225 3754 y Ft(\017)60 b Fp(howmany)28 b Fv(is)i(the)g(n)m(um)m(b)s
 
5744
(er)f(of)i(m)m(ulti-dimensional)26 b(transforms)k Fp(fftwnd)f
 
5745
Fv(will)e(compute.)225 3889 y Ft(\017)60 b Fp(in)p Fv(,)f
 
5746
Fp(istride)52 b Fv(and)h Fp(idist)f Fv(describ)s(e)g(the)i(input)e
 
5747
(arra)m(y\(s\).)112 b(There)53 b(are)h Fp(howmany)e Fv(m)m(ulti-)330
 
5748
3999 y(dimensional)24 b(input)g(arra)m(ys;)29 b(the)e(\014rst)f(one)h
 
5749
(is)e(p)s(oin)m(ted)h(to)h(b)m(y)g Fp(in)p Fv(,)g(the)f(second)h(one)g
 
5750
(is)f(p)s(oin)m(ted)f(to)330 4109 y(b)m(y)i Fp(in)j(+)g(idist)p
 
5751
Fv(,)d(and)f(so)i(on,)g(up)e(to)i Fp(in)i(+)g(\(howmany)e(-)i(1\))g(*)g
 
5752
(idist)p Fv(.)38 b(Eac)m(h)28 b(m)m(ulti-dimensional)330
 
5753
4218 y(input)44 b(arra)m(y)j(consists)f(of)g(complex)g(n)m(um)m(b)s
 
5754
(ers)f(\(see)j(Section)e(3.1)h([Data)h(T)m(yp)s(es],)i(page)d(17\),)330
 
5755
4328 y(stored)62 b(in)e(ro)m(w-ma)5 b(jor)62 b(format)g(\(see)h
 
5756
(Section)e(2.5)i([Multi-dimensional)58 b(Arra)m(y)k(F)-8
 
5757
b(ormat],)330 4437 y(page)49 b(11\),)54 b(whic)m(h)47
 
5758
b(are)h(not)g(necessarily)f(con)m(tiguous)i(in)d(memory)-8
 
5759
b(.)95 b(Sp)s(eci\014cally)-8 b(,)50 b Fp(in[0])d Fv(is)330
 
5760
4547 y(the)d(\014rst)e(elemen)m(t)i(of)g(the)g(\014rst)e(arra)m(y)-8
 
5761
b(,)48 b Fp(in[istride])41 b Fv(is)h(the)i(second)f(elemen)m(t)h(of)g
 
5762
(the)g(\014rst)330 4656 y(arra)m(y)-8 b(,)50 b(and)44
 
5763
b(so)h(on.)85 b(In)44 b(general,)49 b(the)c Fp(i)p Fv(-th)g(elemen)m(t)
 
5764
g(of)g(the)g Fp(j)p Fv(-th)g(input)e(arra)m(y)j(will)c(b)s(e)i(in)330
 
5765
4766 y(p)s(osition)g Fp(in[i)30 b(*)g(istride)e(+)i(j)g(*)g(idist])p
 
5766
Fv(.)87 b(Note)48 b(that,)j(here,)f Fp(i)c Fv(refers)g(to)h(an)f(index)
 
5767
f(in)m(to)330 4876 y(the)c(ro)m(w-ma)5 b(jor)42 b(format)g(for)e(the)i
 
5768
(m)m(ulti-dimensional)37 b(arra)m(y)-8 b(,)45 b(rather)c(than)g(an)g
 
5769
(index)f(in)g(an)m(y)330 4985 y(particular)29 b(dimension.)379
 
5770
5121 y Ft(\000)60 b Fr(In-plac)-5 b(e)30 b(tr)-5 b(ansforms)7
 
5771
b Fv(:)42 b(F)-8 b(or)28 b(plans)d(created)j(with)d(the)i
 
5772
Fp(FFTW_IN_PLACE)c Fv(option,)k(the)g(trans-)510 5230
 
5773
y(form)g(is)g(computed)g(in-place|the)g(output)g(is)g(returned)f(in)h
 
5774
(the)g Fp(in)g Fv(arra)m(y)-8 b(,)30 b(using)c(the)i(same)510
 
5775
5340 y(strides,)h(etcetera,)k(as)e(w)m(ere)g(used)e(in)g(the)i(input.)p
 
5776
eop end
5706
5777
%%Page: 25 27
5707
 
25 26 bop 150 -116 a Fu(Chapter)30 b(3:)41 b(FFTW)31
5708
 
b(Reference)2354 b(25)330 299 y(input)22 b(arra)m(y)j(consists)f(of)g
5709
 
(complex)g(n)m(um)m(b)s(ers)e(\(see)k(Section)e(3.1)h([Data)h(T)m(yp)s
5710
 
(es],)f(page)g(17\),)i(stored)330 408 y(in)20 b(ro)m(w-ma)5
5711
 
b(jor)21 b(format)h(\(see)g(Section)f(2.5)h([Multi-dimensional)17
5712
 
b(Arra)m(y)k(F)-8 b(ormat],)25 b(page)d(11\),)i(whic)m(h)330
5713
 
518 y(are)38 b(not)h(necessarily)e(con)m(tiguous)h(in)f(memory)-8
5714
 
b(.)64 b(Sp)s(eci\014cally)-8 b(,)38 b Fo(in[0])f Fu(is)g(the)h
5715
 
(\014rst)f(elemen)m(t)i(of)330 628 y(the)g(\014rst)f(arra)m(y)-8
5716
 
b(,)43 b Fo(in[istride])36 b Fu(is)i(the)h(second)g(elemen)m(t)h(of)f
5717
 
(the)g(\014rst)f(arra)m(y)-8 b(,)43 b(and)38 b(so)h(on.)67
5718
 
b(In)330 737 y(general,)38 b(the)f Fo(i)p Fu(-th)f(elemen)m(t)h(of)f
5719
 
(the)h Fo(j)p Fu(-th)f(input)e(arra)m(y)j(will)d(b)s(e)h(in)g(p)s
5720
 
(osition)g Fo(in[i)29 b(*)h(istride)330 847 y(+)g(j)g(*)g(idist])p
5721
 
Fu(.)43 b(Note)33 b(that,)g(here,)f Fo(i)f Fu(refers)g(to)i(an)e(index)
5722
 
g(in)m(to)g(the)h(ro)m(w-ma)5 b(jor)32 b(format)g(for)g(the)330
5723
 
956 y(m)m(ulti-dimensional)26 b(arra)m(y)-8 b(,)32 b(rather)e(than)g
5724
 
(an)h(index)e(in)g(an)m(y)h(particular)f(dimension.)379
5725
 
1093 y Fs(\000)60 b Fq(In-plac)-5 b(e)30 b(tr)-5 b(ansforms)7
5726
 
b Fu(:)42 b(F)-8 b(or)28 b(plans)d(created)j(with)d(the)i
5727
 
Fo(FFTW_IN_PLACE)c Fu(option,)k(the)g(trans-)510 1203
5728
 
y(form)g(is)g(computed)g(in-place|the)g(output)g(is)g(returned)f(in)h
5729
 
(the)g Fo(in)g Fu(arra)m(y)-8 b(,)30 b(using)c(the)i(same)510
5730
 
1312 y(strides,)h(etcetera,)k(as)e(w)m(ere)g(used)e(in)g(the)i(input.)
5731
 
225 1449 y Fs(\017)60 b Fo(out)p Fu(,)33 b Fo(ostride)f
5732
 
Fu(and)g Fo(odist)g Fu(describ)s(e)g(the)h(output)g(arra)m(y\(s\).)50
5733
 
b(The)33 b(format)g(is)g(the)g(same)h(as)f(for)330 1559
5734
 
y(the)e(input)d(arra)m(y)-8 b(.)379 1695 y Fs(\000)60
5735
 
b Fq(In-plac)-5 b(e)47 b(tr)-5 b(ansforms)7 b Fu(:)73
5736
 
b(These)45 b(parameters)g(are)h(ignored)e(for)g(plans)g(created)i(with)
5737
 
e(the)510 1805 y Fo(FFTW_IN_PLACE)27 b Fu(option.)275
5738
 
1971 y(The)36 b(function)g Fo(fftwnd_one)e Fu(transforms)i(a)i(single,)
5739
 
g(con)m(tiguous)f(input)e(arra)m(y)i(to)h(a)g(con)m(tiguous)150
5740
 
2080 y(output)30 b(arra)m(y)-8 b(.)42 b(By)30 b(de\014nition,)f(the)h
5741
 
(call)390 2213 y Fo(fftwnd_one\(plan,)43 b(in,)k(out\))275
5742
 
2352 y Fu(is)29 b(equiv)-5 b(alen)m(t)30 b(to)390 2485
5743
 
y Fo(fftwnd\(plan,)44 b(1,)k(in,)f(1,)g(1,)g(out,)g(1,)g(1\))150
5744
 
2718 y Fi(3.3.3)63 b(Destro)m(ying)41 b(a)g(Multi-dimensional)h(Plan)
5745
 
390 2910 y Fo(#include)k(<fftw.h>)390 3117 y(void)h
 
5778
TeXDict begin 25 26 bop 150 -116 a Fv(Chapter)30 b(3:)41
 
5779
b(FFTW)31 b(Reference)2354 b(25)225 299 y Ft(\017)60
 
5780
b Fp(out)p Fv(,)33 b Fp(ostride)f Fv(and)g Fp(odist)g
 
5781
Fv(describ)s(e)g(the)h(output)g(arra)m(y\(s\).)50 b(The)33
 
5782
b(format)g(is)g(the)g(same)h(as)f(for)330 408 y(the)e(input)d(arra)m(y)
 
5783
-8 b(.)379 551 y Ft(\000)60 b Fr(In-plac)-5 b(e)47 b(tr)-5
 
5784
b(ansforms)7 b Fv(:)73 b(These)45 b(parameters)g(are)h(ignored)e(for)g
 
5785
(plans)g(created)i(with)e(the)510 661 y Fp(FFTW_IN_PLACE)27
 
5786
b Fv(option.)275 844 y(The)36 b(function)g Fp(fftwnd_one)e
 
5787
Fv(transforms)i(a)i(single,)g(con)m(tiguous)f(input)e(arra)m(y)i(to)h
 
5788
(a)g(con)m(tiguous)150 954 y(output)30 b(arra)m(y)-8
 
5789
b(.)42 b(By)30 b(de\014nition,)f(the)h(call)390 1099
 
5790
y Fp(fftwnd_one\(plan,)43 b(in,)k(out\))275 1250 y Fv(is)29
 
5791
b(equiv)-5 b(alen)m(t)30 b(to)390 1395 y Fp(fftwnd\(plan,)44
 
5792
b(1,)k(in,)f(1,)g(0,)g(out,)g(1,)g(0\))150 1651 y Fh(3.3.3)63
 
5793
b(Destro)m(ying)41 b(a)g(Multi-dimensional)h(Plan)390
 
5794
1813 y Fp(#include)k(<fftw.h>)390 2021 y(void)h
5746
5795
(fftwnd_destroy_plan\(fft)o(wnd_)o(pla)o(n)42 b(plan\);)275
5747
 
3256 y Fu(The)c(function)g Fo(fftwnd_destroy_plan)c Fu(frees)39
5748
 
b(the)g(plan)e Fo(plan)h Fu(and)h(releases)g(all)f(the)h(memory)150
5749
 
3366 y(asso)s(ciated)31 b(with)e(it.)40 b(After)31 b(destruction,)f(a)g
5750
 
(plan)f(is)h(no)g(longer)g(v)-5 b(alid.)150 3599 y Fi(3.3.4)63
5751
 
b(What)40 b(FFTWND)h(Really)f(Computes)275 3796 y Fu(The)e(con)m(v)m
 
5796
2172 y Fv(The)c(function)g Fp(fftwnd_destroy_plan)c Fv(frees)39
 
5797
b(the)g(plan)e Fp(plan)h Fv(and)h(releases)g(all)f(the)h(memory)150
 
5798
2281 y(asso)s(ciated)31 b(with)e(it.)40 b(After)31 b(destruction,)f(a)g
 
5799
(plan)f(is)h(no)g(longer)g(v)-5 b(alid.)150 2538 y Fh(3.3.4)63
 
5800
b(What)40 b(FFTWND)h(Really)f(Computes)275 2747 y Fv(The)e(con)m(v)m
5752
5801
(en)m(tions)j(that)f(w)m(e)g(follo)m(w)e(for)h(the)h(m)m
5753
5802
(ulti-dimensional)35 b(transform)k(are)h(analogous)f(to)150
5754
 
3905 y(those)23 b(for)f(the)h(one-dimensional)e(transform.)37
 
5803
2856 y(those)23 b(for)f(the)h(one-dimensional)e(transform.)37
5755
5804
b(In)22 b(particular,)h(the)f(forw)m(ard)h(transform)e(has)i(a)g
5756
 
(negativ)m(e)150 4015 y(sign)j(in)f(the)i(exp)s(onen)m(t)f(and)g
 
5805
(negativ)m(e)150 2966 y(sign)j(in)f(the)i(exp)s(onen)m(t)f(and)g
5757
5806
(neither)g(the)h(forw)m(ard)f(nor)g(the)h(bac)m(kw)m(ard)g(transforms)e
5758
 
(will)f(p)s(erform)i(an)m(y)150 4124 y(normalization.)43
 
5807
(will)f(p)s(erform)i(an)m(y)150 3075 y(normalization.)43
5759
5808
b(Computing)30 b(the)i(bac)m(kw)m(ard)h(transform)e(of)g(the)h(forw)m
5760
 
(ard)g(transform)f(will)e(m)m(ultiply)150 4234 y(the)h(arra)m(y)g(b)m
 
5809
(ard)g(transform)f(will)e(m)m(ultiply)150 3185 y(the)h(arra)m(y)g(b)m
5761
5810
(y)f(the)h(pro)s(duct)e(of)i(its)f(dimensions.)37 b(The)29
5762
5811
b(output)h(is)e(in-order,)h(and)g(the)g(zeroth)h(elemen)m(t)150
5763
 
4344 y(of)h(the)f(output)g(is)f(the)i(amplitude)d(of)j(the)f(zero)i
5764
 
(frequency)d(comp)s(onen)m(t.)275 4482 y(The)35 b(exact)j(mathematical)
 
5812
3295 y(of)h(the)f(output)g(is)f(the)i(amplitude)d(of)j(the)f(zero)i
 
5813
(frequency)d(comp)s(onen)m(t.)275 3445 y(The)35 b(exact)j(mathematical)
5765
5814
f(de\014nition)d(of)i(our)g(m)m(ulti-dimensional)c(transform)k(follo)m
5766
 
(ws.)58 b(Let)37 b Fl(X)150 4592 y Fu(b)s(e)d(a)g Fl(d)p
5767
 
Fu(-dimensional)e(complex)i(arra)m(y)h(whose)f(elemen)m(ts)h(are)g
5768
 
Fl(X)7 b Fu([)p Fl(j)2489 4606 y Fn(1)2527 4592 y Fl(;)15
5769
 
b(j)2604 4606 y Fn(2)2642 4592 y Fl(;)g(:)g(:)g(:)i(;)e(j)2881
5770
 
4606 y Fk(d)2920 4592 y Fu(],)36 b(where)e(0)e Fs(\024)g
5771
 
Fl(j)3490 4606 y Fk(s)3557 4592 y Fl(<)g(n)3715 4606
5772
 
y Fk(s)150 4702 y Fu(for)e(all)f Fl(s)c Fs(2)g(f)p Fu(1)p
5773
 
Fl(;)15 b Fu(2)p Fl(;)g(:)g(:)g(:)k(;)c(d)p Fs(g)p Fu(.)41
5774
 
b(Let)31 b(also)g Fl(!)1509 4716 y Fk(s)1569 4702 y Fu(=)25
5775
 
b Fl(e)1707 4669 y Fn(2)p Fk(\031)1781 4624 y Fj(p)p
5776
 
1836 4624 85 3 v 45 x(\000)p Fn(1)p Fk(=n)1996 4677 y
5777
 
Fg(s)2032 4702 y Fu(,)30 b(for)h(all)59 b Fl(s)25 b Fs(2)g(f)p
5778
 
Fu(1)p Fl(;)15 b Fu(2)p Fl(;)g(:)g(:)g(:)j(;)d(d)p Fs(g)p
5779
 
Fu(.)275 4840 y(The)24 b(forw)m(ard)g(transform)g(computes)h(a)g
5780
 
(complex)f(arra)m(y)i Fl(Y)20 b Fu(,)26 b(whose)e(structure)h(is)e(the)
5781
 
i(same)g(as)g(that)150 4950 y(of)31 b Fl(X)7 b Fu(,)31
5782
 
b(de\014ned)e(b)m(y)534 5258 y Fl(Y)20 b Fu([)p Fl(i)663
5783
 
5272 y Fn(1)701 5258 y Fl(;)15 b(i)772 5272 y Fn(2)810
5784
 
5258 y Fl(;)g(:)g(:)g(:)i(;)e(i)1043 5272 y Fk(d)1082
5785
 
5258 y Fu(])26 b(=)1229 5151 y Fk(n)1270 5159 y Ff(1)1302
5786
 
5151 y Fj(\000)p Fn(1)1248 5177 y Fh(X)1236 5354 y Fk(j)1263
5787
 
5362 y Ff(1)1296 5354 y Fn(=0)1402 5151 y Fk(n)1443 5159
5788
 
y Ff(2)1476 5151 y Fj(\000)p Fn(1)1421 5177 y Fh(X)1409
5789
 
5354 y Fk(j)1436 5362 y Ff(2)1469 5354 y Fn(=0)1576 5258
5790
 
y Fs(\001)15 b(\001)g(\001)1697 5150 y Fk(n)1738 5159
5791
 
y Fg(d)1773 5150 y Fj(\000)p Fn(1)1717 5177 y Fh(X)1704
5792
 
5354 y Fk(j)1731 5363 y Fg(d)1767 5354 y Fn(=0)1873 5258
5793
 
y Fl(X)7 b Fu([)p Fl(j)2017 5272 y Fn(1)2055 5258 y Fl(;)15
5794
 
b(j)2132 5272 y Fn(2)2170 5258 y Fl(;)g(:)g(:)g(:)i(;)e(j)2409
5795
 
5272 y Fk(d)2449 5258 y Fu(])p Fl(!)2534 5215 y Fj(\000)p
5796
 
Fk(i)2609 5223 y Ff(1)2642 5215 y Fk(j)2669 5223 y Ff(1)2531
5797
 
5277 y Fn(1)2706 5258 y Fl(!)2766 5215 y Fj(\000)p Fk(i)2841
5798
 
5223 y Ff(2)2873 5215 y Fk(j)2900 5223 y Ff(2)2763 5277
5799
 
y Fn(2)2952 5258 y Fs(\001)g(\001)g(\001)i Fl(!)3134
5800
 
5215 y Fj(\000)p Fk(i)3209 5224 y Fg(d)3244 5215 y Fk(j)3271
5801
 
5224 y Fg(d)3131 5280 y Fk(d)3340 5258 y Fl(:)p eop
 
5815
(ws.)58 b(Let)37 b Fm(X)150 3555 y Fv(b)s(e)d(a)g Fm(d)p
 
5816
Fv(-dimensional)e(complex)i(arra)m(y)h(whose)f(elemen)m(ts)h(are)g
 
5817
Fm(X)7 b Fv([)p Fm(j)2489 3569 y Fo(1)2527 3555 y Fm(;)15
 
5818
b(j)2604 3569 y Fo(2)2642 3555 y Fm(;)g(:)g(:)g(:)i(;)e(j)2881
 
5819
3569 y Fl(d)2920 3555 y Fv(],)36 b(where)e(0)e Ft(\024)g
 
5820
Fm(j)3490 3569 y Fl(s)3557 3555 y Fm(<)g(n)3715 3569
 
5821
y Fl(s)150 3665 y Fv(for)e(all)f Fm(s)c Ft(2)g(f)p Fv(1)p
 
5822
Fm(;)15 b Fv(2)p Fm(;)g(:)g(:)g(:)k(;)c(d)p Ft(g)p Fv(.)41
 
5823
b(Let)31 b(also)g Fm(!)1509 3679 y Fl(s)1569 3665 y Fv(=)25
 
5824
b Fm(e)1707 3632 y Fo(2)p Fl(\031)1781 3587 y Fk(p)p
 
5825
1836 3587 85 3 v 45 x(\000)p Fo(1)p Fl(=n)1996 3640 y
 
5826
Fe(s)2032 3665 y Fv(,)30 b(for)h(all)59 b Fm(s)25 b Ft(2)g(f)p
 
5827
Fv(1)p Fm(;)15 b Fv(2)p Fm(;)g(:)g(:)g(:)j(;)d(d)p Ft(g)p
 
5828
Fv(.)275 3815 y(The)24 b(forw)m(ard)g(transform)g(computes)h(a)g
 
5829
(complex)f(arra)m(y)i Fm(Y)20 b Fv(,)26 b(whose)e(structure)h(is)e(the)
 
5830
i(same)g(as)g(that)150 3925 y(of)31 b Fm(X)7 b Fv(,)31
 
5831
b(de\014ned)e(b)m(y)534 4262 y Fm(Y)20 b Fv([)p Fm(i)663
 
5832
4276 y Fo(1)701 4262 y Fm(;)15 b(i)772 4276 y Fo(2)810
 
5833
4262 y Fm(;)g(:)g(:)g(:)i(;)e(i)1043 4276 y Fl(d)1082
 
5834
4262 y Fv(])26 b(=)1229 4155 y Fl(n)1270 4163 y Fd(1)1302
 
5835
4155 y Fk(\000)p Fo(1)1248 4181 y Fg(X)1236 4358 y Fl(j)1263
 
5836
4366 y Fd(1)1296 4358 y Fo(=0)1402 4155 y Fl(n)1443 4163
 
5837
y Fd(2)1476 4155 y Fk(\000)p Fo(1)1421 4181 y Fg(X)1409
 
5838
4358 y Fl(j)1436 4366 y Fd(2)1469 4358 y Fo(=0)1576 4262
 
5839
y Ft(\001)15 b(\001)g(\001)1697 4155 y Fl(n)1738 4164
 
5840
y Fe(d)1773 4155 y Fk(\000)p Fo(1)1717 4181 y Fg(X)1704
 
5841
4358 y Fl(j)1731 4367 y Fe(d)1767 4358 y Fo(=0)1873 4262
 
5842
y Fm(X)7 b Fv([)p Fm(j)2017 4276 y Fo(1)2055 4262 y Fm(;)15
 
5843
b(j)2132 4276 y Fo(2)2170 4262 y Fm(;)g(:)g(:)g(:)i(;)e(j)2409
 
5844
4276 y Fl(d)2449 4262 y Fv(])p Fm(!)2534 4219 y Fk(\000)p
 
5845
Fl(i)2609 4227 y Fd(1)2642 4219 y Fl(j)2669 4227 y Fd(1)2531
 
5846
4282 y Fo(1)2706 4262 y Fm(!)2766 4219 y Fk(\000)p Fl(i)2841
 
5847
4227 y Fd(2)2873 4219 y Fl(j)2900 4227 y Fd(2)2763 4282
 
5848
y Fo(2)2952 4262 y Ft(\001)g(\001)g(\001)i Fm(!)3134
 
5849
4219 y Fk(\000)p Fl(i)3209 4228 y Fe(d)3244 4219 y Fl(j)3271
 
5850
4228 y Fe(d)3131 4284 y Fl(d)3340 4262 y Fm(:)275 4574
 
5851
y Fv(The)29 b(bac)m(kw)m(ard)i(transform)f(computes)612
 
5852
4877 y Fm(Y)20 b Fv([)p Fm(i)741 4891 y Fo(1)779 4877
 
5853
y Fm(;)15 b(i)850 4891 y Fo(2)888 4877 y Fm(;)g(:)g(:)g(:)i(;)e(i)1121
 
5854
4891 y Fl(d)1160 4877 y Fv(])25 b(=)1306 4771 y Fl(n)1347
 
5855
4779 y Fd(1)1380 4771 y Fk(\000)p Fo(1)1326 4797 y Fg(X)1314
 
5856
4973 y Fl(j)1341 4981 y Fd(1)1374 4973 y Fo(=0)1480 4771
 
5857
y Fl(n)1521 4779 y Fd(2)1553 4771 y Fk(\000)p Fo(1)1499
 
5858
4797 y Fg(X)1487 4973 y Fl(j)1514 4981 y Fd(2)1547 4973
 
5859
y Fo(=0)1654 4877 y Ft(\001)15 b(\001)g(\001)1775 4770
 
5860
y Fl(n)1816 4779 y Fe(d)1851 4770 y Fk(\000)p Fo(1)1795
 
5861
4797 y Fg(X)1782 4973 y Fl(j)1809 4982 y Fe(d)1844 4973
 
5862
y Fo(=0)1951 4877 y Fm(X)7 b Fv([)p Fm(j)2095 4891 y
 
5863
Fo(1)2133 4877 y Fm(;)15 b(j)2210 4891 y Fo(2)2248 4877
 
5864
y Fm(;)g(:)g(:)g(:)i(;)e(j)2487 4891 y Fl(d)2526 4877
 
5865
y Fv(])p Fm(!)2611 4835 y Fl(i)2634 4843 y Fd(1)2667
 
5866
4835 y Fl(j)2694 4843 y Fd(1)2608 4897 y Fo(1)2731 4877
 
5867
y Fm(!)2791 4835 y Fl(i)2814 4843 y Fd(2)2847 4835 y
 
5868
Fl(j)2874 4843 y Fd(2)2788 4897 y Fo(2)2926 4877 y Ft(\001)g(\001)g
 
5869
(\001)i Fm(!)3108 4835 y Fl(i)3131 4844 y Fe(d)3166 4835
 
5870
y Fl(j)3193 4844 y Fe(d)3105 4900 y Fl(d)3263 4877 y
 
5871
Fm(:)275 5230 y Fv(Computing)26 b(the)i(forw)m(ard)f(transform)g(follo)
 
5872
m(w)m(ed)h(b)m(y)g(the)g(bac)m(kw)m(ard)g(transform)f(will)f(m)m
 
5873
(ultiply)f(the)150 5340 y(arra)m(y)31 b(b)m(y)514 5276
 
5874
y Fg(Q)593 5296 y Fl(d)593 5363 y(s)p Fo(=1)727 5340
 
5875
y Fm(n)782 5354 y Fl(d)821 5340 y Fv(.)p eop end
5802
5876
%%Page: 26 28
5803
 
26 27 bop 150 -116 a Fu(26)3232 b(FFTW)275 299 y(The)29
5804
 
b(bac)m(kw)m(ard)i(transform)f(computes)612 583 y Fl(Y)20
5805
 
b Fu([)p Fl(i)741 597 y Fn(1)779 583 y Fl(;)15 b(i)850
5806
 
597 y Fn(2)888 583 y Fl(;)g(:)g(:)g(:)i(;)e(i)1121 597
5807
 
y Fk(d)1160 583 y Fu(])25 b(=)1306 477 y Fk(n)1347 485
5808
 
y Ff(1)1380 477 y Fj(\000)p Fn(1)1326 502 y Fh(X)1314
5809
 
679 y Fk(j)1341 687 y Ff(1)1374 679 y Fn(=0)1480 477
5810
 
y Fk(n)1521 485 y Ff(2)1553 477 y Fj(\000)p Fn(1)1499
5811
 
502 y Fh(X)1487 679 y Fk(j)1514 687 y Ff(2)1547 679 y
5812
 
Fn(=0)1654 583 y Fs(\001)15 b(\001)g(\001)1775 476 y
5813
 
Fk(n)1816 485 y Fg(d)1851 476 y Fj(\000)p Fn(1)1795 502
5814
 
y Fh(X)1782 679 y Fk(j)1809 688 y Fg(d)1844 679 y Fn(=0)1951
5815
 
583 y Fl(X)7 b Fu([)p Fl(j)2095 597 y Fn(1)2133 583 y
5816
 
Fl(;)15 b(j)2210 597 y Fn(2)2248 583 y Fl(;)g(:)g(:)g(:)i(;)e(j)2487
5817
 
597 y Fk(d)2526 583 y Fu(])p Fl(!)2611 541 y Fk(i)2634
5818
 
549 y Ff(1)2667 541 y Fk(j)2694 549 y Ff(1)2608 603 y
5819
 
Fn(1)2731 583 y Fl(!)2791 541 y Fk(i)2814 549 y Ff(2)2847
5820
 
541 y Fk(j)2874 549 y Ff(2)2788 603 y Fn(2)2926 583 y
5821
 
Fs(\001)g(\001)g(\001)i Fl(!)3108 541 y Fk(i)3131 550
5822
 
y Fg(d)3166 541 y Fk(j)3193 550 y Fg(d)3105 606 y Fk(d)3263
5823
 
583 y Fl(:)275 904 y Fu(Computing)26 b(the)i(forw)m(ard)f(transform)g
5824
 
(follo)m(w)m(ed)h(b)m(y)g(the)g(bac)m(kw)m(ard)g(transform)f(will)f(m)m
5825
 
(ultiply)f(the)150 1013 y(arra)m(y)31 b(b)m(y)514 949
5826
 
y Fh(Q)593 970 y Fk(d)593 1036 y(s)p Fn(=1)727 1013 y
5827
 
Fl(n)782 1027 y Fk(d)821 1013 y Fu(.)150 1279 y Ft(3.4)68
5828
 
b(Real)46 b(One-dimensional)g(T)-11 b(ransforms)45 b(Reference)275
5829
 
1475 y Fu(The)29 b(one-dimensional)f(real)h(routines)g(are)h(generally)
5830
 
f(pre\014xed)g(with)f Fo(rfftw_)p Fu(.)3055 1442 y Fn(3)3132
5831
 
1475 y Fu(Programs)h(using)150 1585 y(RFFTW)c(should)d(b)s(e)i(link)m
5832
 
(ed)e(with)h Fo(-lrfftw)29 b(-lfftw)f(-lm)c Fu(on)g(Unix)f(systems,)i
5833
 
(or)g(with)e(the)h(RFFTW,)150 1694 y(the)31 b(FFTW,)g(and)f(the)g
5834
 
(standard)g(math)g(libraries)e(in)h(general.)150 1925
5835
 
y Fi(3.4.1)63 b(Plan)40 b(Creation)h(for)g(Real)f(One-dimensional)i(T)
5836
 
-10 b(ransforms)390 2115 y Fo(#include)46 b(<rfftw.h>)390
5837
 
2323 y(rfftw_plan)f(rfftw_create_plan\(int)d(n,)47 b(fftw_direction)d
5838
 
(dir,)j(int)g(flags\);)390 2530 y(rfftw_plan)e
 
5877
TeXDict begin 26 27 bop 150 -116 a Fv(26)3232 b(FFTW)150
 
5878
299 y Fu(3.4)68 b(Real)46 b(One-dimensional)g(T)-11 b(ransforms)45
 
5879
b(Reference)275 489 y Fv(The)29 b(one-dimensional)f(real)h(routines)g
 
5880
(are)h(generally)f(pre\014xed)g(with)f Fp(rfftw_)p Fv(.)3055
 
5881
457 y Fo(3)3132 489 y Fv(Programs)h(using)150 599 y(RFFTW)c(should)d(b)
 
5882
s(e)i(link)m(ed)e(with)h Fp(-lrfftw)29 b(-lfftw)f(-lm)c
 
5883
Fv(on)g(Unix)f(systems,)i(or)g(with)e(the)h(RFFTW,)150
 
5884
709 y(the)31 b(FFTW,)g(and)f(the)g(standard)g(math)g(libraries)e(in)h
 
5885
(general.)150 927 y Fh(3.4.1)63 b(Plan)40 b(Creation)h(for)g(Real)f
 
5886
(One-dimensional)i(T)-10 b(ransforms)390 1089 y Fp(#include)46
 
5887
b(<rfftw.h>)390 1296 y(rfftw_plan)f(rfftw_create_plan\(int)d(n,)47
 
5888
b(fftw_direction)d(dir,)j(int)g(flags\);)390 1504 y(rfftw_plan)e
5839
5889
(rfftw_create_plan_specif)o(ic\(i)o(nt)d(n,)47 b(fftw_direction)d(dir,)
5840
 
581 2634 y(int)j(flags,)f(fftw_real)f(*in,)i(int)g(istride,)581
5841
 
2738 y(fftw_real)e(*out,)i(int)f(ostride\);)275 2875
5842
 
y Fu(The)23 b(function)f Fo(rfftw_create_plan)d Fu(creates)25
 
5890
581 1608 y(int)j(flags,)f(fftw_real)f(*in,)i(int)g(istride,)581
 
5891
1711 y(fftw_real)e(*out,)i(int)f(ostride\);)275 1844
 
5892
y Fv(The)23 b(function)f Fp(rfftw_create_plan)d Fv(creates)25
5843
5893
b(a)g(plan,)f(whic)m(h)e(is)h(a)h(data)g(structure)g(con)m(taining)f
5844
 
(all)150 2985 y(the)31 b(information)e(that)j Fo(rfftw)e
5845
 
Fu(needs)g(in)g(order)g(to)i(compute)f(the)h(1D)f(real)g(F)-8
5846
 
b(ourier)31 b(transform.)41 b(Y)-8 b(ou)150 3095 y(can)32
 
5894
(all)150 1953 y(the)31 b(information)e(that)j Fp(rfftw)e
 
5895
Fv(needs)g(in)g(order)g(to)i(compute)f(the)h(1D)f(real)g(F)-8
 
5896
b(ourier)31 b(transform.)41 b(Y)-8 b(ou)150 2063 y(can)32
5847
5897
b(create)i(as)f(man)m(y)f(plans)f(as)h(y)m(ou)g(need,)h(but)f(only)f
5848
5898
(one)h(plan)f(for)h(a)h(giv)m(en)f(arra)m(y)h(size)f(is)f(required)150
5849
 
3204 y(\(a)g(plan)e(can)i(b)s(e)f(reused)f(man)m(y)i(times\).)275
5850
 
3342 y Fo(rfftw_create_plan)j Fu(returns)k(a)i(v)-5 b(alid)38
5851
 
b(plan,)j(or)e Fo(NULL)f Fu(if,)j(for)e(some)h(reason,)i(the)e(plan)e
5852
 
(can't)150 3452 y(b)s(e)c(created.)53 b(In)34 b(the)g(default)f
 
5899
2173 y(\(a)g(plan)e(can)i(b)s(e)f(reused)f(man)m(y)i(times\).)275
 
5900
2305 y Fp(rfftw_create_plan)j Fv(returns)k(a)i(v)-5 b(alid)38
 
5901
b(plan,)j(or)e Fp(NULL)f Fv(if,)j(for)e(some)h(reason,)i(the)e(plan)e
 
5902
(can't)150 2415 y(b)s(e)c(created.)53 b(In)34 b(the)g(default)f
5853
5903
(installation,)h(this)f(cannot)i(happ)s(en,)f(but)f(it)h(is)f(p)s
5854
 
(ossible)f(to)j(con\014gure)150 3561 y(RFFTW)e(in)e(suc)m(h)h(a)h(w)m
 
5904
(ossible)f(to)j(con\014gure)150 2524 y(RFFTW)e(in)e(suc)m(h)h(a)h(w)m
5855
5905
(a)m(y)h(that)f(some)f(input)f(sizes)h(are)h(forbidden,)e(and)g(RFFTW)i
5856
 
(cannot)h(create)g(a)150 3671 y(plan.)275 3809 y(The)k
5857
 
Fo(rfftw_create_plan_specifi)o(c)33 b Fu(v)-5 b(arian)m(t)39
 
5906
(cannot)h(create)g(a)150 2634 y(plan.)275 2766 y(The)k
 
5907
Fp(rfftw_create_plan_specifi)o(c)33 b Fv(v)-5 b(arian)m(t)39
5858
5908
b(tak)m(es)i(as)e(additional)e(argumen)m(ts)j(sp)s(eci\014c)e(in-)150
5859
 
3918 y(put/output)43 b(arra)m(ys)g(and)f(their)h(strides.)77
 
5909
2876 y(put/output)43 b(arra)m(ys)g(and)f(their)h(strides.)77
5860
5910
b(F)-8 b(or)44 b(the)f(last)g(four)g(argumen)m(ts,)k(y)m(ou)c(should)e
5861
 
(pass)i(the)150 4028 y(arra)m(ys)36 b(and)f(strides)g(that)h(y)m(ou)g
5862
 
(will)e(ev)m(en)m(tually)h(b)s(e)g(passing)g(to)i Fo(rfftw)p
5863
 
Fu(.)55 b(The)35 b(resulting)f(plans)h(will)150 4137
 
5911
(pass)i(the)150 2986 y(arra)m(ys)36 b(and)f(strides)g(that)h(y)m(ou)g
 
5912
(will)e(ev)m(en)m(tually)h(b)s(e)g(passing)g(to)i Fp(rfftw)p
 
5913
Fv(.)55 b(The)35 b(resulting)f(plans)h(will)150 3095
5864
5914
y(b)s(e)g(optimized)g(for)h(those)g(arra)m(ys)g(and)g(strides,)g
5865
5915
(although)f(they)h(ma)m(y)h(b)s(e)e(used)g(on)h(other)g(arra)m(ys)h(as)
5866
 
150 4247 y(w)m(ell.)i(Note:)j(the)30 b(con)m(ten)m(ts)h(of)f(the)f(in)g
5867
 
(and)f(out)i(arra)m(ys)g(are)g Fq(destr)-5 b(oye)g(d)41
5868
 
b Fu(b)m(y)30 b(the)f(sp)s(eci\014c)g(planner)e(\(the)150
5869
 
4356 y(initial)f(con)m(ten)m(ts)31 b(are)e(ignored,)g(so)g(the)g(arra)m
 
5916
150 3205 y(w)m(ell.)i(Note:)j(the)30 b(con)m(ten)m(ts)h(of)f(the)f(in)g
 
5917
(and)f(out)i(arra)m(ys)g(are)g Fr(destr)-5 b(oye)g(d)41
 
5918
b Fv(b)m(y)30 b(the)f(sp)s(eci\014c)g(planner)e(\(the)150
 
5919
3314 y(initial)f(con)m(ten)m(ts)31 b(are)e(ignored,)g(so)g(the)g(arra)m
5870
5920
(ys)g(need)g(not)g(ha)m(v)m(e)i(b)s(een)d(initialized\).)37
5871
 
b(See)29 b(Section)g(3.2.2)150 4466 y([Discussion)g(on)h(Sp)s(eci\014c)
 
5921
b(See)29 b(Section)g(3.2.2)150 3424 y([Discussion)g(on)h(Sp)s(eci\014c)
5872
5922
f(Plans],)h(page)h(20,)g(for)f(a)h(discussion)d(on)i(sp)s(eci\014c)f
5873
 
(plans.)150 4697 y Fi(Argumen)m(ts)225 4893 y Fs(\017)60
5874
 
b Fo(n)30 b Fu(is)f(the)i(size)f(of)h(the)f(transform.)40
 
5923
(plans.)150 3642 y Fh(Argumen)m(ts)225 3810 y Ft(\017)60
 
5924
b Fp(n)30 b Fv(is)f(the)i(size)f(of)h(the)f(transform.)40
5875
5925
b(It)31 b(can)g(b)s(e)e(an)m(y)i(p)s(ositiv)m(e)e(in)m(teger.)379
5876
 
5029 y Fs(\000)60 b Fu(RFFTW)30 b(is)e(b)s(est)h(at)i(handling)26
5877
 
b(sizes)j(of)h(the)f(form)g(2)2375 4996 y Fk(a)2416 5029
5878
 
y Fu(3)2461 4996 y Fk(b)2495 5029 y Fu(5)2540 4996 y
5879
 
Fk(c)2574 5029 y Fu(7)2619 4996 y Fk(d)2658 5029 y Fu(11)2748
5880
 
4996 y Fk(e)2785 5029 y Fu(13)2875 4996 y Fk(f)2919 5029
5881
 
y Fu(,)h(where)e Fl(e)p Fu(+)p Fl(f)39 b Fu(is)28 b(either)510
5882
 
5138 y(0)h(or)f(1,)h(and)e(the)i(other)f(exp)s(onen)m(ts)g(are)h
 
5926
3942 y Ft(\000)60 b Fv(RFFTW)30 b(is)e(b)s(est)h(at)i(handling)26
 
5927
b(sizes)j(of)h(the)f(form)g(2)2375 3909 y Fl(a)2416 3942
 
5928
y Fv(3)2461 3909 y Fl(b)2495 3942 y Fv(5)2540 3909 y
 
5929
Fl(c)2574 3942 y Fv(7)2619 3909 y Fl(d)2658 3942 y Fv(11)2748
 
5930
3909 y Fl(e)2785 3942 y Fv(13)2875 3909 y Fl(f)2919 3942
 
5931
y Fv(,)h(where)e Fm(e)p Fv(+)p Fm(f)39 b Fv(is)28 b(either)510
 
5932
4052 y(0)h(or)f(1,)h(and)e(the)i(other)f(exp)s(onen)m(ts)g(are)h
5883
5933
(arbitrary)-8 b(.)39 b(Other)28 b(sizes)f(are)i(computed)f(b)m(y)g
5884
 
(means)p 150 5241 1200 4 v 192 5307 a Fn(3)275 5340 y
5885
 
Fu(The)h(et)m(ymologically-correct)j(sp)s(elling)c(w)m(ould)h(b)s(e)h
5886
 
Fo(frftw_)p Fu(,)f(but)g(it)h(is)g(hard)f(to)i(remem)m(b)s(er.)p
5887
 
eop
 
5934
(means)510 4162 y(of)c(a)g(slo)m(w,)i(general-purp)s(ose)c(routine)h
 
5935
(\(reducing)g(to)i Fm(O)s Fv(\()p Fm(n)2517 4129 y Fo(2)2554
 
5936
4162 y Fv(\))f(p)s(erformance)f(for)h(prime)f(sizes\).)510
 
5937
4271 y(\(It)35 b(is)e(p)s(ossible)f(to)j(customize)f(RFFTW)h(for)f
 
5938
(di\013eren)m(t)g(arra)m(y)g(sizes.)52 b(See)35 b(Chapter)f(6)g([In-)
 
5939
510 4381 y(stallation)c(and)g(Customization],)g(page)i(55,)g(for)e
 
5940
(more)h(information.\))41 b(T)-8 b(ransforms)29 b(whose)510
 
5941
4490 y(sizes)36 b(are)g(p)s(o)m(w)m(ers)g(of)g(2)g(are)h(esp)s(ecially)
 
5942
d(fast.)58 b(If)35 b(y)m(ou)i(ha)m(v)m(e)g(large)f(prime)e(factors,)39
 
5943
b(it)c(ma)m(y)510 4600 y(b)s(e)j(faster)h(to)h(switc)m(h)e(o)m(v)m(er)i
 
5944
(to)g(the)f(complex)f(FFTW)i(routines,)g(whic)m(h)e(ha)m(v)m(e)i
 
5945
Fm(O)s Fv(\()p Fm(n)15 b Fv(log)h Fm(n)p Fv(\))510 4709
 
5946
y(p)s(erformance)25 b(ev)m(en)i(for)f(prime)f(sizes)g(\(w)m(e)i(don't)f
 
5947
(kno)m(w)g(of)h(a)f(similar)e(algorithm)h(sp)s(ecialized)510
 
5948
4819 y(for)30 b(real)g(data,)h(unfortunately\).)225 4952
 
5949
y Ft(\017)60 b Fp(dir)31 b Fv(is)g(the)g(direction)g(of)h(the)g
 
5950
(desired)e(transform,)h(either)h Fp(FFTW_REAL_TO_COMPLEX)26
 
5951
b Fv(or)31 b Fp(FFTW_)330 5061 y(COMPLEX_TO_REAL)p Fv(,)h(corresp)s
 
5952
(onding)g(to)j Fp(FFTW_FORWARD)c Fv(or)k Fp(FFTW_BACKWARD)p
 
5953
Fv(,)d(resp)s(ectiv)m(ely)-8 b(.)p 150 5241 1200 4 v
 
5954
199 5308 a Fo(3)275 5340 y Fj(The)25 b(et)n(ymologically-correct)i(sp)r
 
5955
(elling)g(w)n(ould)f(b)r(e)g Fi(frftw_)p Fj(,)h(but)e(it)h(is)g(hard)f
 
5956
(to)h(remem)n(b)r(er.)p eop end
5888
5957
%%Page: 27 29
5889
 
27 28 bop 150 -116 a Fu(Chapter)30 b(3:)41 b(FFTW)31
5890
 
b(Reference)2354 b(27)510 299 y(of)24 b(a)g(slo)m(w,)i(general-purp)s
5891
 
(ose)c(routine)h(\(reducing)g(to)i Fl(O)s Fu(\()p Fl(n)2517
5892
 
266 y Fn(2)2554 299 y Fu(\))f(p)s(erformance)f(for)h(prime)f(sizes\).)
5893
 
510 408 y(\(It)35 b(is)e(p)s(ossible)f(to)j(customize)f(RFFTW)h(for)f
5894
 
(di\013eren)m(t)g(arra)m(y)g(sizes.)52 b(See)35 b(Chapter)f(6)g([In-)
5895
 
510 518 y(stallation)c(and)g(Customization],)g(page)i(55,)g(for)e(more)
5896
 
h(information.\))41 b(T)-8 b(ransforms)29 b(whose)510
5897
 
628 y(sizes)h(are)h(p)s(o)m(w)m(ers)f(of)h(2)f(are)h(esp)s(ecially)e
5898
 
(fast.)225 766 y Fs(\017)60 b Fo(dir)31 b Fu(is)g(the)g(direction)g(of)
5899
 
h(the)g(desired)e(transform,)h(either)h Fo(FFTW_REAL_TO_COMPLEX)26
5900
 
b Fu(or)31 b Fo(FFTW_)330 876 y(COMPLEX_TO_REAL)p Fu(,)h(corresp)s
5901
 
(onding)g(to)j Fo(FFTW_FORWARD)c Fu(or)k Fo(FFTW_BACKWARD)p
5902
 
Fu(,)d(resp)s(ectiv)m(ely)-8 b(.)225 1123 y Fs(\017)60
5903
 
b Fo(flags)29 b Fu(is)g(a)i(b)s(o)s(olean)e(OR)h(\(`)p
5904
 
Fo(|)p Fu('\))i(of)e(zero)h(or)g(more)f(of)h(the)f(follo)m(wing:)379
5905
 
1262 y Fs(\000)60 b Fo(FFTW_MEASURE)p Fu(:)43 b(this)32
 
5958
TeXDict begin 27 28 bop 150 -116 a Fv(Chapter)30 b(3:)41
 
5959
b(FFTW)31 b(Reference)2354 b(27)225 299 y Ft(\017)60
 
5960
b Fp(flags)29 b Fv(is)g(a)i(b)s(o)s(olean)e(OR)h(\(`)p
 
5961
Fp(|)p Fv('\))i(of)e(zero)h(or)g(more)f(of)h(the)f(follo)m(wing:)379
 
5962
439 y Ft(\000)60 b Fp(FFTW_MEASURE)p Fv(:)43 b(this)32
5906
5963
b(\015ag)i(tells)e(RFFTW)i(to)g(\014nd)e(the)h(optimal)g(plan)e(b)m(y)j
5907
 
(actually)e Fq(c)-5 b(om-)510 1371 y(puting)45 b Fu(sev)m(eral)37
 
5964
(actually)e Fr(c)-5 b(om-)510 548 y(puting)45 b Fv(sev)m(eral)37
5908
5965
b(FFTs)g(and)f(measuring)g(their)g(execution)g(time.)60
5909
 
b(Dep)s(ending)35 b(on)i(the)g(in-)510 1481 y(stallation,)30
5910
 
b(this)f(can)i(tak)m(e)h(some)e(time.)379 1619 y Fs(\000)60
5911
 
b Fo(FFTW_ESTIMATE)p Fu(:)34 b(do)25 b(not)g(run)f(an)m(y)h(FFT)h(and)e
 
5966
b(Dep)s(ending)35 b(on)i(the)g(in-)510 658 y(stallation,)30
 
5967
b(this)f(can)i(tak)m(e)h(some)e(time.)379 798 y Ft(\000)60
 
5968
b Fp(FFTW_ESTIMATE)p Fv(:)34 b(do)25 b(not)g(run)f(an)m(y)h(FFT)h(and)e
5912
5969
(pro)m(vide)g(a)h(\\reasonable")h(plan)e(\(for)h(a)g(RISC)510
5913
 
1729 y(pro)s(cessor)37 b(with)g(man)m(y)h(registers\).)62
5914
 
b(If)38 b(neither)e Fo(FFTW_ESTIMATE)e Fu(nor)k Fo(FFTW_MEASURE)c
5915
 
Fu(is)510 1838 y(pro)m(vided,)29 b(the)i(default)e(is)h
5916
 
Fo(FFTW_ESTIMATE)p Fu(.)379 1977 y Fs(\000)60 b Fo(FFTW_OUT_OF_PLACE)p
5917
 
Fu(:)41 b(pro)s(duce)32 b(a)i(plan)e(assuming)f(that)j(the)f(input)e
5918
 
(and)i(output)g(arra)m(ys)510 2086 y(will)28 b(b)s(e)h(distinct)g
5919
 
(\(this)h(is)f(the)i(default\).)379 2225 y Fs(\000)60
5920
 
b Fo(FFTW_IN_PLACE)p Fu(:)39 b(pro)s(duce)31 b(a)h(plan)e(assuming)g
 
5970
908 y(pro)s(cessor)37 b(with)g(man)m(y)h(registers\).)62
 
5971
b(If)38 b(neither)e Fp(FFTW_ESTIMATE)e Fv(nor)k Fp(FFTW_MEASURE)c
 
5972
Fv(is)510 1017 y(pro)m(vided,)29 b(the)i(default)e(is)h
 
5973
Fp(FFTW_ESTIMATE)p Fv(.)379 1157 y Ft(\000)60 b Fp(FFTW_OUT_OF_PLACE)p
 
5974
Fv(:)41 b(pro)s(duce)32 b(a)i(plan)e(assuming)f(that)j(the)f(input)e
 
5975
(and)i(output)g(arra)m(ys)510 1267 y(will)28 b(b)s(e)h(distinct)g
 
5976
(\(this)h(is)f(the)i(default\).)379 1407 y Ft(\000)60
 
5977
b Fp(FFTW_IN_PLACE)p Fv(:)39 b(pro)s(duce)31 b(a)h(plan)e(assuming)g
5921
5978
(that)i(y)m(ou)g(w)m(an)m(t)g(the)g(output)f(in)f(the)i(input)510
5922
 
2334 y(arra)m(y)-8 b(.)41 b(The)28 b(algorithm)g(used)f(is)h(not)g
 
5979
1516 y(arra)m(y)-8 b(.)41 b(The)28 b(algorithm)g(used)f(is)h(not)g
5923
5980
(necessarily)g(in)f(place:)40 b(RFFTW)29 b(is)e(able)h(to)i(compute)510
5924
 
2444 y(true)43 b(in-place)f(transforms)g(only)g(for)g(small)g(v)-5
5925
 
b(alues)42 b(of)h Fo(n)p Fu(.)78 b(If)42 b(RFFTW)i(is)e(not)h(able)f
5926
 
(to)510 2553 y(compute)g(the)h(transform)e(in-place,)j(it)e(will)d
 
5981
1626 y(true)43 b(in-place)f(transforms)g(only)g(for)g(small)g(v)-5
 
5982
b(alues)42 b(of)h Fp(n)p Fv(.)78 b(If)42 b(RFFTW)i(is)e(not)h(able)f
 
5983
(to)510 1735 y(compute)g(the)h(transform)e(in-place,)j(it)e(will)d
5927
5984
(allo)s(cate)j(a)h(temp)s(orary)e(arra)m(y)i(\(unless)e(y)m(ou)510
5928
 
2663 y(pro)m(vide)d(one)i(y)m(ourself)7 b(\),)41 b(compute)f(the)g
 
5985
1845 y(pro)m(vide)d(one)i(y)m(ourself)7 b(\),)41 b(compute)f(the)g
5929
5986
(transform)e(out)i(of)f(place,)j(and)c(cop)m(y)j(the)e(result)510
5930
 
2773 y(bac)m(k.)60 b Fq(Warning:)54 b(This)39 b(option)h(changes)f(the)
 
5987
1955 y(bac)m(k.)60 b Fr(Warning:)54 b(This)39 b(option)h(changes)f(the)
5931
5988
g(me)-5 b(aning)39 b(of)g(some)g(p)-5 b(ar)g(ameters)42
5932
 
b(of)c Fo(rfftw)510 2882 y Fu(\(see)31 b(Section)g(3.4.2)h([Computing)d
 
5989
b(of)c Fp(rfftw)510 2064 y Fv(\(see)31 b(Section)g(3.4.2)h([Computing)d
5933
5990
(the)h(Real)h(One-dimensional)c(T)-8 b(ransform],)30
5934
 
b(page)h(27\).)510 3021 y(The)f(default)f(mo)s(de)h(of)h(op)s(eration)f
5935
 
(is)f Fo(FFTW_OUT_OF_PLACE)p Fu(.)379 3159 y Fs(\000)60
5936
 
b Fo(FFTW_USE_WISDOM)p Fu(:)37 b(use)30 b(an)m(y)h Fo(wisdom)e
5937
 
Fu(that)i(is)e(a)m(v)-5 b(ailable)30 b(to)h(help)e(in)h(the)g(creation)
5938
 
h(of)g(the)510 3268 y(plan.)50 b(\(See)34 b(Section)g(2.6)h([W)-8
 
5991
b(page)h(27\).)510 2204 y(The)f(default)f(mo)s(de)h(of)h(op)s(eration)f
 
5992
(is)f Fp(FFTW_OUT_OF_PLACE)p Fv(.)379 2344 y Ft(\000)60
 
5993
b Fp(FFTW_USE_WISDOM)p Fv(:)37 b(use)30 b(an)m(y)h Fp(wisdom)e
 
5994
Fv(that)i(is)e(a)m(v)-5 b(ailable)30 b(to)h(help)e(in)h(the)g(creation)
 
5995
h(of)g(the)510 2454 y(plan.)50 b(\(See)34 b(Section)g(2.6)h([W)-8
5939
5996
b(ords)34 b(of)h(Wisdom],)f(page)h(13.\))52 b(This)32
5940
 
b(can)i(greatly)h(sp)s(eed)e(the)510 3378 y(creation)27
5941
 
b(of)g(plans,)g(esp)s(ecially)e(with)g(the)j Fo(FFTW_MEASURE)23
5942
 
b Fu(option.)39 b Fo(FFTW_ESTIMATE)23 b Fu(plans)510
5943
 
3488 y(can)28 b(also)g(tak)m(e)i(adv)-5 b(an)m(tage)30
5944
 
b(of)e Fo(wisdom)e Fu(to)j(pro)s(duce)e(a)h(more)g(optimal)f(plan)g
5945
 
(\(based)h(on)g(past)510 3597 y(measuremen)m(ts\))i(than)g(the)g
 
5997
b(can)i(greatly)h(sp)s(eed)e(the)510 2563 y(creation)27
 
5998
b(of)g(plans,)g(esp)s(ecially)e(with)g(the)j Fp(FFTW_MEASURE)23
 
5999
b Fv(option.)39 b Fp(FFTW_ESTIMATE)23 b Fv(plans)510
 
6000
2673 y(can)28 b(also)g(tak)m(e)i(adv)-5 b(an)m(tage)30
 
6001
b(of)e Fp(wisdom)e Fv(to)j(pro)s(duce)e(a)h(more)g(optimal)f(plan)g
 
6002
(\(based)h(on)g(past)510 2783 y(measuremen)m(ts\))i(than)g(the)g
5946
6003
(estimation)f(heuristic)f(w)m(ould)g(normally)g(generate.)42
5947
 
b(When)30 b(the)510 3707 y Fo(FFTW_MEASURE)36 b Fu(option)j(is)f(used,)
5948
 
j(new)d Fo(wisdom)g Fu(will)f(also)i(b)s(e)f(generated)j(if)d(the)h
5949
 
(curren)m(t)510 3816 y(transform)30 b(size)g(is)f(not)i(completely)f
5950
 
(understo)s(o)s(d)f(b)m(y)h(existing)f Fo(wisdom)p Fu(.)225
5951
 
3955 y Fs(\017)60 b Fo(in)p Fu(,)29 b Fo(out)p Fu(,)f
5952
 
Fo(istride)p Fu(,)g Fo(ostride)f Fu(\(only)h(for)h Fo
5953
 
(rfftw_create_plan_specifi)o(c)p Fu(\):)34 b(see)c(corresp)s(ond-)330
5954
 
4064 y(ing)38 b(argumen)m(ts)i(in)e(the)h(description)e(of)i
5955
 
Fo(rfftw)p Fu(.)66 b(\(See)40 b(Section)f(3.4.2)i([Computing)d(the)h
5956
 
(Real)330 4174 y(One-dimensional)23 b(T)-8 b(ransform],)27
5957
 
b(page)g(27.\))40 b(In)26 b(particular,)f(the)i Fo(out)e
5958
 
Fu(and)g Fo(ostride)g Fu(parameters)330 4283 y(ha)m(v)m(e)31
5959
 
b(the)f(same)h(sp)s(ecial)d(meaning)h(for)h Fo(FFTW_IN_PLACE)c
5960
 
Fu(transforms)j(as)h(they)g(ha)m(v)m(e)h(for)f Fo(rfftw)p
5961
 
Fu(.)150 4523 y Fi(3.4.2)63 b(Computing)41 b(the)f(Real)h
5962
 
(One-dimensional)g(T)-10 b(ransform)390 4717 y Fo(#include)46
5963
 
b(<rfftw.h>)390 4925 y(void)h(rfftw\(rfftw_plan)c(plan,)j(int)h
5964
 
(howmany,)915 5029 y(fftw_real)e(*in,)i(int)g(istride,)e(int)i(idist,)
5965
 
915 5132 y(fftw_real)e(*out,)i(int)g(ostride,)e(int)i(odist\);)390
5966
 
5340 y(void)g(rfftw_one\(rfftw_plan)42 b(plan,)k(fftw_real)g(*in,)g
5967
 
(fftw_real)f(*out\);)p eop
5968
 
%%Page: 28 30
5969
 
28 29 bop 150 -116 a Fu(28)3232 b(FFTW)275 299 y(The)23
5970
 
b(function)h Fo(rfftw)f Fu(computes)h(the)h(Real)f(One-dimensional)e(F)
5971
 
-8 b(ourier)24 b(T)-8 b(ransform,)25 b(using)d(a)j(plan)150
5972
 
408 y(created)k(b)m(y)e Fo(rfftw_create_plan)c Fu(\(see)29
 
6004
b(When)30 b(the)510 2892 y Fp(FFTW_MEASURE)36 b Fv(option)j(is)f(used,)
 
6005
j(new)d Fp(wisdom)g Fv(will)f(also)i(b)s(e)f(generated)j(if)d(the)h
 
6006
(curren)m(t)510 3002 y(transform)30 b(size)g(is)f(not)i(completely)f
 
6007
(understo)s(o)s(d)f(b)m(y)h(existing)f Fp(wisdom)p Fv(.)225
 
6008
3142 y Ft(\017)60 b Fp(in)p Fv(,)29 b Fp(out)p Fv(,)f
 
6009
Fp(istride)p Fv(,)g Fp(ostride)f Fv(\(only)h(for)h Fp
 
6010
(rfftw_create_plan_specifi)o(c)p Fv(\):)34 b(see)c(corresp)s(ond-)330
 
6011
3251 y(ing)38 b(argumen)m(ts)i(in)e(the)h(description)e(of)i
 
6012
Fp(rfftw)p Fv(.)66 b(\(See)40 b(Section)f(3.4.2)i([Computing)d(the)h
 
6013
(Real)330 3361 y(One-dimensional)23 b(T)-8 b(ransform],)27
 
6014
b(page)g(27.\))40 b(In)26 b(particular,)f(the)i Fp(out)e
 
6015
Fv(and)g Fp(ostride)g Fv(parameters)330 3470 y(ha)m(v)m(e)31
 
6016
b(the)f(same)h(sp)s(ecial)d(meaning)h(for)h Fp(FFTW_IN_PLACE)c
 
6017
Fv(transforms)j(as)h(they)g(ha)m(v)m(e)h(for)f Fp(rfftw)p
 
6018
Fv(.)150 3717 y Fh(3.4.2)63 b(Computing)41 b(the)f(Real)h
 
6019
(One-dimensional)g(T)-10 b(ransform)390 3878 y Fp(#include)46
 
6020
b(<rfftw.h>)390 4086 y(void)h(rfftw\(rfftw_plan)c(plan,)j(int)h
 
6021
(howmany,)915 4190 y(fftw_real)e(*in,)i(int)g(istride,)e(int)i(idist,)
 
6022
915 4294 y(fftw_real)e(*out,)i(int)g(ostride,)e(int)i(odist\);)390
 
6023
4501 y(void)g(rfftw_one\(rfftw_plan)42 b(plan,)k(fftw_real)g(*in,)g
 
6024
(fftw_real)f(*out\);)275 4647 y Fv(The)23 b(function)h
 
6025
Fp(rfftw)f Fv(computes)h(the)h(Real)f(One-dimensional)e(F)-8
 
6026
b(ourier)24 b(T)-8 b(ransform,)25 b(using)d(a)j(plan)150
 
6027
4756 y(created)k(b)m(y)e Fp(rfftw_create_plan)c Fv(\(see)29
5973
6028
b(Section)f(3.4.1)h([Plan)e(Creation)h(for)f(Real)h(One-dimensional)150
5974
 
518 y(T)-8 b(ransforms],)47 b(page)f(26\).)84 b(The)44
5975
 
b(function)f Fo(rfftw_one)f Fu(pro)m(vides)i(a)h(simpli\014ed)40
5976
 
b(in)m(terface)45 b(for)g(the)150 628 y(common)31 b(case)g(of)g(single)
5977
 
e(input)f(arra)m(y)j(of)f(stride)g(1.)275 777 y Fq(Imp)-5
5978
 
b(ortant:)63 b Fu(When)37 b(in)m(v)m(ok)m(ed)h(for)g(an)g
5979
 
(out-of-place,)i Fo(FFTW_COMPLEX_TO_REAL)33 b Fu(transform,)39
5980
 
b(the)150 886 y(input)d(arra)m(y)j(is)e(o)m(v)m(erwritten)h(with)f
 
6029
4866 y(T)-8 b(ransforms],)47 b(page)f(26\).)84 b(The)44
 
6030
b(function)f Fp(rfftw_one)f Fv(pro)m(vides)i(a)h(simpli\014ed)40
 
6031
b(in)m(terface)45 b(for)g(the)150 4975 y(common)31 b(case)g(of)g
 
6032
(single)e(input)f(arra)m(y)j(of)f(stride)g(1.)275 5121
 
6033
y Fr(Imp)-5 b(ortant:)63 b Fv(When)37 b(in)m(v)m(ok)m(ed)h(for)g(an)g
 
6034
(out-of-place,)i Fp(FFTW_COMPLEX_TO_REAL)33 b Fv(transform,)39
 
6035
b(the)150 5230 y(input)d(arra)m(y)j(is)e(o)m(v)m(erwritten)h(with)f
5981
6036
(scratc)m(h)i(v)-5 b(alues)37 b(b)m(y)h(these)h(routines.)62
5982
 
b(The)38 b(input)e(arra)m(y)j(is)e(not)150 996 y(mo)s(di\014ed)28
5983
 
b(for)i Fo(FFTW_REAL_TO_COMPLEX)25 b Fu(transforms.)150
5984
 
1250 y Fi(Argumen)m(ts)225 1457 y Fs(\017)60 b Fo(plan)31
5985
 
b Fu(is)h(the)g(plan)f(created)i(b)m(y)g Fo(rfftw_create_plan)27
5986
 
b Fu(\(see)33 b(Section)g(3.4.1)h([Plan)d(Creation)h(for)330
5987
 
1567 y(Real)e(One-dimensional)e(T)-8 b(ransforms],)30
5988
 
b(page)h(26\).)225 1709 y Fs(\017)60 b Fo(howmany)30
5989
 
b Fu(is)h(the)g(n)m(um)m(b)s(er)g(of)h(transforms)f Fo(rfftw)f
5990
 
Fu(will)f(compute.)45 b(It)32 b(is)f(faster)h(to)g(tell)f(RFFTW)330
5991
 
1818 y(to)g(compute)g(man)m(y)f(transforms,)g(instead)g(of)g(simply)e
5992
 
(calling)h Fo(rfftw)g Fu(man)m(y)h(times.)225 1960 y
5993
 
Fs(\017)60 b Fo(in)p Fu(,)49 b Fo(istride)43 b Fu(and)i
5994
 
Fo(idist)f Fu(describ)s(e)g(the)i(input)e(arra)m(y\(s\).)87
5995
 
b(There)45 b(are)h(t)m(w)m(o)h(cases.)87 b(If)45 b(the)330
5996
 
2070 y Fo(plan)36 b Fu(de\014nes)h(a)h Fo(FFTW_REAL_TO_COMPLEX)32
5997
 
b Fu(transform,)39 b Fo(in)e Fu(is)g(a)g(real)h(arra)m(y)-8
5998
 
b(.)63 b(Otherwise,)38 b(for)330 2179 y Fo(FFTW_COMPLEX_TO_REAL)30
5999
 
b Fu(transforms,)35 b Fo(in)g Fu(is)f(a)h(halfcomplex)f(arra)m(y)i
6000
 
Fq(whose)i(c)-5 b(ontents)38 b(wil)5 b(l)37 b(b)-5 b(e)330
6001
 
2289 y(destr)g(oye)g(d)p Fu(.)225 2431 y Fs(\017)60 b
6002
 
Fo(out)p Fu(,)32 b Fo(ostride)d Fu(and)j Fo(odist)e Fu(describ)s(e)g
 
6037
b(The)38 b(input)e(arra)m(y)j(is)e(not)150 5340 y(mo)s(di\014ed)28
 
6038
b(for)i Fp(FFTW_REAL_TO_COMPLEX)25 b Fv(transforms.)p
 
6039
eop end
 
6040
%%Page: 28 30
 
6041
TeXDict begin 28 29 bop 150 -116 a Fv(28)3232 b(FFTW)150
 
6042
299 y Fh(Argumen)m(ts)225 467 y Ft(\017)60 b Fp(plan)31
 
6043
b Fv(is)h(the)g(plan)f(created)i(b)m(y)g Fp(rfftw_create_plan)27
 
6044
b Fv(\(see)33 b(Section)g(3.4.1)h([Plan)d(Creation)h(for)330
 
6045
576 y(Real)e(One-dimensional)e(T)-8 b(ransforms],)30
 
6046
b(page)h(26\).)225 713 y Ft(\017)60 b Fp(howmany)30 b
 
6047
Fv(is)h(the)g(n)m(um)m(b)s(er)g(of)h(transforms)f Fp(rfftw)f
 
6048
Fv(will)f(compute.)45 b(It)32 b(is)f(faster)h(to)g(tell)f(RFFTW)330
 
6049
822 y(to)g(compute)g(man)m(y)f(transforms,)g(instead)g(of)g(simply)e
 
6050
(calling)h Fp(rfftw)g Fv(man)m(y)h(times.)225 959 y Ft(\017)60
 
6051
b Fp(in)p Fv(,)49 b Fp(istride)43 b Fv(and)i Fp(idist)f
 
6052
Fv(describ)s(e)g(the)i(input)e(arra)m(y\(s\).)87 b(There)45
 
6053
b(are)h(t)m(w)m(o)h(cases.)87 b(If)45 b(the)330 1069
 
6054
y Fp(plan)36 b Fv(de\014nes)h(a)h Fp(FFTW_REAL_TO_COMPLEX)32
 
6055
b Fv(transform,)39 b Fp(in)e Fv(is)g(a)g(real)h(arra)m(y)-8
 
6056
b(.)63 b(Otherwise,)38 b(for)330 1178 y Fp(FFTW_COMPLEX_TO_REAL)30
 
6057
b Fv(transforms,)35 b Fp(in)g Fv(is)f(a)h(halfcomplex)f(arra)m(y)i
 
6058
Fr(whose)i(c)-5 b(ontents)38 b(wil)5 b(l)37 b(b)-5 b(e)330
 
6059
1288 y(destr)g(oye)g(d)p Fv(.)225 1424 y Ft(\017)60 b
 
6060
Fp(out)p Fv(,)32 b Fp(ostride)d Fv(and)j Fp(odist)e Fv(describ)s(e)g
6003
6061
(the)i(output)g(arra)m(y\(s\),)h(and)e(ha)m(v)m(e)i(the)f(same)h
6004
 
(meaning)e(as)330 2540 y(the)g(corresp)s(onding)d(parameters)j(for)f
6005
 
(the)g(input)f(arra)m(y)-8 b(.)379 2682 y Fs(\000)60
6006
 
b Fq(In-plac)-5 b(e)46 b(tr)-5 b(ansforms)7 b Fu(:)73
6007
 
b(If)44 b(the)h Fo(plan)e Fu(sp)s(eci\014es)g(an)i(in-place)e
6008
 
(transform,)48 b Fo(ostride)43 b Fu(and)510 2792 y Fo(odist)33
6009
 
b Fu(are)h(alw)m(a)m(ys)h(ignored.)51 b(If)33 b Fo(out)g
6010
 
Fu(is)h Fo(NULL)p Fu(,)g Fo(out)f Fu(is)g(ignored,)h(to)s(o.)53
6011
 
b(Otherwise,)34 b Fo(out)f Fu(is)510 2901 y(in)m(terpreted)h(as)h(a)g
6012
 
(p)s(oin)m(ter)f(to)i(an)e(arra)m(y)i(of)f Fo(n)f Fu(complex)h(n)m(um)m
6013
 
(b)s(ers,)f(that)i(FFTW)f(will)d(use)510 3011 y(as)j(temp)s(orary)f
 
6062
(meaning)e(as)330 1534 y(the)g(corresp)s(onding)d(parameters)j(for)f
 
6063
(the)g(input)f(arra)m(y)-8 b(.)379 1671 y Ft(\000)60
 
6064
b Fr(In-plac)-5 b(e)46 b(tr)-5 b(ansforms)7 b Fv(:)73
 
6065
b(If)44 b(the)h Fp(plan)e Fv(sp)s(eci\014es)g(an)i(in-place)e
 
6066
(transform,)48 b Fp(ostride)43 b Fv(and)510 1780 y Fp(odist)33
 
6067
b Fv(are)h(alw)m(a)m(ys)h(ignored.)51 b(If)33 b Fp(out)g
 
6068
Fv(is)h Fp(NULL)p Fv(,)g Fp(out)f Fv(is)g(ignored,)h(to)s(o.)53
 
6069
b(Otherwise,)34 b Fp(out)f Fv(is)510 1890 y(in)m(terpreted)h(as)h(a)g
 
6070
(p)s(oin)m(ter)f(to)i(an)e(arra)m(y)i(of)f Fp(n)f Fv(complex)h(n)m(um)m
 
6071
(b)s(ers,)f(that)i(FFTW)f(will)d(use)510 2000 y(as)j(temp)s(orary)f
6014
6072
(space)h(to)h(p)s(erform)d(the)i(in-place)e(computation.)54
6015
 
b Fo(out)34 b Fu(is)f(used)h(as)h(scratc)m(h)510 3120
 
6073
b Fp(out)34 b Fv(is)f(used)h(as)h(scratc)m(h)510 2109
6016
6074
y(space)41 b(and)g(its)f(con)m(ten)m(ts)j(destro)m(y)m(ed.)73
6017
 
b(In)40 b(this)g(case,)k Fo(out)c Fu(m)m(ust)h(b)s(e)f(an)h(ordinary)e
6018
 
(arra)m(y)510 3230 y(whose)30 b(elemen)m(ts)h(are)g(con)m(tiguous)f(in)
6019
 
f(memory)i(\(no)f(striding\).)275 3412 y(The)40 b(function)g
6020
 
Fo(rfftw_one)f Fu(transforms)i(a)g(single,)i(con)m(tiguous)f(input)d
6021
 
(arra)m(y)j(to)g(a)g(con)m(tiguous)150 3521 y(output)30
 
6075
b(In)40 b(this)g(case,)k Fp(out)c Fv(m)m(ust)h(b)s(e)f(an)h(ordinary)e
 
6076
(arra)m(y)510 2219 y(whose)30 b(elemen)m(ts)h(are)g(con)m(tiguous)f(in)
 
6077
f(memory)i(\(no)f(striding\).)275 2385 y(The)40 b(function)g
 
6078
Fp(rfftw_one)f Fv(transforms)i(a)g(single,)i(con)m(tiguous)f(input)d
 
6079
(arra)m(y)j(to)g(a)g(con)m(tiguous)150 2494 y(output)30
6022
6080
b(arra)m(y)-8 b(.)42 b(By)30 b(de\014nition,)f(the)h(call)390
6023
 
3665 y Fo(rfftw_one\(plan,)44 b(in,)j(out\))275 3814
6024
 
y Fu(is)29 b(equiv)-5 b(alen)m(t)30 b(to)390 3957 y Fo(rfftw\(plan,)45
6025
 
b(1,)i(in,)g(1,)g(1,)g(out,)g(1,)g(1\))150 4211 y Fi(3.4.3)63
 
6081
2627 y Fp(rfftw_one\(plan,)44 b(in,)j(out\))275 2766
 
6082
y Fv(is)29 b(equiv)-5 b(alen)m(t)30 b(to)390 2899 y Fp(rfftw\(plan,)45
 
6083
b(1,)i(in,)g(1,)g(0,)g(out,)g(1,)g(0\))150 3132 y Fh(3.4.3)63
6026
6084
b(Destro)m(ying)41 b(a)g(Real)f(One-dimensional)i(Plan)390
6027
 
4413 y Fo(#include)k(<rfftw.h>)390 4620 y(void)h
 
6085
3294 y Fp(#include)k(<rfftw.h>)390 3501 y(void)h
6028
6086
(rfftw_destroy_plan\(rfft)o(w_pl)o(an)41 b(plan\);)275
6029
 
4769 y Fu(The)h(function)g Fo(rfftw_destroy_plan)d Fu(frees)k(the)h
6030
 
(plan)e Fo(plan)g Fu(and)h(releases)g(all)f(the)i(memory)150
6031
 
4879 y(asso)s(ciated)31 b(with)e(it.)40 b(After)31 b(destruction,)f(a)g
6032
 
(plan)f(is)h(no)g(longer)g(v)-5 b(alid.)150 5133 y Fi(3.4.4)63
6033
 
b(What)40 b(RFFTW)h(Really)f(Computes)275 5340 y Fu(In)29
 
6087
3640 y Fv(The)h(function)g Fp(rfftw_destroy_plan)d Fv(frees)k(the)h
 
6088
(plan)e Fp(plan)g Fv(and)h(releases)g(all)f(the)i(memory)150
 
6089
3750 y(asso)s(ciated)31 b(with)e(it.)40 b(After)31 b(destruction,)f(a)g
 
6090
(plan)f(is)h(no)g(longer)g(v)-5 b(alid.)150 3982 y Fh(3.4.4)63
 
6091
b(What)40 b(RFFTW)h(Really)f(Computes)275 4179 y Fv(In)29
6034
6092
b(this)g(section,)i(w)m(e)g(de\014ne)f(precisely)f(what)h(RFFTW)h
6035
 
(computes.)p eop
 
6093
(computes.)275 4318 y(The)40 b(real)g(to)h(complex)g(\()p
 
6094
Fp(FFTW_REAL_TO_COMPLEX)p Fv(\))36 b(transform)k(of)g(a)i(real)e(arra)m
 
6095
(y)h Fm(X)48 b Fv(of)41 b(size)f Fm(n)150 4428 y Fv(computes)31
 
6096
b(an)f(hermitian)e(arra)m(y)j Fm(Y)20 b Fv(,)31 b(where)1503
 
6097
4712 y Fm(Y)1556 4726 y Fl(i)1608 4712 y Fv(=)1704 4606
 
6098
y Fl(n)p Fk(\000)p Fo(1)1707 4631 y Fg(X)1710 4808 y
 
6099
Fl(j)s Fo(=0)1846 4712 y Fm(X)1921 4726 y Fl(j)1956 4712
 
6100
y Fm(e)1998 4675 y Fk(\000)p Fo(2)p Fl(\031)r(ij)2177
 
6101
4630 y Fk(p)p 2233 4630 85 3 v 2233 4675 a(\000)p Fo(1)p
 
6102
Fl(=n)150 5011 y Fv(\(That)26 b Fm(Y)45 b Fv(is)24 b(a)i(hermitian)d
 
6103
(arra)m(y)j(is)e(not)h(in)m(tended)g(to)h(b)s(e)e(ob)m(vious,)i
 
6104
(although)f(the)g(pro)s(of)g(is)f(easy)-8 b(.\))40 b(The)150
 
6105
5121 y(hermitian)32 b(arra)m(y)i Fm(Y)54 b Fv(is)32 b(stored)i(in)f
 
6106
(halfcomplex)f(order)h(\(see)i(Section)e(3.1)i([Data)h(T)m(yp)s(es],)e
 
6107
(page)g(17\).)150 5230 y(Curren)m(tly)-8 b(,)24 b(RFFTW)h(pro)m(vides)f
 
6108
(no)g(w)m(a)m(y)h(to)g(compute)g(a)g(real)f(to)h(complex)f(transform)f
 
6109
(with)g(a)i(p)s(ositiv)m(e)150 5340 y(sign)k(in)g(the)i(exp)s(onen)m
 
6110
(t.)p eop end
6036
6111
%%Page: 29 31
6037
 
29 30 bop 150 -116 a Fu(Chapter)30 b(3:)41 b(FFTW)31
6038
 
b(Reference)2354 b(29)275 299 y(The)40 b(real)g(to)h(complex)g(\()p
6039
 
Fo(FFTW_REAL_TO_COMPLEX)p Fu(\))36 b(transform)k(of)g(a)i(real)e(arra)m
6040
 
(y)h Fl(X)48 b Fu(of)41 b(size)f Fl(n)150 408 y Fu(computes)31
6041
 
b(an)f(hermitian)e(arra)m(y)j Fl(Y)20 b Fu(,)31 b(where)1503
6042
 
692 y Fl(Y)1556 706 y Fk(i)1608 692 y Fu(=)1704 586 y
6043
 
Fk(n)p Fj(\000)p Fn(1)1707 611 y Fh(X)1710 788 y Fk(j)s
6044
 
Fn(=0)1846 692 y Fl(X)1921 706 y Fk(j)1956 692 y Fl(e)1998
6045
 
654 y Fj(\000)p Fn(2)p Fk(\031)r(ij)2177 609 y Fj(p)p
6046
 
2233 609 85 3 v 2233 654 a(\000)p Fn(1)p Fk(=n)150 990
6047
 
y Fu(\(That)26 b Fl(Y)45 b Fu(is)24 b(a)i(hermitian)d(arra)m(y)j(is)e
6048
 
(not)h(in)m(tended)g(to)h(b)s(e)e(ob)m(vious,)i(although)f(the)g(pro)s
6049
 
(of)g(is)f(easy)-8 b(.\))40 b(The)150 1100 y(hermitian)32
6050
 
b(arra)m(y)i Fl(Y)54 b Fu(is)32 b(stored)i(in)f(halfcomplex)f(order)h
6051
 
(\(see)i(Section)e(3.1)i([Data)h(T)m(yp)s(es],)e(page)g(17\).)150
6052
 
1209 y(Curren)m(tly)-8 b(,)24 b(RFFTW)h(pro)m(vides)f(no)g(w)m(a)m(y)h
6053
 
(to)g(compute)g(a)g(real)f(to)h(complex)f(transform)f(with)g(a)i(p)s
6054
 
(ositiv)m(e)150 1319 y(sign)k(in)g(the)i(exp)s(onen)m(t.)275
6055
 
1457 y(The)d(complex)g(to)i(real)e(\()p Fo(FFTW_COMPLEX_TO_REAL)p
6056
 
Fu(\))c(transform)29 b(of)g(a)g(hermitian)e(arra)m(y)i
6057
 
Fl(X)36 b Fu(of)29 b(size)150 1567 y Fl(n)h Fu(computes)g(a)h(real)f
6058
 
(arra)m(y)h Fl(Y)20 b Fu(,)31 b(where)1529 1850 y Fl(Y)1582
6059
 
1864 y Fk(i)1634 1850 y Fu(=)1730 1745 y Fk(n)p Fj(\000)p
6060
 
Fn(1)1733 1769 y Fh(X)1736 1946 y Fk(j)s Fn(=0)1872 1850
6061
 
y Fl(X)1947 1864 y Fk(j)1982 1850 y Fl(e)2024 1813 y
6062
 
Fn(2)p Fk(\031)r(ij)2151 1768 y Fj(p)p 2207 1768 V 2207
6063
 
1813 a(\000)p Fn(1)p Fk(=n)150 2148 y Fu(\(That)39 b
6064
 
Fl(Y)58 b Fu(is)37 b(a)i(real)f(arra)m(y)h(is)f(not)g(in)m(tended)g(to)
6065
 
h(b)s(e)f(ob)m(vious,)i(although)d(the)i(pro)s(of)f(is)f(easy)-8
6066
 
b(.\))66 b(The)150 2258 y(hermitian)43 b(input)g(arra)m(y)i
6067
 
Fl(X)53 b Fu(is)44 b(stored)h(in)e(halfcomplex)h(order)h(\(see)g
6068
 
(Section)g(3.1)h([Data)h(T)m(yp)s(es],)150 2368 y(page)40
6069
 
b(17\).)68 b(Curren)m(tly)-8 b(,)40 b(RFFTW)f(pro)m(vides)f(no)h(w)m(a)
6070
 
m(y)h(to)g(compute)f(a)h(complex)e(to)i(real)f(transform)150
6071
 
2477 y(with)29 b(a)i(negativ)m(e)g(sign)f(in)f(the)h(exp)s(onen)m(t.)
6072
 
275 2615 y(Lik)m(e)j(FFTW,)i(RFFTW)f(computes)g(an)g(unnormalized)d
6073
 
(transform.)50 b(In)33 b(other)h(w)m(ords,)g(applying)150
6074
 
2725 y(the)44 b(real)f(to)i(complex)e(\(forw)m(ard\))h(and)f(then)h
6075
 
(the)f(complex)h(to)g(real)g(\(bac)m(kw)m(ard\))h(transform)e(will)150
6076
 
2835 y(m)m(ultiply)28 b(the)i(input)f(b)m(y)h Fl(n)p
6077
 
Fu(.)150 3101 y Ft(3.5)68 b(Real)46 b(Multi-dimensional)h(T)-11
6078
 
b(ransforms)44 b(Reference)275 3298 y Fu(The)39 b(m)m(ulti-dimensional)
6079
 
c(real)40 b(routines)e(are)i(generally)f(pre\014xed)f(with)h
6080
 
Fo(rfftwnd_)p Fu(.)66 b(Programs)150 3407 y(using)25
6081
 
b(RFFTWND)i(should)e(b)s(e)g(link)m(ed)g(with)g Fo(-lrfftw)j(-lfftw)h
6082
 
(-lm)d Fu(on)g(Unix)f(systems,)i(or)g(with)e(the)150
6083
 
3517 y(FFTW,)31 b(RFFTW,)h(and)d(standard)h(math)g(libraries)e(in)h
6084
 
(general.)150 3748 y Fi(3.5.1)63 b(Plan)40 b(Creation)h(for)g(Real)f
6085
 
(Multi-dimensional)i(T)-10 b(ransforms)390 3939 y Fo(#include)46
6086
 
b(<rfftw.h>)390 4147 y(rfftwnd_plan)e(rfftwnd_create_plan\(int)e(rank,)
6087
 
k(const)h(int)g(*n,)1965 4250 y(fftw_direction)d(dir,)j(int)f(flags\);)
6088
 
390 4458 y(rfftwnd_plan)e(rfftw2d_create_plan\(int)e(nx,)47
6089
 
b(int)g(ny,)1965 4562 y(fftw_direction)d(dir,)j(int)f(flags\);)390
6090
 
4769 y(rfftwnd_plan)e(rfftw3d_create_plan\(int)e(nx,)47
6091
 
b(int)g(ny,)g(int)g(nz,)1965 4873 y(fftw_direction)d(dir,)j(int)f
6092
 
(flags\);)275 5011 y Fu(The)25 b(function)g Fo(rfftwnd_create_plan)20
6093
 
b Fu(creates)28 b(a)e(plan,)g(whic)m(h)f(is)g(a)h(data)h(structure)e
6094
 
(con)m(taining)150 5121 y(all)e(the)h(information)e(that)i
6095
 
Fo(rfftwnd)e Fu(needs)h(in)f(order)i(to)g(compute)g(a)g(m)m
6096
 
(ulti-dimensional)c(real)j(F)-8 b(ourier)150 5230 y(transform.)42
 
6112
TeXDict begin 29 30 bop 150 -116 a Fv(Chapter)30 b(3:)41
 
6113
b(FFTW)31 b(Reference)2354 b(29)275 299 y(The)28 b(complex)g(to)i(real)
 
6114
e(\()p Fp(FFTW_COMPLEX_TO_REAL)p Fv(\))c(transform)29
 
6115
b(of)g(a)g(hermitian)e(arra)m(y)i Fm(X)36 b Fv(of)29
 
6116
b(size)150 408 y Fm(n)h Fv(computes)g(a)h(real)f(arra)m(y)h
 
6117
Fm(Y)20 b Fv(,)31 b(where)1529 688 y Fm(Y)1582 702 y
 
6118
Fl(i)1634 688 y Fv(=)1730 583 y Fl(n)p Fk(\000)p Fo(1)1733
 
6119
607 y Fg(X)1736 784 y Fl(j)s Fo(=0)1872 688 y Fm(X)1947
 
6120
702 y Fl(j)1982 688 y Fm(e)2024 651 y Fo(2)p Fl(\031)r(ij)2151
 
6121
606 y Fk(p)p 2207 606 85 3 v 2207 651 a(\000)p Fo(1)p
 
6122
Fl(=n)150 983 y Fv(\(That)39 b Fm(Y)58 b Fv(is)37 b(a)i(real)f(arra)m
 
6123
(y)h(is)f(not)g(in)m(tended)g(to)h(b)s(e)f(ob)m(vious,)i(although)d
 
6124
(the)i(pro)s(of)f(is)f(easy)-8 b(.\))66 b(The)150 1092
 
6125
y(hermitian)43 b(input)g(arra)m(y)i Fm(X)53 b Fv(is)44
 
6126
b(stored)h(in)e(halfcomplex)h(order)h(\(see)g(Section)g(3.1)h([Data)h
 
6127
(T)m(yp)s(es],)150 1202 y(page)40 b(17\).)68 b(Curren)m(tly)-8
 
6128
b(,)40 b(RFFTW)f(pro)m(vides)f(no)h(w)m(a)m(y)h(to)g(compute)f(a)h
 
6129
(complex)e(to)i(real)f(transform)150 1311 y(with)29 b(a)i(negativ)m(e)g
 
6130
(sign)f(in)f(the)h(exp)s(onen)m(t.)275 1447 y(Lik)m(e)j(FFTW,)i(RFFTW)f
 
6131
(computes)g(an)g(unnormalized)d(transform.)50 b(In)33
 
6132
b(other)h(w)m(ords,)g(applying)150 1557 y(the)44 b(real)f(to)i(complex)
 
6133
e(\(forw)m(ard\))h(and)f(then)h(the)f(complex)h(to)g(real)g(\(bac)m(kw)
 
6134
m(ard\))h(transform)e(will)150 1666 y(m)m(ultiply)28
 
6135
b(the)i(input)f(b)m(y)h Fm(n)p Fv(.)150 1927 y Fu(3.5)68
 
6136
b(Real)46 b(Multi-dimensional)h(T)-11 b(ransforms)44
 
6137
b(Reference)275 2121 y Fv(The)39 b(m)m(ulti-dimensional)c(real)40
 
6138
b(routines)e(are)i(generally)f(pre\014xed)f(with)h Fp(rfftwnd_)p
 
6139
Fv(.)66 b(Programs)150 2230 y(using)25 b(RFFTWND)i(should)e(b)s(e)g
 
6140
(link)m(ed)g(with)g Fp(-lrfftw)j(-lfftw)h(-lm)d Fv(on)g(Unix)f
 
6141
(systems,)i(or)g(with)e(the)150 2340 y(FFTW,)31 b(RFFTW,)h(and)d
 
6142
(standard)h(math)g(libraries)e(in)h(general.)150 2566
 
6143
y Fh(3.5.1)63 b(Plan)40 b(Creation)h(for)g(Real)f(Multi-dimensional)i
 
6144
(T)-10 b(ransforms)390 2728 y Fp(#include)46 b(<rfftw.h>)390
 
6145
2936 y(rfftwnd_plan)e(rfftwnd_create_plan\(int)e(rank,)k(const)h(int)g
 
6146
(*n,)1965 3040 y(fftw_direction)d(dir,)j(int)f(flags\);)390
 
6147
3247 y(rfftwnd_plan)e(rfftw2d_create_plan\(int)e(nx,)47
 
6148
b(int)g(ny,)1965 3351 y(fftw_direction)d(dir,)j(int)f(flags\);)390
 
6149
3559 y(rfftwnd_plan)e(rfftw3d_create_plan\(int)e(nx,)47
 
6150
b(int)g(ny,)g(int)g(nz,)1965 3662 y(fftw_direction)d(dir,)j(int)f
 
6151
(flags\);)275 3798 y Fv(The)25 b(function)g Fp(rfftwnd_create_plan)20
 
6152
b Fv(creates)28 b(a)e(plan,)g(whic)m(h)f(is)g(a)h(data)h(structure)e
 
6153
(con)m(taining)150 3908 y(all)e(the)h(information)e(that)i
 
6154
Fp(rfftwnd)e Fv(needs)h(in)f(order)i(to)g(compute)g(a)g(m)m
 
6155
(ulti-dimensional)c(real)j(F)-8 b(ourier)150 4017 y(transform.)42
6097
6156
b(Y)-8 b(ou)32 b(can)f(create)i(as)e(man)m(y)g(plans)f(as)h(y)m(ou)g
6098
6157
(need,)h(but)e(only)g(one)h(plan)f(for)h(a)g(giv)m(en)g(arra)m(y)150
6099
 
5340 y(size)j(is)e(required)g(\(a)i(plan)f(can)h(b)s(e)f(reused)g(man)m
6100
 
(y)g(times\).)51 b(The)33 b(functions)f Fo(rfftw2d_create_plan)p
6101
 
eop
 
6158
4127 y(size)j(is)e(required)g(\(a)i(plan)f(can)h(b)s(e)f(reused)g(man)m
 
6159
(y)g(times\).)51 b(The)33 b(functions)f Fp(rfftw2d_create_plan)150
 
6160
4236 y Fv(and)41 b Fp(rfftw3d_create_plan)35 b Fv(are)42
 
6161
b(optional,)h(alternativ)m(e)f(in)m(terfaces)f(to)h Fp
 
6162
(rfftwnd_create_plan)150 4346 y Fv(for)30 b(t)m(w)m(o)i(and)e(three)g
 
6163
(dimensions,)e(resp)s(ectiv)m(ely)-8 b(.)275 4482 y Fp
 
6164
(rfftwnd_create_plan)27 b Fv(returns)k(a)i(v)-5 b(alid)31
 
6165
b(plan,)h(or)g Fp(NULL)f Fv(if,)i(for)f(some)h(reason,)g(the)g(plan)e
 
6166
(can't)150 4591 y(b)s(e)f(created.)42 b(This)28 b(can)j(happ)s(en)d(if)
 
6167
i(the)g(argumen)m(ts)h(are)g(in)m(v)-5 b(alid)28 b(in)h(some)i(w)m(a)m
 
6168
(y)g(\(e.g.)42 b(if)29 b Fp(rank)h(<)g Fv(0\).)150 4818
 
6169
y Fh(Argumen)m(ts)225 4986 y Ft(\017)60 b Fp(rank)27
 
6170
b Fv(is)g(the)h(dimensionalit)m(y)d(of)j(the)g(arra)m(ys)g(to)g(b)s(e)f
 
6171
(transformed.)40 b(It)27 b(can)i(b)s(e)e(an)m(y)h(non-negativ)m(e)330
 
6172
5095 y(in)m(teger.)225 5230 y Ft(\017)60 b Fp(n)22 b
 
6173
Fv(is)g(a)g(p)s(oin)m(ter)g(to)h(an)f(arra)m(y)h(of)g
 
6174
Fp(rank)e Fv(in)m(tegers,)k(giving)c(the)i(size)f(of)h(eac)m(h)g
 
6175
(dimension)d(of)j(the)g(arra)m(ys)330 5340 y(to)35 b(b)s(e)e
 
6176
(transformed.)51 b(Note)36 b(that)e(these)h(are)f(alw)m(a)m(ys)h(the)f
 
6177
(dimensions)e(of)i(the)g Fr(r)-5 b(e)g(al)45 b Fv(arra)m(ys;)37
 
6178
b(the)p eop end
6102
6179
%%Page: 30 32
6103
 
30 31 bop 150 -116 a Fu(30)3232 b(FFTW)150 299 y(and)41
6104
 
b Fo(rfftw3d_create_plan)35 b Fu(are)42 b(optional,)h(alternativ)m(e)f
6105
 
(in)m(terfaces)f(to)h Fo(rfftwnd_create_plan)150 408
6106
 
y Fu(for)30 b(t)m(w)m(o)i(and)e(three)g(dimensions,)e(resp)s(ectiv)m
6107
 
(ely)-8 b(.)275 546 y Fo(rfftwnd_create_plan)27 b Fu(returns)k(a)i(v)-5
6108
 
b(alid)31 b(plan,)h(or)g Fo(NULL)f Fu(if,)i(for)f(some)h(reason,)g(the)
6109
 
g(plan)e(can't)150 655 y(b)s(e)f(created.)42 b(This)28
6110
 
b(can)j(happ)s(en)d(if)i(the)g(argumen)m(ts)h(are)g(in)m(v)-5
6111
 
b(alid)28 b(in)h(some)i(w)m(a)m(y)g(\(e.g.)42 b(if)29
6112
 
b Fo(rank)h(<)g Fu(0\).)150 885 y Fi(Argumen)m(ts)225
6113
 
1080 y Fs(\017)60 b Fo(rank)27 b Fu(is)g(the)h(dimensionalit)m(y)d(of)j
6114
 
(the)g(arra)m(ys)g(to)g(b)s(e)f(transformed.)40 b(It)27
6115
 
b(can)i(b)s(e)e(an)m(y)h(non-negativ)m(e)330 1190 y(in)m(teger.)225
6116
 
1325 y Fs(\017)60 b Fo(n)22 b Fu(is)g(a)g(p)s(oin)m(ter)g(to)h(an)f
6117
 
(arra)m(y)h(of)g Fo(rank)e Fu(in)m(tegers,)k(giving)c(the)i(size)f(of)h
6118
 
(eac)m(h)g(dimension)d(of)j(the)g(arra)m(ys)330 1435
6119
 
y(to)35 b(b)s(e)e(transformed.)51 b(Note)36 b(that)e(these)h(are)f(alw)
6120
 
m(a)m(ys)h(the)f(dimensions)e(of)i(the)g Fq(r)-5 b(e)g(al)45
6121
 
b Fu(arra)m(ys;)37 b(the)330 1545 y(complex)26 b(arra)m(ys)g(ha)m(v)m
6122
 
(e)h(di\013eren)m(t)e(dimensions)f(\(see)i(Section)g(3.5.3)i([Arra)m(y)
6123
 
e(Dimensions)e(for)i(Real)330 1654 y(Multi-dimensional)d(T)-8
6124
 
b(ransforms],)26 b(page)h(32\).)41 b(These)26 b(sizes,)i(whic)m(h)d(m)m
6125
 
(ust)h(b)s(e)g(p)s(ositiv)m(e)f(in)m(tegers,)330 1764
6126
 
y(corresp)s(ond)20 b(to)i(the)g(dimensions)c(of)k(ro)m(w-ma)5
6127
 
b(jor)22 b(arra)m(ys|i.e.)37 b Fo(n[0])21 b Fu(is)f(the)h(size)h(of)f
6128
 
(the)h(dimension)330 1873 y(whose)32 b(indices)d(v)-5
6129
 
b(ary)32 b(most)g(slo)m(wly)-8 b(,)32 b(and)f(so)h(on.)45
6130
 
b(\(See)32 b(Section)g(2.5)h([Multi-dimensional)28 b(Arra)m(y)330
6131
 
1983 y(F)-8 b(ormat],)32 b(page)f(11,)h(for)e(more)g(information.\))379
6132
 
2119 y Fs(\000)60 b Fu(See)34 b(Section)g(3.4.1)h([Plan)e(Creation)h
6133
 
(for)f(Real)h(One-dimensional)d(T)-8 b(ransforms],)34
6134
 
b(page)h(26,)510 2228 y(for)30 b(more)h(information)d(regarding)i
6135
 
(optimal)f(arra)m(y)i(sizes.)225 2364 y Fs(\017)60 b
6136
 
Fo(nx)30 b Fu(and)h Fo(ny)f Fu(in)g Fo(rfftw2d_create_plan)25
6137
 
b Fu(are)32 b(p)s(ositiv)m(e)d(in)m(tegers)j(sp)s(ecifying)c(the)k
6138
 
(dimensions)c(of)330 2474 y(the)34 b(rank)f(2)h(arra)m(y)g(to)h(b)s(e)e
6139
 
(transformed.)50 b(i.e.)h(they)34 b(sp)s(ecify)e(that)i(the)g
6140
 
(transform)f(will)e(op)s(erate)330 2583 y(on)k Fo(nx)30
6141
 
b(x)g(ny)35 b Fu(arra)m(ys)g(in)f(ro)m(w-ma)5 b(jor)36
6142
 
b(order,)g(where)f Fo(nx)g Fu(is)f(the)h(n)m(um)m(b)s(er)f(of)i(ro)m
6143
 
(ws)f(and)g Fo(ny)f Fu(is)h(the)330 2693 y(n)m(um)m(b)s(er)29
6144
 
b(of)i(columns.)225 2829 y Fs(\017)60 b Fo(nx)p Fu(,)27
6145
 
b Fo(ny)f Fu(and)g Fo(nz)g Fu(in)f Fo(rfftw3d_create_plan)c
6146
 
Fu(are)27 b(p)s(ositiv)m(e)f(in)m(tegers)h(sp)s(ecifying)d(the)j
6147
 
(dimensions)330 2938 y(of)g(the)h(rank)e(3)i(arra)m(y)g(to)g(b)s(e)e
6148
 
(transformed.)39 b(i.e.)h(they)27 b(sp)s(ecify)f(that)i(the)f
6149
 
(transform)g(will)d(op)s(erate)330 3048 y(on)30 b Fo(nx)g(x)g(ny)g(x)g
6150
 
(nz)g Fu(arra)m(ys)g(in)f(ro)m(w-ma)5 b(jor)31 b(order.)225
6151
 
3184 y Fs(\017)60 b Fo(dir)31 b Fu(is)g(the)g(direction)g(of)h(the)g
6152
 
(desired)e(transform,)h(either)h Fo(FFTW_REAL_TO_COMPLEX)26
6153
 
b Fu(or)31 b Fo(FFTW_)330 3293 y(COMPLEX_TO_REAL)p Fu(,)26
6154
 
b(corresp)s(onding)j(to)i Fo(FFTW_FORWARD)c Fu(or)j Fo(FFTW_BACKWARD)p
6155
 
Fu(,)d(resp)s(ectiv)m(ely)-8 b(.)225 3429 y Fs(\017)60
6156
 
b Fo(flags)29 b Fu(is)g(a)i(b)s(o)s(olean)e(OR)h(\(`)p
6157
 
Fo(|)p Fu('\))i(of)e(zero)h(or)g(more)f(of)h(the)f(follo)m(wing:)379
6158
 
3565 y Fs(\000)60 b Fo(FFTW_MEASURE)p Fu(:)33 b(this)19
6159
 
b(\015ag)j(tells)e(FFTW)h(to)h(\014nd)d(the)i(optimal)f(plan)g(b)m(y)g
6160
 
(actually)h Fq(c)-5 b(omputing)510 3675 y Fu(sev)m(eral)31
 
6180
TeXDict begin 30 31 bop 150 -116 a Fv(30)3232 b(FFTW)330
 
6181
299 y(complex)26 b(arra)m(ys)g(ha)m(v)m(e)h(di\013eren)m(t)e
 
6182
(dimensions)f(\(see)i(Section)g(3.5.3)i([Arra)m(y)e(Dimensions)e(for)i
 
6183
(Real)330 408 y(Multi-dimensional)d(T)-8 b(ransforms],)26
 
6184
b(page)h(32\).)41 b(These)26 b(sizes,)i(whic)m(h)d(m)m(ust)h(b)s(e)g(p)
 
6185
s(ositiv)m(e)f(in)m(tegers,)330 518 y(corresp)s(ond)20
 
6186
b(to)i(the)g(dimensions)c(of)k(ro)m(w-ma)5 b(jor)22 b(arra)m(ys|i.e.)37
 
6187
b Fp(n[0])21 b Fv(is)f(the)h(size)h(of)f(the)h(dimension)330
 
6188
628 y(whose)32 b(indices)d(v)-5 b(ary)32 b(most)g(slo)m(wly)-8
 
6189
b(,)32 b(and)f(so)h(on.)45 b(\(See)32 b(Section)g(2.5)h
 
6190
([Multi-dimensional)28 b(Arra)m(y)330 737 y(F)-8 b(ormat],)32
 
6191
b(page)f(11,)h(for)e(more)g(information.\))379 872 y
 
6192
Ft(\000)60 b Fv(See)34 b(Section)g(3.4.1)h([Plan)e(Creation)h(for)f
 
6193
(Real)h(One-dimensional)d(T)-8 b(ransforms],)34 b(page)h(26,)510
 
6194
982 y(for)30 b(more)h(information)d(regarding)i(optimal)f(arra)m(y)i
 
6195
(sizes.)225 1116 y Ft(\017)60 b Fp(nx)30 b Fv(and)h Fp(ny)f
 
6196
Fv(in)g Fp(rfftw2d_create_plan)25 b Fv(are)32 b(p)s(ositiv)m(e)d(in)m
 
6197
(tegers)j(sp)s(ecifying)c(the)k(dimensions)c(of)330 1226
 
6198
y(the)34 b(rank)f(2)h(arra)m(y)g(to)h(b)s(e)e(transformed.)50
 
6199
b(i.e.)h(they)34 b(sp)s(ecify)e(that)i(the)g(transform)f(will)e(op)s
 
6200
(erate)330 1335 y(on)k Fp(nx)30 b(x)g(ny)35 b Fv(arra)m(ys)g(in)f(ro)m
 
6201
(w-ma)5 b(jor)36 b(order,)g(where)f Fp(nx)g Fv(is)f(the)h(n)m(um)m(b)s
 
6202
(er)f(of)i(ro)m(ws)f(and)g Fp(ny)f Fv(is)h(the)330 1445
 
6203
y(n)m(um)m(b)s(er)29 b(of)i(columns.)225 1580 y Ft(\017)60
 
6204
b Fp(nx)p Fv(,)27 b Fp(ny)f Fv(and)g Fp(nz)g Fv(in)f
 
6205
Fp(rfftw3d_create_plan)c Fv(are)27 b(p)s(ositiv)m(e)f(in)m(tegers)h(sp)
 
6206
s(ecifying)d(the)j(dimensions)330 1689 y(of)g(the)h(rank)e(3)i(arra)m
 
6207
(y)g(to)g(b)s(e)e(transformed.)39 b(i.e.)h(they)27 b(sp)s(ecify)f(that)
 
6208
i(the)f(transform)g(will)d(op)s(erate)330 1799 y(on)30
 
6209
b Fp(nx)g(x)g(ny)g(x)g(nz)g Fv(arra)m(ys)g(in)f(ro)m(w-ma)5
 
6210
b(jor)31 b(order.)225 1934 y Ft(\017)60 b Fp(dir)31 b
 
6211
Fv(is)g(the)g(direction)g(of)h(the)g(desired)e(transform,)h(either)h
 
6212
Fp(FFTW_REAL_TO_COMPLEX)26 b Fv(or)31 b Fp(FFTW_)330
 
6213
2043 y(COMPLEX_TO_REAL)p Fv(,)26 b(corresp)s(onding)j(to)i
 
6214
Fp(FFTW_FORWARD)c Fv(or)j Fp(FFTW_BACKWARD)p Fv(,)d(resp)s(ectiv)m(ely)
 
6215
-8 b(.)225 2178 y Ft(\017)60 b Fp(flags)29 b Fv(is)g(a)i(b)s(o)s(olean)
 
6216
e(OR)h(\(`)p Fp(|)p Fv('\))i(of)e(zero)h(or)g(more)f(of)h(the)f(follo)m
 
6217
(wing:)379 2313 y Ft(\000)60 b Fp(FFTW_MEASURE)p Fv(:)33
 
6218
b(this)19 b(\015ag)j(tells)e(FFTW)h(to)h(\014nd)d(the)i(optimal)f(plan)
 
6219
g(b)m(y)g(actually)h Fr(c)-5 b(omputing)510 2422 y Fv(sev)m(eral)31
6161
6220
b(FFTs)f(and)g(measuring)f(their)h(execution)g(time.)379
6162
 
3810 y Fs(\000)60 b Fo(FFTW_ESTIMATE)p Fu(:)34 b(do)25
 
6221
2557 y Ft(\000)60 b Fp(FFTW_ESTIMATE)p Fv(:)34 b(do)25
6163
6222
b(not)g(run)f(an)m(y)h(FFT)h(and)e(pro)m(vide)g(a)h(\\reasonable")h
6164
 
(plan)e(\(for)h(a)g(RISC)510 3920 y(pro)s(cessor)37 b(with)g(man)m(y)h
6165
 
(registers\).)62 b(If)38 b(neither)e Fo(FFTW_ESTIMATE)e
6166
 
Fu(nor)k Fo(FFTW_MEASURE)c Fu(is)510 4030 y(pro)m(vided,)29
6167
 
b(the)i(default)e(is)h Fo(FFTW_ESTIMATE)p Fu(.)379 4165
6168
 
y Fs(\000)60 b Fo(FFTW_OUT_OF_PLACE)p Fu(:)41 b(pro)s(duce)32
 
6223
(plan)e(\(for)h(a)g(RISC)510 2667 y(pro)s(cessor)37 b(with)g(man)m(y)h
 
6224
(registers\).)62 b(If)38 b(neither)e Fp(FFTW_ESTIMATE)e
 
6225
Fv(nor)k Fp(FFTW_MEASURE)c Fv(is)510 2776 y(pro)m(vided,)29
 
6226
b(the)i(default)e(is)h Fp(FFTW_ESTIMATE)p Fv(.)379 2911
 
6227
y Ft(\000)60 b Fp(FFTW_OUT_OF_PLACE)p Fv(:)41 b(pro)s(duce)32
6169
6228
b(a)i(plan)e(assuming)f(that)j(the)f(input)e(and)i(output)g(arra)m(ys)
6170
 
510 4275 y(will)28 b(b)s(e)h(distinct)g(\(this)h(is)f(the)i(default\).)
6171
 
379 4411 y Fs(\000)60 b Fo(FFTW_IN_PLACE)p Fu(:)54 b(pro)s(duce)37
 
6229
510 3020 y(will)28 b(b)s(e)h(distinct)g(\(this)h(is)f(the)i(default\).)
 
6230
379 3155 y Ft(\000)60 b Fp(FFTW_IN_PLACE)p Fv(:)54 b(pro)s(duce)37
6172
6231
b(a)i(plan)e(assuming)h(that)h(y)m(ou)g(w)m(an)m(t)g(to)h(p)s(erform)d
6173
 
(the)i(trans-)510 4520 y(form)33 b(in-place.)51 b(\(Unlik)m(e)33
 
6232
(the)i(trans-)510 3265 y(form)33 b(in-place.)51 b(\(Unlik)m(e)33
6174
6233
b(the)h(one-dimensional)e(transform,)i(this)f(\\really")g(p)s(erforms)g
6175
 
(the)510 4630 y(transform)j(in-place.\))58 b(Note)37
 
6234
(the)510 3374 y(transform)j(in-place.\))58 b(Note)37
6176
6235
b(that,)i(if)c(y)m(ou)i(w)m(an)m(t)g(to)g(p)s(erform)e(in-place)g
6177
 
(transforms,)j(y)m(ou)510 4740 y Fq(must)45 b Fu(use)35
 
6236
(transforms,)j(y)m(ou)510 3484 y Fr(must)45 b Fv(use)35
6178
6237
b(a)h(plan)f(created)i(with)d(this)h(option.)56 b(The)35
6179
6238
b(use)h(of)f(this)g(option)g(has)h(imp)s(ortan)m(t)510
6180
 
4849 y(implications)29 b(for)j(the)h(size)f(of)g(the)h(input/output)d
 
6239
3594 y(implications)29 b(for)j(the)h(size)f(of)g(the)h(input/output)d
6181
6240
(arra)m(y)j(\(see)g(Section)f(3.5.2)i([Computing)510
6182
 
4959 y(the)d(Real)f(Multi-dimensional)c(T)-8 b(ransform],)30
6183
 
b(page)h(31\).)510 5095 y(The)f(default)f(mo)s(de)h(of)h(op)s(eration)f
6184
 
(is)f Fo(FFTW_OUT_OF_PLACE)p Fu(.)379 5230 y Fs(\000)60
6185
 
b Fo(FFTW_USE_WISDOM)p Fu(:)h(use)42 b(an)m(y)h Fo(wisdom)e
6186
 
Fu(that)i(is)f(a)m(v)-5 b(ailable)41 b(to)j(help)d(in)g(the)i(creation)
6187
 
g(of)510 5340 y(the)34 b(plan.)50 b(\(See)35 b(Section)e(2.6)i([W)-8
 
6241
3703 y(the)d(Real)f(Multi-dimensional)c(T)-8 b(ransform],)30
 
6242
b(page)h(30\).)510 3838 y(The)f(default)f(mo)s(de)h(of)h(op)s(eration)f
 
6243
(is)f Fp(FFTW_OUT_OF_PLACE)p Fv(.)379 3973 y Ft(\000)60
 
6244
b Fp(FFTW_USE_WISDOM)p Fv(:)h(use)42 b(an)m(y)h Fp(wisdom)e
 
6245
Fv(that)i(is)f(a)m(v)-5 b(ailable)41 b(to)j(help)d(in)g(the)i(creation)
 
6246
g(of)510 4082 y(the)34 b(plan.)50 b(\(See)35 b(Section)e(2.6)i([W)-8
6188
6247
b(ords)35 b(of)f(Wisdom],)g(page)h(13.\))52 b(This)32
6189
 
b(can)i(greatly)h(sp)s(eed)p eop
 
6248
b(can)i(greatly)h(sp)s(eed)510 4192 y(the)g(creation)h(of)f(plans,)g
 
6249
(esp)s(ecially)e(with)h(the)h Fp(FFTW_MEASURE)d Fv(option.)55
 
6250
b Fp(FFTW_ESTIMATE)510 4301 y Fv(plans)33 b(can)i(also)f(tak)m(e)i(adv)
 
6251
-5 b(an)m(tage)36 b(of)f Fp(wisdom)e Fv(to)i(pro)s(duce)e(a)i(more)f
 
6252
(optimal)g(plan)f(\(based)510 4411 y(on)42 b(past)g(measuremen)m(ts\))g
 
6253
(than)f(the)h(estimation)g(heuristic)e(w)m(ould)g(normally)g(generate.)
 
6254
510 4520 y(When)30 b(the)h Fp(FFTW_MEASURE)c Fv(option)j(is)g(used,)g
 
6255
(new)g Fp(wisdom)f Fv(will)f(also)j(b)s(e)f(generated)h(if)f(the)510
 
6256
4630 y(curren)m(t)42 b(transform)g(size)g(is)g(not)g(completely)g
 
6257
(understo)s(o)s(d)f(b)m(y)h(existing)g Fp(wisdom)p Fv(.)75
 
6258
b(Note)510 4740 y(that)37 b(the)g(same)g Fp(wisdom)e
 
6259
Fv(is)h(shared)g(b)s(et)m(w)m(een)h(one-dimensional)e(and)h(m)m
 
6260
(ulti-dimensional)510 4849 y(transforms.)150 5074 y Fh(3.5.2)63
 
6261
b(Computing)41 b(the)f(Real)h(Multi-dimensional)g(T)-10
 
6262
b(ransform)390 5236 y Fp(#include)46 b(<rfftw.h>)p eop
 
6263
end
6190
6264
%%Page: 31 33
6191
 
31 32 bop 150 -116 a Fu(Chapter)30 b(3:)41 b(FFTW)31
6192
 
b(Reference)2354 b(31)510 299 y(the)35 b(creation)h(of)f(plans,)g(esp)s
6193
 
(ecially)e(with)h(the)h Fo(FFTW_MEASURE)d Fu(option.)55
6194
 
b Fo(FFTW_ESTIMATE)510 408 y Fu(plans)33 b(can)i(also)f(tak)m(e)i(adv)
6195
 
-5 b(an)m(tage)36 b(of)f Fo(wisdom)e Fu(to)i(pro)s(duce)e(a)i(more)f
6196
 
(optimal)g(plan)f(\(based)510 518 y(on)42 b(past)g(measuremen)m(ts\))g
6197
 
(than)f(the)h(estimation)g(heuristic)e(w)m(ould)g(normally)g(generate.)
6198
 
510 628 y(When)30 b(the)h Fo(FFTW_MEASURE)c Fu(option)j(is)g(used,)g
6199
 
(new)g Fo(wisdom)f Fu(will)f(also)j(b)s(e)f(generated)h(if)f(the)510
6200
 
737 y(curren)m(t)42 b(transform)g(size)g(is)g(not)g(completely)g
6201
 
(understo)s(o)s(d)f(b)m(y)h(existing)g Fo(wisdom)p Fu(.)75
6202
 
b(Note)510 847 y(that)37 b(the)g(same)g Fo(wisdom)e Fu(is)h(shared)g(b)
6203
 
s(et)m(w)m(een)h(one-dimensional)e(and)h(m)m(ulti-dimensional)510
6204
 
956 y(transforms.)150 1179 y Fi(3.5.2)63 b(Computing)41
6205
 
b(the)f(Real)h(Multi-dimensional)g(T)-10 b(ransform)390
6206
 
1366 y Fo(#include)46 b(<rfftw.h>)390 1573 y(void)h
6207
 
(rfftwnd_real_to_complex)o(\(rff)o(twn)o(d_pl)o(an)42
6208
 
b(plan,)k(int)h(howmany,)1774 1677 y(fftw_real)e(*in,)i(int)g(istride,)
6209
 
e(int)i(idist,)1774 1781 y(fftw_complex)e(*out,)h(int)h(ostride,)e(int)
6210
 
i(odist\);)390 1884 y(void)g(rfftwnd_complex_to_real)o(\(rff)o(twn)o
6211
 
(d_pl)o(an)42 b(plan,)k(int)h(howmany,)1774 1988 y(fftw_complex)e(*in,)
6212
 
h(int)h(istride,)f(int)h(idist,)1774 2092 y(fftw_real)e(*out,)i(int)g
6213
 
(ostride,)e(int)i(odist\);)390 2300 y(void)g(rfftwnd_one_real_to_com)o
 
6265
TeXDict begin 31 32 bop 150 -116 a Fv(Chapter)30 b(3:)41
 
6266
b(FFTW)31 b(Reference)2354 b(31)390 299 y Fp(void)47
 
6267
b(rfftwnd_real_to_complex)o(\(rff)o(twn)o(d_pl)o(an)42
 
6268
b(plan,)k(int)h(howmany,)1774 403 y(fftw_real)e(*in,)i(int)g(istride,)e
 
6269
(int)i(idist,)1774 506 y(fftw_complex)e(*out,)h(int)h(ostride,)e(int)i
 
6270
(odist\);)390 610 y(void)g(rfftwnd_complex_to_real)o(\(rff)o(twn)o
 
6271
(d_pl)o(an)42 b(plan,)k(int)h(howmany,)1774 714 y(fftw_complex)e(*in,)h
 
6272
(int)h(istride,)f(int)h(idist,)1774 818 y(fftw_real)e(*out,)i(int)g
 
6273
(ostride,)e(int)i(odist\);)390 1025 y(void)g(rfftwnd_one_real_to_com)o
6214
6274
(plex)o(\(rf)o(ftwn)o(d_pl)o(an)41 b(p,)48 b(fftw_real)d(*in,)1965
6215
 
2403 y(fftw_complex)f(*out\);)390 2507 y(void)j
 
6275
1129 y(fftw_complex)f(*out\);)390 1233 y(void)j
6216
6276
(rfftwnd_one_complex_to_)o(real)o(\(rf)o(ftwn)o(d_pl)o(an)41
6217
 
b(p,)48 b(fftw_complex)c(*in,)1965 2611 y(fftw_real)h(*out\);)275
6218
 
2745 y Fu(These)24 b(functions)f(compute)h(the)h(real)f(m)m
 
6277
b(p,)48 b(fftw_complex)c(*in,)1965 1337 y(fftw_real)h(*out\);)275
 
6278
1464 y Fv(These)24 b(functions)f(compute)h(the)h(real)f(m)m
6219
6279
(ulti-dimensional)c(F)-8 b(ourier)24 b(T)-8 b(ransform,)25
6220
 
b(using)e(a)h(plan)f(cre-)150 2855 y(ated)i(b)m(y)f Fo
6221
 
(rfftwnd_create_plan)18 b Fu(\(see)25 b(Section)f(3.5.1)i([Plan)d
6222
 
(Creation)h(for)g(Real)g(Multi-dimensional)150 2964 y(T)-8
 
6280
b(using)e(a)h(plan)f(cre-)150 1574 y(ated)i(b)m(y)f Fp
 
6281
(rfftwnd_create_plan)18 b Fv(\(see)25 b(Section)f(3.5.1)i([Plan)d
 
6282
(Creation)h(for)g(Real)g(Multi-dimensional)150 1683 y(T)-8
6223
6283
b(ransforms],)31 b(page)h(29\).)46 b(\(Note)33 b(that)f(the)f(plan)g
6224
6284
(determines)f(the)i(rank)f(and)g(dimensions)d(of)k(the)g(ar-)150
6225
 
3074 y(ra)m(y)27 b(to)g(b)s(e)e(transformed.\))39 b(The)26
6226
 
b(`)p Fo(rfftwnd_one_)p Fu(')d(functions)i(pro)m(vide)h(a)g
6227
 
(simpli\014ed)d(in)m(terface)k(for)f(the)150 3183 y(common)33
 
6285
1793 y(ra)m(y)27 b(to)g(b)s(e)e(transformed.\))39 b(The)26
 
6286
b(`)p Fp(rfftwnd_one_)p Fv(')d(functions)i(pro)m(vide)h(a)g
 
6287
(simpli\014ed)d(in)m(terface)k(for)f(the)150 1903 y(common)33
6228
6288
b(case)h(of)f(single)f(input)f(arra)m(y)i(of)g(stride)f(1.)48
6229
6289
b(Unlik)m(e)32 b(other)h(transform)g(routines)e(in)h(FFTW,)150
6230
 
3293 y(w)m(e)f(here)f(use)g(separate)h(functions)e(for)h(the)h(t)m(w)m
 
6290
2012 y(w)m(e)f(here)f(use)g(separate)h(functions)e(for)h(the)h(t)m(w)m
6231
6291
(o)g(directions)e(of)i(the)f(transform)g(in)f(order)g(to)j(correctly)
6232
 
150 3402 y(express)e(the)g(datat)m(yp)s(es)i(of)e(the)h(parameters.)275
6233
 
3537 y Fq(Imp)-5 b(ortant:)58 b Fu(When)35 b(in)m(v)m(ok)m(ed)g(for)g
6234
 
(an)g(out-of-place,)j Fo(FFTW_COMPLEX_TO_REAL)30 b Fu(transform)k(with)
6235
 
150 3646 y Fo(rank)29 b(>)h(1)p Fu(,)35 b(the)f(input)e(arra)m(y)j(is)e
 
6292
150 2122 y(express)e(the)g(datat)m(yp)s(es)i(of)e(the)h(parameters.)275
 
6293
2249 y Fr(Imp)-5 b(ortant:)58 b Fv(When)35 b(in)m(v)m(ok)m(ed)g(for)g
 
6294
(an)g(out-of-place,)j Fp(FFTW_COMPLEX_TO_REAL)30 b Fv(transform)k(with)
 
6295
150 2359 y Fp(rank)29 b(>)h(1)p Fv(,)35 b(the)f(input)e(arra)m(y)j(is)e
6236
6296
(o)m(v)m(erwritten)h(with)f(scratc)m(h)i(v)-5 b(alues)33
6237
 
b(b)m(y)h(these)g(routines.)50 b(The)34 b(input)150 3756
6238
 
y(arra)m(y)26 b(is)e(not)i(mo)s(di\014ed)d(for)i Fo
6239
 
(FFTW_REAL_TO_COMPLEX)20 b Fu(transforms)25 b(or)g(for)g
6240
 
Fo(FFTW_COMPLEX_TO_REAL)150 3865 y Fu(with)k Fo(rank)g(==)h(1)p
6241
 
Fu(.)150 4088 y Fi(Argumen)m(ts)225 4280 y Fs(\017)60
6242
 
b Fo(plan)20 b Fu(is)g(the)h(plan)e(created)j(b)m(y)f
6243
 
Fo(rfftwnd_create_plan)p Fu(.)32 b(\(see)22 b(Section)f(3.5.1)i([Plan)d
6244
 
(Creation)g(for)330 4390 y(Real)i(Multi-dimensional)c(T)-8
 
6297
b(b)m(y)h(these)g(routines.)50 b(The)34 b(input)150 2468
 
6298
y(arra)m(y)26 b(is)e(not)i(mo)s(di\014ed)d(for)i Fp
 
6299
(FFTW_REAL_TO_COMPLEX)20 b Fv(transforms)25 b(or)g(for)g
 
6300
Fp(FFTW_COMPLEX_TO_REAL)150 2578 y Fv(with)k Fp(rank)g(==)h(1)p
 
6301
Fv(.)150 2781 y Fh(Argumen)m(ts)225 2949 y Ft(\017)60
 
6302
b Fp(plan)20 b Fv(is)g(the)h(plan)e(created)j(b)m(y)f
 
6303
Fp(rfftwnd_create_plan)p Fv(.)32 b(\(see)22 b(Section)f(3.5.1)i([Plan)d
 
6304
(Creation)g(for)330 3059 y(Real)i(Multi-dimensional)c(T)-8
6245
6305
b(ransforms],)23 b(page)f(29\).)39 b(In)21 b(the)h(case)h(of)f(t)m(w)m
6246
 
(o)h(and)e(three-dimensional)330 4499 y(transforms,)45
 
6306
(o)h(and)e(three-dimensional)330 3168 y(transforms,)45
6247
6307
b(it)d(could)g(also)h(ha)m(v)m(e)h(b)s(een)e(created)h(b)m(y)g
6248
 
Fo(rfftw2d_create_plan)37 b Fu(or)43 b Fo(rfftw3d_)330
6249
 
4609 y(create_plan)p Fu(,)28 b(resp)s(ectiv)m(ely)-8
6250
 
b(.)330 4743 y Fo(FFTW_REAL_TO_COMPLEX)15 b Fu(plans)20
6251
 
b(m)m(ust)h(b)s(e)f(used)g(with)g(the)h(`)p Fo(real_to_complex)p
6252
 
Fu(')c(functions,)22 b(and)330 4853 y Fo(FFTW_COMPLEX_TO_REAL)h
6253
 
Fu(plans)k(m)m(ust)i(b)s(e)e(used)h(with)f(the)i(`)p
6254
 
Fo(complex_to_real)p Fu(')c(functions.)39 b(It)330 4962
 
6308
Fp(rfftw2d_create_plan)37 b Fv(or)43 b Fp(rfftw3d_)330
 
6309
3278 y(create_plan)p Fv(,)28 b(resp)s(ectiv)m(ely)-8
 
6310
b(.)330 3405 y Fp(FFTW_REAL_TO_COMPLEX)15 b Fv(plans)20
 
6311
b(m)m(ust)h(b)s(e)f(used)g(with)g(the)h(`)p Fp(real_to_complex)p
 
6312
Fv(')c(functions,)22 b(and)330 3515 y Fp(FFTW_COMPLEX_TO_REAL)h
 
6313
Fv(plans)k(m)m(ust)i(b)s(e)e(used)h(with)f(the)i(`)p
 
6314
Fp(complex_to_real)p Fv(')c(functions.)39 b(It)330 3624
6255
6315
y(is)29 b(an)i(error)f(to)h(mismatc)m(h)f(the)g(plan)f(direction)h(and)
6256
 
f(the)i(transform)f(function.)225 5096 y Fs(\017)60 b
6257
 
Fo(howmany)28 b Fu(is)i(the)g(n)m(um)m(b)s(er)f(of)i(transforms)f(to)h
6258
 
(b)s(e)e(computed.)225 5230 y Fs(\017)60 b Fo(in)p Fu(,)33
6259
 
b Fo(istride)e Fu(and)i Fo(idist)e Fu(describ)s(e)h(the)h(input)e(arra)
6260
 
m(y\(s\).)50 b(There)33 b(are)g Fo(howmany)e Fu(input)g(arra)m(ys;)330
6261
 
5340 y(the)42 b(\014rst)e(one)i(is)f(p)s(oin)m(ted)f(to)i(b)m(y)g
6262
 
Fo(in)p Fu(,)h(the)f(second)g(one)f(is)g(p)s(oin)m(ted)f(to)i(b)m(y)g
6263
 
Fo(in)29 b(+)h(idist)p Fu(,)44 b(and)p eop
6264
 
%%Page: 32 34
6265
 
32 33 bop 150 -116 a Fu(32)3232 b(FFTW)330 299 y(so)40
6266
 
b(on,)i(up)d(to)i Fo(in)30 b(+)g(\(howmany)e(-)i(1\))g(*)g(idist)p
6267
 
Fu(.)67 b(Eac)m(h)41 b(input)d(arra)m(y)i(is)f(stored)h(in)e(ro)m(w-ma)
6268
 
5 b(jor)330 408 y(format)31 b(\(see)h(Section)f(2.5)i
 
6316
f(the)i(transform)f(function.)225 3752 y Ft(\017)60 b
 
6317
Fp(howmany)28 b Fv(is)i(the)g(n)m(um)m(b)s(er)f(of)i(transforms)f(to)h
 
6318
(b)s(e)e(computed.)225 3879 y Ft(\017)60 b Fp(in)p Fv(,)33
 
6319
b Fp(istride)e Fv(and)i Fp(idist)e Fv(describ)s(e)h(the)h(input)e(arra)
 
6320
m(y\(s\).)50 b(There)33 b(are)g Fp(howmany)e Fv(input)g(arra)m(ys;)330
 
6321
3989 y(the)42 b(\014rst)e(one)i(is)f(p)s(oin)m(ted)f(to)i(b)m(y)g
 
6322
Fp(in)p Fv(,)h(the)f(second)g(one)f(is)g(p)s(oin)m(ted)f(to)i(b)m(y)g
 
6323
Fp(in)29 b(+)h(idist)p Fv(,)44 b(and)330 4099 y(so)c(on,)i(up)d(to)i
 
6324
Fp(in)30 b(+)g(\(howmany)e(-)i(1\))g(*)g(idist)p Fv(.)67
 
6325
b(Eac)m(h)41 b(input)d(arra)m(y)i(is)f(stored)h(in)e(ro)m(w-ma)5
 
6326
b(jor)330 4208 y(format)31 b(\(see)h(Section)f(2.5)i
6269
6327
([Multi-dimensional)27 b(Arra)m(y)k(F)-8 b(ormat],)34
6270
 
b(page)d(11\),)i(and)e(is)f(not)h(neces-)330 518 y(sarily)g(con)m
 
6328
b(page)d(11\),)i(and)e(is)f(not)h(neces-)330 4318 y(sarily)g(con)m
6271
6329
(tiguous)h(in)f(memory)-8 b(.)47 b(Sp)s(eci\014cally)-8
6272
 
b(,)31 b Fo(in[0])g Fu(is)h(the)g(\014rst)g(elemen)m(t)h(of)g(the)f
6273
 
(\014rst)g(arra)m(y)-8 b(,)330 628 y Fo(in[istride])31
6274
 
b Fu(is)i(the)h(second)g(elemen)m(t)g(of)g(the)g(\014rst)f(arra)m(y)-8
 
6330
b(,)31 b Fp(in[0])g Fv(is)h(the)g(\014rst)g(elemen)m(t)h(of)g(the)f
 
6331
(\014rst)g(arra)m(y)-8 b(,)330 4427 y Fp(in[istride])31
 
6332
b Fv(is)i(the)h(second)g(elemen)m(t)g(of)g(the)g(\014rst)f(arra)m(y)-8
6275
6333
b(,)36 b(and)e(so)g(on.)51 b(In)33 b(general,)i(the)f
6276
 
Fo(i)p Fu(-th)330 737 y(elemen)m(t)27 b(of)g(the)g Fo(j)p
6277
 
Fu(-th)g(input)d(arra)m(y)k(will)c(b)s(e)i(in)f(p)s(osition)g
6278
 
Fo(in[i)k(*)h(istride)f(+)h(j)g(*)g(idist])p Fu(.)38
6279
 
b(Note)330 847 y(that,)h(here,)g Fo(i)d Fu(refers)h(to)g(an)g(index)e
 
6334
Fp(i)p Fv(-th)330 4537 y(elemen)m(t)27 b(of)g(the)g Fp(j)p
 
6335
Fv(-th)g(input)d(arra)m(y)k(will)c(b)s(e)i(in)f(p)s(osition)g
 
6336
Fp(in[i)k(*)h(istride)f(+)h(j)g(*)g(idist])p Fv(.)38
 
6337
b(Note)330 4647 y(that,)h(here,)g Fp(i)d Fv(refers)h(to)g(an)g(index)e
6280
6338
(in)m(to)i(the)g(ro)m(w-ma)5 b(jor)37 b(format)g(for)g(the)g(m)m
6281
 
(ulti-dimensional)330 956 y(arra)m(y)-8 b(,)32 b(rather)e(than)g(an)g
6282
 
(index)f(in)g(an)m(y)i(particular)e(dimension.)330 1089
 
6339
(ulti-dimensional)330 4756 y(arra)m(y)-8 b(,)32 b(rather)e(than)g(an)g
 
6340
(index)f(in)g(an)m(y)i(particular)e(dimension.)330 4884
6283
6341
y(The)d(dimensions)e(of)j(the)g(arra)m(ys)g(are)g(di\013eren)m(t)f(for)
6284
6342
g(real)g(and)g(complex)h(data,)h(and)e(are)h(discussed)330
6285
 
1199 y(in)k(more)h(detail)g(b)s(elo)m(w)f(\(see)i(Section)f(3.5.3)i
 
6343
4993 y(in)k(more)h(detail)g(b)s(elo)m(w)f(\(see)i(Section)f(3.5.3)i
6286
6344
([Arra)m(y)f(Dimensions)e(for)h(Real)g(Multi-dimensional)330
6287
 
1309 y(T)-8 b(ransforms],)30 b(page)h(32\).)379 1442
6288
 
y Fs(\000)60 b Fq(In-plac)-5 b(e)30 b(tr)-5 b(ansforms)7
6289
 
b Fu(:)42 b(F)-8 b(or)28 b(plans)d(created)j(with)d(the)i
6290
 
Fo(FFTW_IN_PLACE)c Fu(option,)k(the)g(trans-)510 1551
 
6345
5103 y(T)-8 b(ransforms],)30 b(page)h(32\).)379 5230
 
6346
y Ft(\000)60 b Fr(In-plac)-5 b(e)30 b(tr)-5 b(ansforms)7
 
6347
b Fv(:)42 b(F)-8 b(or)28 b(plans)d(created)j(with)d(the)i
 
6348
Fp(FFTW_IN_PLACE)c Fv(option,)k(the)g(trans-)510 5340
6291
6349
y(form)32 b(is)f(computed)h(in-place|the)g(output)f(is)h(returned)f(in)
6292
 
g(the)h Fo(in)g Fu(arra)m(y)-8 b(.)47 b(The)32 b(meaning)510
6293
 
1661 y(of)j(the)h Fo(stride)d Fu(and)i Fo(dist)f Fu(parameters)i(in)e
 
6350
g(the)h Fp(in)g Fv(arra)m(y)-8 b(.)47 b(The)32 b(meaning)p
 
6351
eop end
 
6352
%%Page: 32 34
 
6353
TeXDict begin 32 33 bop 150 -116 a Fv(32)3232 b(FFTW)510
 
6354
299 y(of)35 b(the)h Fp(stride)d Fv(and)i Fp(dist)f Fv(parameters)i(in)e
6294
6355
(this)g(case)i(is)e(subtle)g(and)h(is)f(discussed)g(b)s(elo)m(w)510
6295
 
1770 y(\(see)d(Section)g(3.5.4)h([Strides)d(in)g(In-place)h(RFFTWND],)i
6296
 
(page)f(33\).)225 1904 y Fs(\017)60 b Fo(out)p Fu(,)29
6297
 
b Fo(ostride)f Fu(and)g Fo(odist)g Fu(describ)s(e)g(the)i(output)f
 
6356
408 y(\(see)d(Section)g(3.5.4)h([Strides)d(in)g(In-place)h(RFFTWND],)i
 
6357
(page)f(32\).)225 545 y Ft(\017)60 b Fp(out)p Fv(,)29
 
6358
b Fp(ostride)f Fv(and)g Fp(odist)g Fv(describ)s(e)g(the)i(output)f
6298
6359
(arra)m(y\(s\).)42 b(The)29 b(format)g(is)g(the)h(same)g(as)f(that)330
6299
 
2013 y(for)34 b(the)g(input)f(arra)m(y)-8 b(.)53 b(See)34
 
6360
654 y(for)34 b(the)g(input)f(arra)m(y)-8 b(.)53 b(See)34
6300
6361
b(b)s(elo)m(w)g(for)g(a)g(discussion)e(of)i(the)g(dimensions)e(of)i
6301
 
(the)h(output)e(arra)m(y)330 2123 y(for)d(real)g(and)g(complex)g(data.)
6302
 
379 2256 y Fs(\000)60 b Fq(In-plac)-5 b(e)47 b(tr)-5
6303
 
b(ansforms)7 b Fu(:)73 b(These)45 b(parameters)g(are)h(ignored)e(for)g
6304
 
(plans)g(created)i(with)e(the)510 2365 y Fo(FFTW_IN_PLACE)27
6305
 
b Fu(option.)275 2522 y(The)32 b(function)f Fo(rfftwnd_one)f
6306
 
Fu(transforms)i(a)h(single,)g(con)m(tiguous)g(input)e(arra)m(y)i(to)h
6307
 
(a)f(con)m(tiguous)150 2631 y(output)d(arra)m(y)-8 b(.)42
6308
 
b(By)30 b(de\014nition,)f(the)h(call)390 2759 y Fo
6309
 
(rfftwnd_one_...\(plan,)42 b(in,)47 b(out\))275 2892
6310
 
y Fu(is)29 b(equiv)-5 b(alen)m(t)30 b(to)390 3019 y Fo
6311
 
(rfftwnd_...\(plan,)43 b(1,)k(in,)g(1,)g(1,)h(out,)e(1,)h(1\))150
6312
 
3239 y Fi(3.5.3)63 b(Arra)m(y)39 b(Dimensions)j(for)f(Real)g
6313
 
(Multi-dimensional)g(T)-10 b(ransforms)275 3430 y Fu(The)32
 
6362
(the)h(output)e(arra)m(y)330 764 y(for)d(real)g(and)g(complex)g(data.)
 
6363
379 900 y Ft(\000)60 b Fr(In-plac)-5 b(e)47 b(tr)-5 b(ansforms)7
 
6364
b Fv(:)73 b(These)45 b(parameters)g(are)h(ignored)e(for)g(plans)g
 
6365
(created)i(with)e(the)510 1010 y Fp(FFTW_IN_PLACE)27
 
6366
b Fv(option.)275 1175 y(The)32 b(function)f Fp(rfftwnd_one)f
 
6367
Fv(transforms)i(a)h(single,)g(con)m(tiguous)g(input)e(arra)m(y)i(to)h
 
6368
(a)f(con)m(tiguous)150 1284 y(output)d(arra)m(y)-8 b(.)42
 
6369
b(By)30 b(de\014nition,)f(the)h(call)390 1417 y Fp
 
6370
(rfftwnd_one_...\(plan,)42 b(in,)47 b(out\))275 1555
 
6371
y Fv(is)29 b(equiv)-5 b(alen)m(t)30 b(to)390 1687 y Fp
 
6372
(rfftwnd_...\(plan,)43 b(1,)k(in,)g(1,)g(0,)h(out,)e(1,)h(0\))150
 
6373
1919 y Fh(3.5.3)63 b(Arra)m(y)39 b(Dimensions)j(for)f(Real)g
 
6374
(Multi-dimensional)g(T)-10 b(ransforms)275 2115 y Fv(The)32
6314
6375
b(output)h(of)g(a)g(m)m(ulti-dimensional)c(transform)k(of)g(real)g
6315
 
(data)g(con)m(tains)h(symmetries)e(that,)i(in)150 3540
 
6376
(data)g(con)m(tains)h(symmetries)e(that,)i(in)150 2225
6316
6377
y(principle,)27 b(mak)m(e)j(half)f(of)h(the)f(outputs)h(redundan)m(t)e
6317
6378
(\(see)i(Section)g(3.5.6)h([What)g(RFFTWND)g(Really)150
6318
 
3649 y(Computes],)36 b(page)f(33\).)55 b(In)34 b(practice,)i(it)f(is)e
 
6379
2334 y(Computes],)36 b(page)f(33\).)55 b(In)34 b(practice,)i(it)f(is)e
6319
6380
(not)i(p)s(ossible)d(to)k(en)m(tirely)e(realize)g(these)h(sa)m(vings)g
6320
 
(in)e(an)150 3759 y(e\016cien)m(t)24 b(and)e(understandable)f(format.)
 
6381
(in)e(an)150 2444 y(e\016cien)m(t)24 b(and)e(understandable)f(format.)
6321
6382
39 b(Instead,)25 b(the)e(output)g(of)g(the)h(r\013t)m(wnd)e(transforms)
6322
 
h(is)f Fq(slightly)150 3868 y Fu(o)m(v)m(er)27 b(half)e(of)h(the)g
 
6383
h(is)f Fr(slightly)150 2554 y Fv(o)m(v)m(er)27 b(half)e(of)h(the)g
6323
6384
(output)f(of)h(the)g(corresp)s(onding)d(complex)j(transform.)38
6324
6385
b(W)-8 b(e)27 b(do)f(not)g(\\pac)m(k")h(the)f(data)150
6325
 
3978 y(in)k(an)m(y)i(w)m(a)m(y)-8 b(,)33 b(but)e(store)h(it)f(as)h(an)f
6326
 
(ordinary)f(arra)m(y)i(of)f Fo(fftw_complex)e Fu(v)-5
6327
 
b(alues.)43 b(In)31 b(fact,)h(this)f(data)h(is)150 4088
 
6386
2663 y(in)k(an)m(y)i(w)m(a)m(y)-8 b(,)33 b(but)e(store)h(it)f(as)h(an)f
 
6387
(ordinary)f(arra)m(y)i(of)f Fp(fftw_complex)e Fv(v)-5
 
6388
b(alues.)43 b(In)31 b(fact,)h(this)f(data)h(is)150 2773
6328
6389
y(simply)c(a)j(subsection)e(of)i(what)f(w)m(ould)f(b)s(e)h(the)g(arra)m
6329
6390
(y)h(in)e(the)i(corresp)s(onding)d(complex)i(transform.)275
6330
 
4221 y(Sp)s(eci\014cally)-8 b(,)32 b(for)i(a)g(real)f(transform)g(of)h
6331
 
(dimensions)d Fl(n)2225 4235 y Fn(1)2284 4221 y Fs(\002)22
6332
 
b Fl(n)2432 4235 y Fn(2)2491 4221 y Fs(\002)h(\001)15
6333
 
b(\001)g(\001)23 b(\002)f Fl(n)2861 4235 y Fk(d)2899
6334
 
4221 y Fu(,)35 b(the)f(complex)f(data)i(is)150 4330 y(an)e
6335
 
Fl(n)334 4344 y Fn(1)393 4330 y Fs(\002)21 b Fl(n)540
6336
 
4344 y Fn(2)599 4330 y Fs(\002)h(\001)15 b(\001)g(\001)23
6337
 
b(\002)f Fu(\()p Fl(n)1003 4344 y Fk(d)1041 4330 y Fl(=)p
6338
 
Fu(2)h(+)f(1\))34 b(arra)m(y)f(of)h Fo(fftw_complex)29
6339
 
b Fu(v)-5 b(alues)33 b(in)e(ro)m(w-ma)5 b(jor)34 b(order)f(\(with)f
6340
 
(the)150 4440 y(division)j(rounded)h(do)m(wn\).)63 b(That)38
 
6391
2911 y(Sp)s(eci\014cally)-8 b(,)32 b(for)i(a)g(real)f(transform)g(of)h
 
6392
(dimensions)d Fm(n)2225 2925 y Fo(1)2284 2911 y Ft(\002)22
 
6393
b Fm(n)2432 2925 y Fo(2)2491 2911 y Ft(\002)h(\001)15
 
6394
b(\001)g(\001)23 b(\002)f Fm(n)2861 2925 y Fl(d)2899
 
6395
2911 y Fv(,)35 b(the)f(complex)f(data)i(is)150 3021 y(an)e
 
6396
Fm(n)334 3035 y Fo(1)393 3021 y Ft(\002)21 b Fm(n)540
 
6397
3035 y Fo(2)599 3021 y Ft(\002)h(\001)15 b(\001)g(\001)23
 
6398
b(\002)f Fv(\()p Fm(n)1003 3035 y Fl(d)1041 3021 y Fm(=)p
 
6399
Fv(2)h(+)f(1\))34 b(arra)m(y)f(of)h Fp(fftw_complex)29
 
6400
b Fv(v)-5 b(alues)33 b(in)e(ro)m(w-ma)5 b(jor)34 b(order)f(\(with)f
 
6401
(the)150 3130 y(division)j(rounded)h(do)m(wn\).)63 b(That)38
6341
6402
b(is,)h(w)m(e)g(only)e(store)h(the)g(lo)m(w)m(er)h(half)d(\(plus)h(one)
6342
 
h(elemen)m(t\))h(of)f(the)150 4549 y(last)c(dimension)e(of)i(the)g
 
6403
h(elemen)m(t\))h(of)f(the)150 3240 y(last)c(dimension)e(of)i(the)g
6343
6404
(data)h(from)f(the)g(ordinary)e(complex)i(transform.)51
6344
 
b(\(W)-8 b(e)36 b(could)d(ha)m(v)m(e)j(instead)150 4659
 
6405
b(\(W)-8 b(e)36 b(could)d(ha)m(v)m(e)j(instead)150 3349
6345
6406
y(tak)m(en)d(half)f(of)g(an)m(y)h(other)f(dimension,)f(but)g(implemen)m
6346
6407
(tation)g(turns)g(out)i(to)g(b)s(e)e(simpler)f(if)h(the)i(last,)150
6347
 
4769 y(con)m(tiguous,)e(dimension)d(is)h(used.\))275
6348
 
4902 y(Since)c(the)i(complex)g(data)h(is)d(sligh)m(tly)h(larger)g(than)
 
6408
3459 y(con)m(tiguous,)e(dimension)d(is)h(used.\))275
 
6409
3597 y(Since)c(the)i(complex)g(data)h(is)d(sligh)m(tly)h(larger)g(than)
6349
6410
h(the)g(real)f(data,)j(some)e(complications)f(arise)g(for)150
6350
 
5011 y(in-place)j(transforms.)39 b(In)29 b(this)f(case,)j(the)f
 
6411
3707 y(in-place)j(transforms.)39 b(In)29 b(this)f(case,)j(the)f
6351
6412
(\014nal)e(dimension)f(of)j(the)f(real)g(data)i(m)m(ust)e(b)s(e)g
6352
 
(padded)f(with)150 5121 y(extra)c(v)-5 b(alues)22 b(to)h(accommo)s
 
6413
(padded)f(with)150 3816 y(extra)c(v)-5 b(alues)22 b(to)h(accommo)s
6353
6414
(date)i(the)e(size)g(of)g(the)g(complex)g(data|t)m(w)m(o)h(extra)g(if)e
6354
 
(the)h(last)g(dimension)d(is)150 5230 y(ev)m(en)k(and)f(one)h(if)f(it)g
 
6415
(the)h(last)g(dimension)d(is)150 3926 y(ev)m(en)k(and)f(one)h(if)f(it)g
6355
6416
(is)g(o)s(dd.)37 b(That)24 b(is,)g(the)g(last)f(dimension)e(of)j(the)g
6356
6417
(real)f(data)i(m)m(ust)e(ph)m(ysically)f(con)m(tain)150
6357
 
5340 y(2\()p Fl(n)285 5354 y Fk(d)324 5340 y Fl(=)p Fu(2)j(+)g(1\))37
6358
 
b Fo(fftw_real)d Fu(v)-5 b(alues)36 b(\(exactly)i(enough)f(to)g(hold)e
6359
 
(the)i(complex)g(data\).)60 b(This)35 b(ph)m(ysical)p
6360
 
eop
 
6418
4035 y(2\()p Fm(n)285 4049 y Fl(d)324 4035 y Fm(=)p Fv(2)j(+)g(1\))37
 
6419
b Fp(fftw_real)d Fv(v)-5 b(alues)36 b(\(exactly)i(enough)f(to)g(hold)e
 
6420
(the)i(complex)g(data\).)60 b(This)35 b(ph)m(ysical)150
 
6421
4145 y(arra)m(y)42 b(size)g(do)s(es)f(not,)k(ho)m(w)m(ev)m(er,)i(c)m
 
6422
(hange)c(the)e Fr(lo)-5 b(gic)g(al)53 b Fv(arra)m(y)42
 
6423
b(size|only)f Fm(n)2909 4159 y Fl(d)2989 4145 y Fv(v)-5
 
6424
b(alues)41 b(are)h(actually)150 4255 y(stored)27 b(in)e(the)i(last)g
 
6425
(dimension,)e(and)h Fm(n)1524 4269 y Fl(d)1589 4255 y
 
6426
Fv(is)g(the)h(last)g(dimension)d(passed)i(to)i Fp(rfftwnd_create_plan)p
 
6427
Fv(.)150 4486 y Fh(3.5.4)63 b(Strides)41 b(in)g(In-place)g(RFFTWND)275
 
6428
4682 y Fv(The)32 b(fact)i(that)g(the)f(input)f(and)g(output)h(datat)m
 
6429
(yp)s(es)h(are)g(di\013eren)m(t)e(for)h(r\013t)m(wnd)g(complicates)g
 
6430
(the)150 4792 y(meaning)i(of)h(the)g Fp(stride)e Fv(and)i
 
6431
Fp(dist)e Fv(parameters)j(of)f(in-place)e(transforms|are)i(they)g(in)e
 
6432
(units)h(of)150 4902 y Fp(fftw_real)g Fv(or)i Fp(fftw_complex)c
 
6433
Fv(elemen)m(ts?)61 b(When)37 b(reading)f(the)h(input,)g(they)g(are)h
 
6434
(in)m(terpreted)e(in)150 5011 y(units)28 b(of)i(the)g(datat)m(yp)s(e)h
 
6435
(of)f(the)g(input)d(data.)42 b(When)29 b(writing)f(the)i(output,)g(the)
 
6436
f Fp(istride)f Fv(and)h Fp(idist)150 5121 y Fv(are)36
 
6437
b(translated)g(to)g(the)g(output)f(datat)m(yp)s(e's)i(\\units")e(in)f
 
6438
(one)i(of)g(t)m(w)m(o)h(w)m(a)m(ys,)i(corresp)s(onding)33
 
6439
b(to)k(the)150 5230 y(t)m(w)m(o)d(most)e(common)h(situations)e(in)g
 
6440
(whic)m(h)g Fp(stride)g Fv(and)g Fp(dist)h Fv(parameters)g(are)h
 
6441
(useful.)45 b(Belo)m(w,)33 b(w)m(e)150 5340 y(refer)k(to)h(these)g
 
6442
(\\translated")h(parameters)e(as)h Fp(ostride_t)d Fv(and)i
 
6443
Fp(odist_t)p Fv(.)60 b(\(Note)39 b(that)g(these)f(are)p
 
6444
eop end
6361
6445
%%Page: 33 35
6362
 
33 34 bop 150 -116 a Fu(Chapter)30 b(3:)41 b(FFTW)31
6363
 
b(Reference)2354 b(33)150 299 y(arra)m(y)42 b(size)g(do)s(es)f(not,)k
6364
 
(ho)m(w)m(ev)m(er,)i(c)m(hange)c(the)e Fq(lo)-5 b(gic)g(al)53
6365
 
b Fu(arra)m(y)42 b(size|only)f Fl(n)2909 313 y Fk(d)2989
6366
 
299 y Fu(v)-5 b(alues)41 b(are)h(actually)150 408 y(stored)27
6367
 
b(in)e(the)i(last)g(dimension,)e(and)h Fl(n)1524 422
6368
 
y Fk(d)1589 408 y Fu(is)g(the)h(last)g(dimension)d(passed)i(to)i
6369
 
Fo(rfftwnd_create_plan)p Fu(.)150 647 y Fi(3.5.4)63 b(Strides)41
6370
 
b(in)g(In-place)g(RFFTWND)275 847 y Fu(The)32 b(fact)i(that)g(the)f
6371
 
(input)f(and)g(output)h(datat)m(yp)s(es)h(are)g(di\013eren)m(t)e(for)h
6372
 
(r\013t)m(wnd)g(complicates)g(the)150 956 y(meaning)i(of)h(the)g
6373
 
Fo(stride)e Fu(and)i Fo(dist)e Fu(parameters)j(of)f(in-place)e
6374
 
(transforms|are)i(they)g(in)e(units)h(of)150 1066 y Fo(fftw_real)g
6375
 
Fu(or)i Fo(fftw_complex)c Fu(elemen)m(ts?)61 b(When)37
6376
 
b(reading)f(the)h(input,)g(they)g(are)h(in)m(terpreted)e(in)150
6377
 
1175 y(units)28 b(of)i(the)g(datat)m(yp)s(e)h(of)f(the)g(input)d(data.)
6378
 
42 b(When)29 b(writing)f(the)i(output,)g(the)f Fo(istride)f
6379
 
Fu(and)h Fo(idist)150 1285 y Fu(are)36 b(translated)g(to)g(the)g
6380
 
(output)f(datat)m(yp)s(e's)i(\\units")e(in)f(one)i(of)g(t)m(w)m(o)h(w)m
6381
 
(a)m(ys,)i(corresp)s(onding)33 b(to)k(the)150 1394 y(t)m(w)m(o)d(most)e
6382
 
(common)h(situations)e(in)g(whic)m(h)g Fo(stride)g Fu(and)g
6383
 
Fo(dist)h Fu(parameters)g(are)h(useful.)45 b(Belo)m(w,)33
6384
 
b(w)m(e)150 1504 y(refer)k(to)h(these)g(\\translated")h(parameters)e
6385
 
(as)h Fo(ostride_t)d Fu(and)i Fo(odist_t)p Fu(.)60 b(\(Note)39
6386
 
b(that)g(these)f(are)150 1614 y(computed)31 b(in)m(ternally)e(b)m(y)i
6387
 
(r\013t)m(wnd;)f(the)h(actual)h Fo(ostride)d Fu(and)h
6388
 
Fo(odist)f Fu(parameters)j(are)f(ignored)f(for)150 1723
6389
 
y(in-place)f(transforms.\))275 1865 y(First,)24 b(there)g(is)f(the)h
6390
 
(case)h(where)e(y)m(ou)h(are)g(transforming)e(a)i(n)m(um)m(b)s(er)f(of)
6391
 
h(con)m(tiguous)g(arra)m(ys)g(lo)s(cated)150 1974 y(one)34
6392
 
b(after)g(another)g(in)f(memory)-8 b(.)51 b(In)33 b(this)g(situation,)h
6393
 
Fo(istride)e Fu(is)g Fo(1)i Fu(and)f Fo(idist)f Fu(is)h(the)h(pro)s
6394
 
(duct)f(of)150 2084 y(the)f(ph)m(ysical)e(dimensions)e(of)k(the)g(arra)
6395
 
m(y)-8 b(.)45 b Fo(ostride_t)29 b Fu(and)h Fo(odist_t)g
6396
 
Fu(are)i(then)f(c)m(hosen)h(so)g(that)g(the)150 2194
6397
 
y(output)38 b(arra)m(ys)g(are)h(con)m(tiguous)f(and)f(lie)g(on)h(top)h
6398
 
(of)f(the)g(input)e(arra)m(ys.)65 b Fo(ostride_t)35 b
6399
 
Fu(is)i(therefore)150 2303 y Fo(1)p Fu(.)50 b(F)-8 b(or)34
6400
 
b(a)g(real-to-complex)g(transform,)g Fo(odist_t)e Fu(is)g
6401
 
Fo(idist/2)p Fu(;)i(for)f(a)h(complex-to-real)g(transform,)150
6402
 
2413 y Fo(odist_t)28 b Fu(is)i Fo(idist*2)p Fu(.)275
6403
 
2554 y(The)k(second)g(case)i(is)e(when)f(y)m(ou)i(ha)m(v)m(e)h(an)f
6404
 
(arra)m(y)g(in)e(whic)m(h)g(eac)m(h)j(elemen)m(t)f(has)g
6405
 
Fo(nc)f Fu(comp)s(onen)m(ts)150 2664 y(\(e.g.)42 b(a)30
6406
 
b(structure)g(with)e Fo(nc)i Fu(n)m(umeric)e(\014elds\),)i(and)f(y)m
 
6446
TeXDict begin 33 34 bop 150 -116 a Fv(Chapter)30 b(3:)41
 
6447
b(FFTW)31 b(Reference)2354 b(33)150 299 y(computed)31
 
6448
b(in)m(ternally)e(b)m(y)i(r\013t)m(wnd;)f(the)h(actual)h
 
6449
Fp(ostride)d Fv(and)h Fp(odist)f Fv(parameters)j(are)f(ignored)f(for)
 
6450
150 408 y(in-place)f(transforms.\))275 548 y(First,)24
 
6451
b(there)g(is)f(the)h(case)h(where)e(y)m(ou)h(are)g(transforming)e(a)i
 
6452
(n)m(um)m(b)s(er)f(of)h(con)m(tiguous)g(arra)m(ys)g(lo)s(cated)150
 
6453
658 y(one)34 b(after)g(another)g(in)f(memory)-8 b(.)51
 
6454
b(In)33 b(this)g(situation,)h Fp(istride)e Fv(is)g Fp(1)i
 
6455
Fv(and)f Fp(idist)f Fv(is)h(the)h(pro)s(duct)f(of)150
 
6456
768 y(the)f(ph)m(ysical)e(dimensions)e(of)k(the)g(arra)m(y)-8
 
6457
b(.)45 b Fp(ostride_t)29 b Fv(and)h Fp(odist_t)g Fv(are)i(then)f(c)m
 
6458
(hosen)h(so)g(that)g(the)150 877 y(output)38 b(arra)m(ys)g(are)h(con)m
 
6459
(tiguous)f(and)f(lie)g(on)h(top)h(of)f(the)g(input)e(arra)m(ys.)65
 
6460
b Fp(ostride_t)35 b Fv(is)i(therefore)150 987 y Fp(1)p
 
6461
Fv(.)50 b(F)-8 b(or)34 b(a)g(real-to-complex)g(transform,)g
 
6462
Fp(odist_t)e Fv(is)g Fp(idist/2)p Fv(;)i(for)f(a)h(complex-to-real)g
 
6463
(transform,)150 1096 y Fp(odist_t)28 b Fv(is)i Fp(idist*2)p
 
6464
Fv(.)275 1236 y(The)k(second)g(case)i(is)e(when)f(y)m(ou)i(ha)m(v)m(e)h
 
6465
(an)f(arra)m(y)g(in)e(whic)m(h)g(eac)m(h)j(elemen)m(t)f(has)g
 
6466
Fp(nc)f Fv(comp)s(onen)m(ts)150 1346 y(\(e.g.)42 b(a)30
 
6467
b(structure)g(with)e Fp(nc)i Fv(n)m(umeric)e(\014elds\),)i(and)f(y)m
6407
6468
(ou)h(w)m(an)m(t)h(to)g(transform)e(all)g(of)h(the)g(comp)s(onen)m(ts)
6408
 
150 2774 y(at)f(once.)40 b(Here,)29 b Fo(istride)d Fu(is)h
6409
 
Fo(nc)h Fu(and)f Fo(idist)g Fu(is)g Fo(1)p Fu(.)39 b(F)-8
 
6469
150 1456 y(at)f(once.)40 b(Here,)29 b Fp(istride)d Fv(is)h
 
6470
Fp(nc)h Fv(and)f Fp(idist)g Fv(is)g Fp(1)p Fv(.)39 b(F)-8
6410
6471
b(or)29 b(this)e(case,)i(it)f(is)f(natural)g(to)h(w)m(an)m(t)h(the)f
6411
 
(output)150 2883 y(to)j(also)g(ha)m(v)m(e)g Fo(nc)f Fu(consecutiv)m(e)i
 
6472
(output)150 1565 y(to)j(also)g(ha)m(v)m(e)g Fp(nc)f Fv(consecutiv)m(e)i
6412
6473
(comp)s(onen)m(ts,)f(no)m(w)f(of)h(the)g(output)f(data)h(t)m(yp)s(e;)g
6413
 
(this)e(is)h(exactly)h(what)150 2993 y(r\013t)m(wnd)k(do)s(es.)58
6414
 
b(Sp)s(eci\014cally)-8 b(,)35 b(it)h(uses)f(an)h Fo(ostride_t)e
6415
 
Fu(equal)i(to)g Fo(istride)p Fu(,)g(and)g(an)g Fo(odist_t)e
6416
 
Fu(of)i Fo(1)p Fu(.)150 3102 y(\(Astute)26 b(readers)e(will)e(realize)j
 
6474
(this)e(is)h(exactly)h(what)150 1675 y(r\013t)m(wnd)k(do)s(es.)58
 
6475
b(Sp)s(eci\014cally)-8 b(,)35 b(it)h(uses)f(an)h Fp(ostride_t)e
 
6476
Fv(equal)i(to)g Fp(istride)p Fv(,)g(and)g(an)g Fp(odist_t)e
 
6477
Fv(of)i Fp(1)p Fv(.)150 1784 y(\(Astute)26 b(readers)e(will)e(realize)j
6417
6478
(that)g(some)g(extra)g(bu\013er)f(space)h(is)f(required)f(in)g(order)i
6418
 
(to)g(p)s(erform)e(suc)m(h)150 3212 y(a)31 b(transform;)f(this)f(is)g
 
6479
(to)g(p)s(erform)e(suc)m(h)150 1894 y(a)31 b(transform;)f(this)f(is)g
6419
6480
(handled)g(automatically)h(b)m(y)g(r\013t)m(wnd.\))275
6420
 
3353 y(The)k(general)h(rule)e(is)h(as)h(follo)m(ws.)53
6421
 
b Fo(ostride_t)33 b Fu(equals)h Fo(istride)p Fu(.)52
6422
 
b(If)34 b Fo(idist)g Fu(is)g Fo(1)g Fu(and)g Fo(idist)g
6423
 
Fu(is)150 3463 y(less)29 b(than)g Fo(istride)p Fu(,)e(then)i
6424
 
Fo(odist_t)f Fu(is)g Fo(1)p Fu(.)40 b(Otherwise,)28 b(for)i(a)f
6425
 
(real-to-complex)h(transform)f Fo(odist_t)150 3573 y
6426
 
Fu(is)g Fo(idist/2)g Fu(and)h(for)g(a)g(complex-to-real)i(transform)d
6427
 
Fo(odist_t)g Fu(is)g Fo(idist*2)p Fu(.)150 3811 y Fi(3.5.5)63
 
6481
2034 y(The)k(general)h(rule)e(is)h(as)h(follo)m(ws.)53
 
6482
b Fp(ostride_t)33 b Fv(equals)h Fp(istride)p Fv(.)52
 
6483
b(If)34 b Fp(idist)g Fv(is)g Fp(1)g Fv(and)g Fp(idist)g
 
6484
Fv(is)150 2144 y(less)29 b(than)g Fp(istride)p Fv(,)e(then)i
 
6485
Fp(odist_t)f Fv(is)g Fp(1)p Fv(.)40 b(Otherwise,)28 b(for)i(a)f
 
6486
(real-to-complex)h(transform)f Fp(odist_t)150 2253 y
 
6487
Fv(is)g Fp(idist/2)g Fv(and)h(for)g(a)g(complex-to-real)i(transform)d
 
6488
Fp(odist_t)g Fv(is)g Fp(idist*2)p Fv(.)150 2488 y Fh(3.5.5)63
6428
6489
b(Destro)m(ying)41 b(a)g(Multi-dimensional)h(Plan)390
6429
 
4005 y Fo(#include)k(<rfftw.h>)390 4212 y(void)h
 
6490
2650 y Fp(#include)k(<rfftw.h>)390 2858 y(void)h
6430
6491
(rfftwnd_destroy_plan\(rf)o(ftwn)o(d_p)o(lan)41 b(plan\);)275
6431
 
4354 y Fu(The)34 b(function)f Fo(rfftwnd_destroy_plan)c
6432
 
Fu(frees)35 b(the)g(plan)e Fo(plan)h Fu(and)g(releases)h(all)e(the)i
6433
 
(memory)150 4464 y(asso)s(ciated)c(with)e(it.)40 b(After)31
 
6492
2998 y Fv(The)34 b(function)f Fp(rfftwnd_destroy_plan)c
 
6493
Fv(frees)35 b(the)g(plan)e Fp(plan)h Fv(and)g(releases)h(all)e(the)i
 
6494
(memory)150 3107 y(asso)s(ciated)c(with)e(it.)40 b(After)31
6434
6495
b(destruction,)f(a)g(plan)f(is)h(no)g(longer)g(v)-5 b(alid.)150
6435
 
4702 y Fi(3.5.6)63 b(What)40 b(RFFTWND)h(Really)f(Computes)275
6436
 
4902 y Fu(The)23 b(con)m(v)m(en)m(tions)i(that)f(w)m(e)h(follo)m(w)e
 
6496
3343 y Fh(3.5.6)63 b(What)40 b(RFFTWND)h(Really)f(Computes)275
 
6497
3541 y Fv(The)23 b(con)m(v)m(en)m(tions)i(that)f(w)m(e)h(follo)m(w)e
6437
6498
(for)g(the)h(real)g(m)m(ulti-dimensional)c(transform)j(are)h(analogous)
6438
 
g(to)150 5011 y(those)f(for)f(the)g(complex)g(m)m(ulti-dimensional)d
 
6499
g(to)150 3650 y(those)f(for)f(the)g(complex)g(m)m(ulti-dimensional)d
6439
6500
(transform.)37 b(In)22 b(particular,)g(the)h(forw)m(ard)f(transform)f
6440
 
(has)150 5121 y(a)33 b(negativ)m(e)g(sign)f(in)f(the)i(exp)s(onen)m(t)f
 
6501
(has)150 3760 y(a)33 b(negativ)m(e)g(sign)f(in)f(the)i(exp)s(onen)m(t)f
6441
6502
(and)g(neither)f(the)i(forw)m(ard)f(nor)g(the)g(bac)m(kw)m(ard)h
6442
 
(transforms)f(will)150 5230 y(p)s(erform)g(an)m(y)i(normalization.)48
 
6503
(transforms)f(will)150 3870 y(p)s(erform)g(an)m(y)i(normalization.)48
6443
6504
b(Computing)32 b(the)h(bac)m(kw)m(ard)h(transform)f(of)g(the)h(forw)m
6444
 
(ard)f(transform)150 5340 y(will)c(m)m(ultiply)g(the)j(arra)m(y)h(b)m
 
6505
(ard)f(transform)150 3979 y(will)c(m)m(ultiply)g(the)j(arra)m(y)h(b)m
6445
6506
(y)f(the)g(pro)s(duct)e(of)i(its)g(dimensions)d(\(that)k(is,)e(the)h
6446
 
(logical)g(dimensions)d(of)p eop
 
6507
(logical)g(dimensions)d(of)150 4089 y(the)37 b(real)g(data\).)63
 
6508
b(The)36 b(forw)m(ard)h(transform)f(is)h(real-to-complex)h(and)e(the)i
 
6509
(bac)m(kw)m(ard)f(transform)g(is)150 4198 y(complex-to-real.)275
 
6510
4338 y(The)29 b(exact)j(mathematical)f(de\014nition)d(of)j(our)e(real)h
 
6511
(m)m(ulti-dimensional)d(transform)j(follo)m(ws.)150 4478
 
6512
y Fr(R)-5 b(e)g(al)35 b(to)e(c)-5 b(omplex)35 b(\(forwar)-5
 
6513
b(d\))36 b(tr)-5 b(ansform.)44 b Fv(Let)32 b Fm(X)38
 
6514
b Fv(b)s(e)30 b(a)h Fm(d)p Fv(-dimensional)e(real)i(arra)m(y)g(whose)g
 
6515
(elemen)m(ts)150 4588 y(are)g Fm(X)7 b Fv([)p Fm(j)446
 
6516
4602 y Fo(1)484 4588 y Fm(;)15 b(j)561 4602 y Fo(2)599
 
6517
4588 y Fm(;)g(:)g(:)g(:)i(;)e(j)838 4602 y Fl(d)877 4588
 
6518
y Fv(],)31 b(where)f(0)c Ft(\024)f Fm(j)1425 4602 y Fl(s)1486
 
6519
4588 y Fm(<)g(n)1637 4602 y Fl(s)1703 4588 y Fv(for)30
 
6520
b(all)f Fm(s)c Ft(2)g(f)p Fv(1)p Fm(;)15 b Fv(2)p Fm(;)g(:)g(:)g(:)k(;)
 
6521
c(d)p Ft(g)p Fv(.)42 b(Let)31 b(also)f Fm(!)3062 4602
 
6522
y Fl(s)3122 4588 y Fv(=)c Fm(e)3261 4555 y Fo(2)p Fl(\031)3335
 
6523
4510 y Fk(p)p 3389 4510 85 3 v 3389 4555 a(\000)p Fo(1)p
 
6524
Fl(=n)3549 4563 y Fe(s)3586 4588 y Fv(,)k(for)150 4698
 
6525
y(all)60 b Fm(s)24 b Ft(2)h(f)p Fv(1)p Fm(;)15 b Fv(2)p
 
6526
Fm(;)g(:)g(:)g(:)k(;)c(d)p Ft(g)p Fv(.)275 4838 y(The)24
 
6527
b(real)g(to)h(complex)g(transform)e(computes)i(a)g(complex)f(arra)m(y)i
 
6528
Fm(Y)20 b Fv(,)26 b(whose)e(structure)g(is)g(the)h(same)150
 
6529
4947 y(as)31 b(that)g(of)f Fm(X)7 b Fv(,)31 b(de\014ned)e(b)m(y)534
 
6530
5258 y Fm(Y)20 b Fv([)p Fm(i)663 5272 y Fo(1)701 5258
 
6531
y Fm(;)15 b(i)772 5272 y Fo(2)810 5258 y Fm(;)g(:)g(:)g(:)i(;)e(i)1043
 
6532
5272 y Fl(d)1082 5258 y Fv(])26 b(=)1229 5151 y Fl(n)1270
 
6533
5159 y Fd(1)1302 5151 y Fk(\000)p Fo(1)1248 5177 y Fg(X)1236
 
6534
5354 y Fl(j)1263 5362 y Fd(1)1296 5354 y Fo(=0)1402 5151
 
6535
y Fl(n)1443 5159 y Fd(2)1476 5151 y Fk(\000)p Fo(1)1421
 
6536
5177 y Fg(X)1409 5354 y Fl(j)1436 5362 y Fd(2)1469 5354
 
6537
y Fo(=0)1576 5258 y Ft(\001)15 b(\001)g(\001)1697 5150
 
6538
y Fl(n)1738 5159 y Fe(d)1773 5150 y Fk(\000)p Fo(1)1717
 
6539
5177 y Fg(X)1704 5354 y Fl(j)1731 5363 y Fe(d)1767 5354
 
6540
y Fo(=0)1873 5258 y Fm(X)7 b Fv([)p Fm(j)2017 5272 y
 
6541
Fo(1)2055 5258 y Fm(;)15 b(j)2132 5272 y Fo(2)2170 5258
 
6542
y Fm(;)g(:)g(:)g(:)i(;)e(j)2409 5272 y Fl(d)2449 5258
 
6543
y Fv(])p Fm(!)2534 5215 y Fk(\000)p Fl(i)2609 5223 y
 
6544
Fd(1)2642 5215 y Fl(j)2669 5223 y Fd(1)2531 5277 y Fo(1)2706
 
6545
5258 y Fm(!)2766 5215 y Fk(\000)p Fl(i)2841 5223 y Fd(2)2873
 
6546
5215 y Fl(j)2900 5223 y Fd(2)2763 5277 y Fo(2)2952 5258
 
6547
y Ft(\001)g(\001)g(\001)i Fm(!)3134 5215 y Fk(\000)p
 
6548
Fl(i)3209 5224 y Fe(d)3244 5215 y Fl(j)3271 5224 y Fe(d)3131
 
6549
5280 y Fl(d)3340 5258 y Fm(:)p eop end
6447
6550
%%Page: 34 36
6448
 
34 35 bop 150 -116 a Fu(34)3232 b(FFTW)150 299 y(the)37
6449
 
b(real)g(data\).)63 b(The)36 b(forw)m(ard)h(transform)f(is)h
6450
 
(real-to-complex)h(and)e(the)i(bac)m(kw)m(ard)f(transform)g(is)150
6451
 
408 y(complex-to-real.)275 541 y(The)29 b(exact)j(mathematical)f
6452
 
(de\014nition)d(of)j(our)e(real)h(m)m(ulti-dimensional)d(transform)j
6453
 
(follo)m(ws.)150 674 y Fq(R)-5 b(e)g(al)35 b(to)e(c)-5
6454
 
b(omplex)35 b(\(forwar)-5 b(d\))36 b(tr)-5 b(ansform.)44
6455
 
b Fu(Let)32 b Fl(X)38 b Fu(b)s(e)30 b(a)h Fl(d)p Fu(-dimensional)e
6456
 
(real)i(arra)m(y)g(whose)g(elemen)m(ts)150 784 y(are)g
6457
 
Fl(X)7 b Fu([)p Fl(j)446 798 y Fn(1)484 784 y Fl(;)15
6458
 
b(j)561 798 y Fn(2)599 784 y Fl(;)g(:)g(:)g(:)i(;)e(j)838
6459
 
798 y Fk(d)877 784 y Fu(],)31 b(where)f(0)c Fs(\024)f
6460
 
Fl(j)1425 798 y Fk(s)1486 784 y Fl(<)g(n)1637 798 y Fk(s)1703
6461
 
784 y Fu(for)30 b(all)f Fl(s)c Fs(2)g(f)p Fu(1)p Fl(;)15
6462
 
b Fu(2)p Fl(;)g(:)g(:)g(:)k(;)c(d)p Fs(g)p Fu(.)42 b(Let)31
6463
 
b(also)f Fl(!)3062 798 y Fk(s)3122 784 y Fu(=)c Fl(e)3261
6464
 
751 y Fn(2)p Fk(\031)3335 706 y Fj(p)p 3389 706 85 3
6465
 
v 3389 751 a(\000)p Fn(1)p Fk(=n)3549 759 y Fg(s)3586
6466
 
784 y Fu(,)k(for)150 893 y(all)60 b Fl(s)24 b Fs(2)h(f)p
6467
 
Fu(1)p Fl(;)15 b Fu(2)p Fl(;)g(:)g(:)g(:)k(;)c(d)p Fs(g)p
6468
 
Fu(.)275 1026 y(The)24 b(real)g(to)h(complex)g(transform)e(computes)i
6469
 
(a)g(complex)f(arra)m(y)i Fl(Y)20 b Fu(,)26 b(whose)e(structure)g(is)g
6470
 
(the)h(same)150 1136 y(as)31 b(that)g(of)f Fl(X)7 b Fu(,)31
6471
 
b(de\014ned)e(b)m(y)534 1430 y Fl(Y)20 b Fu([)p Fl(i)663
6472
 
1444 y Fn(1)701 1430 y Fl(;)15 b(i)772 1444 y Fn(2)810
6473
 
1430 y Fl(;)g(:)g(:)g(:)i(;)e(i)1043 1444 y Fk(d)1082
6474
 
1430 y Fu(])26 b(=)1229 1324 y Fk(n)1270 1332 y Ff(1)1302
6475
 
1324 y Fj(\000)p Fn(1)1248 1350 y Fh(X)1236 1526 y Fk(j)1263
6476
 
1534 y Ff(1)1296 1526 y Fn(=0)1402 1324 y Fk(n)1443 1332
6477
 
y Ff(2)1476 1324 y Fj(\000)p Fn(1)1421 1350 y Fh(X)1409
6478
 
1526 y Fk(j)1436 1534 y Ff(2)1469 1526 y Fn(=0)1576 1430
6479
 
y Fs(\001)15 b(\001)g(\001)1697 1323 y Fk(n)1738 1332
6480
 
y Fg(d)1773 1323 y Fj(\000)p Fn(1)1717 1350 y Fh(X)1704
6481
 
1526 y Fk(j)1731 1535 y Fg(d)1767 1526 y Fn(=0)1873 1430
6482
 
y Fl(X)7 b Fu([)p Fl(j)2017 1444 y Fn(1)2055 1430 y Fl(;)15
6483
 
b(j)2132 1444 y Fn(2)2170 1430 y Fl(;)g(:)g(:)g(:)i(;)e(j)2409
6484
 
1444 y Fk(d)2449 1430 y Fu(])p Fl(!)2534 1388 y Fj(\000)p
6485
 
Fk(i)2609 1396 y Ff(1)2642 1388 y Fk(j)2669 1396 y Ff(1)2531
6486
 
1450 y Fn(1)2706 1430 y Fl(!)2766 1388 y Fj(\000)p Fk(i)2841
6487
 
1396 y Ff(2)2873 1388 y Fk(j)2900 1396 y Ff(2)2763 1450
6488
 
y Fn(2)2952 1430 y Fs(\001)g(\001)g(\001)i Fl(!)3134
6489
 
1388 y Fj(\000)p Fk(i)3209 1397 y Fg(d)3244 1388 y Fk(j)3271
6490
 
1397 y Fg(d)3131 1453 y Fk(d)3340 1430 y Fl(:)275 1693
6491
 
y Fu(The)26 b(output)g(arra)m(y)h Fl(Y)46 b Fu(enjo)m(ys)27
 
6551
TeXDict begin 34 35 bop 150 -116 a Fv(34)3232 b(FFTW)275
 
6552
299 y(The)26 b(output)g(arra)m(y)h Fm(Y)46 b Fv(enjo)m(ys)27
6492
6553
b(a)g(m)m(ultidimensional)22 b(hermitian)j(symmetry)-8
6493
 
b(,)27 b(that)g(is,)g(the)f(iden)m(tit)m(y)150 1802 y
6494
 
Fl(Y)20 b Fu([)p Fl(i)279 1816 y Fn(1)317 1802 y Fl(;)15
6495
 
b(i)388 1816 y Fn(2)426 1802 y Fl(;)g(:)g(:)g(:)h(;)f(i)658
6496
 
1816 y Fk(d)698 1802 y Fu(])28 b(=)g Fl(Y)20 b Fu([)p
6497
 
Fl(n)1003 1816 y Fn(1)1062 1802 y Fs(\000)h Fl(i)1185
6498
 
1816 y Fn(1)1223 1802 y Fl(;)15 b(n)1318 1816 y Fn(2)1376
6499
 
1802 y Fs(\000)21 b Fl(i)1499 1816 y Fn(2)1537 1802 y
6500
 
Fl(;)15 b(:)g(:)g(:)h(;)f(n)1793 1816 y Fk(d)1854 1802
6501
 
y Fs(\000)21 b Fl(i)1977 1816 y Fk(d)2016 1802 y Fu(])2041
6502
 
1769 y Fj(\003)2111 1802 y Fu(holds)31 b(for)h(all)f(0)e
6503
 
Fs(\024)f Fl(i)2824 1816 y Fk(s)2888 1802 y Fl(<)g(n)3042
6504
 
1816 y Fk(s)3077 1802 y Fu(.)46 b(Because)34 b(of)e(this)150
6505
 
1912 y(symmetry)-8 b(,)29 b Fl(Y)48 b Fu(is)28 b(stored)g(in)g(the)g(p)
6506
 
s(eculiar)e(w)m(a)m(y)k(describ)s(ed)c(in)h(Section)h(3.5.3)j([Arra)m
6507
 
(y)e(Dimensions)d(for)150 2021 y(Real)k(Multi-dimensional)d(T)-8
6508
 
b(ransforms],)30 b(page)h(32.)150 2154 y Fq(Complex)44
6509
 
b(to)f(r)-5 b(e)g(al)44 b(\(b)-5 b(ackwar)g(d\))45 b(tr)-5
6510
 
b(ansform.)74 b Fu(Let)42 b Fl(X)48 b Fu(b)s(e)41 b(a)g
6511
 
Fl(d)p Fu(-dimensional)e(complex)i(arra)m(y)g(whose)150
6512
 
2264 y(elemen)m(ts)f(are)h Fl(X)7 b Fu([)p Fl(j)837 2278
6513
 
y Fn(1)875 2264 y Fl(;)15 b(j)952 2278 y Fn(2)990 2264
6514
 
y Fl(;)g(:)g(:)g(:)i(;)e(j)1229 2278 y Fk(d)1268 2264
6515
 
y Fu(],)43 b(where)c(0)j Fs(\024)f Fl(j)1869 2278 y Fk(s)1946
6516
 
2264 y Fl(<)g(n)2113 2278 y Fk(s)2188 2264 y Fu(for)e(all)g
6517
 
Fl(s)i Fs(2)g(f)p Fu(1)p Fl(;)15 b Fu(2)p Fl(;)g(:)g(:)g(:)j(;)d(d)p
6518
 
Fs(g)p Fu(.)71 b(The)39 b(arra)m(y)i Fl(X)150 2373 y
6519
 
Fu(m)m(ust)33 b(b)s(e)f(hermitian,)f(that)i(is,)g(the)g(iden)m(tit)m(y)
6520
 
f Fl(X)7 b Fu([)p Fl(j)1906 2387 y Fn(1)1944 2373 y Fl(;)15
6521
 
b(j)2021 2387 y Fn(2)2059 2373 y Fl(;)g(:)g(:)g(:)i(;)e(j)2298
6522
 
2387 y Fk(d)2337 2373 y Fu(])30 b(=)f Fl(X)7 b Fu([)p
6523
 
Fl(n)2654 2387 y Fn(1)2713 2373 y Fs(\000)21 b Fl(j)2842
6524
 
2387 y Fn(1)2880 2373 y Fl(;)15 b(n)2975 2387 y Fn(2)3034
6525
 
2373 y Fs(\000)21 b Fl(j)3163 2387 y Fn(2)3201 2373 y
6526
 
Fl(;)15 b(:)g(:)g(:)i(;)e(n)3458 2387 y Fk(d)3518 2373
6527
 
y Fs(\000)21 b Fl(j)3647 2387 y Fk(d)3687 2373 y Fu(])3712
6528
 
2340 y Fj(\003)150 2483 y Fu(m)m(ust)39 b(hold)e(for)i(all)f(0)i
6529
 
Fs(\024)f Fl(j)1109 2497 y Fk(s)1184 2483 y Fl(<)g(n)1349
6530
 
2497 y Fk(s)1384 2483 y Fu(.)66 b(Moreo)m(v)m(er,)44
6531
 
b Fl(X)i Fu(m)m(ust)39 b(b)s(e)f(stored)h(in)f(memory)g(in)g(the)h(p)s
6532
 
(eculiar)150 2593 y(w)m(a)m(y)32 b(describ)s(ed)d(in)h(Section)h(3.5.3)
6533
 
i([Arra)m(y)e(Dimensions)e(for)i(Real)g(Multi-dimensional)d(T)-8
6534
 
b(ransforms],)150 2702 y(page)31 b(32.)275 2835 y(Let)f
6535
 
Fl(!)494 2849 y Fk(s)554 2835 y Fu(=)25 b Fl(e)692 2802
6536
 
y Fn(2)p Fk(\031)766 2757 y Fj(p)p 821 2757 V 45 x(\000)p
6537
 
Fn(1)p Fk(=n)981 2810 y Fg(s)1017 2835 y Fu(,)30 b(for)f(all)58
6538
 
b Fl(s)25 b Fs(2)g(f)p Fu(1)p Fl(;)15 b Fu(2)p Fl(;)g(:)g(:)g(:)j(;)d
6539
 
(d)p Fs(g)p Fu(.)42 b(The)29 b(complex)g(to)i(real)e(transform)g
6540
 
(computes)h(a)150 2944 y(real)g(arra)m(y)h Fl(Y)20 b
6541
 
Fu(,)30 b(whose)h(structure)f(is)f(the)h(same)h(as)g(that)g(of)f
6542
 
Fl(X)7 b Fu(,)31 b(de\014ned)f(b)m(y)612 3239 y Fl(Y)20
6543
 
b Fu([)p Fl(i)741 3253 y Fn(1)779 3239 y Fl(;)15 b(i)850
6544
 
3253 y Fn(2)888 3239 y Fl(;)g(:)g(:)g(:)i(;)e(i)1121
6545
 
3253 y Fk(d)1160 3239 y Fu(])25 b(=)1306 3133 y Fk(n)1347
6546
 
3141 y Ff(1)1380 3133 y Fj(\000)p Fn(1)1326 3158 y Fh(X)1314
6547
 
3335 y Fk(j)1341 3343 y Ff(1)1374 3335 y Fn(=0)1480 3133
6548
 
y Fk(n)1521 3141 y Ff(2)1553 3133 y Fj(\000)p Fn(1)1499
6549
 
3158 y Fh(X)1487 3335 y Fk(j)1514 3343 y Ff(2)1547 3335
6550
 
y Fn(=0)1654 3239 y Fs(\001)15 b(\001)g(\001)1775 3132
6551
 
y Fk(n)1816 3141 y Fg(d)1851 3132 y Fj(\000)p Fn(1)1795
6552
 
3158 y Fh(X)1782 3335 y Fk(j)1809 3344 y Fg(d)1844 3335
6553
 
y Fn(=0)1951 3239 y Fl(X)7 b Fu([)p Fl(j)2095 3253 y
6554
 
Fn(1)2133 3239 y Fl(;)15 b(j)2210 3253 y Fn(2)2248 3239
6555
 
y Fl(;)g(:)g(:)g(:)i(;)e(j)2487 3253 y Fk(d)2526 3239
6556
 
y Fu(])p Fl(!)2611 3197 y Fk(i)2634 3205 y Ff(1)2667
6557
 
3197 y Fk(j)2694 3205 y Ff(1)2608 3259 y Fn(1)2731 3239
6558
 
y Fl(!)2791 3197 y Fk(i)2814 3205 y Ff(2)2847 3197 y
6559
 
Fk(j)2874 3205 y Ff(2)2788 3259 y Fn(2)2926 3239 y Fs(\001)g(\001)g
6560
 
(\001)i Fl(!)3108 3197 y Fk(i)3131 3206 y Fg(d)3166 3197
6561
 
y Fk(j)3193 3206 y Fg(d)3105 3262 y Fk(d)3263 3239 y
6562
 
Fl(:)275 3507 y Fu(\(That)30 b Fl(Y)50 b Fu(is)30 b(real)g(is)f(not)i
6563
 
(mean)m(t)g(to)g(b)s(e)f(ob)m(vious,)g(although)g(the)g(pro)s(of)g(is)f
6564
 
(easy)-8 b(.\))275 3639 y(Computing)26 b(the)i(forw)m(ard)f(transform)g
6565
 
(follo)m(w)m(ed)h(b)m(y)g(the)g(bac)m(kw)m(ard)g(transform)f(will)f(m)m
6566
 
(ultiply)f(the)150 3749 y(arra)m(y)31 b(b)m(y)514 3685
6567
 
y Fh(Q)593 3705 y Fk(d)593 3772 y(s)p Fn(=1)727 3749
6568
 
y Fl(n)782 3763 y Fk(d)821 3749 y Fu(.)150 4001 y Ft(3.6)68
6569
 
b(Wisdom)45 b(Reference)150 4278 y Fi(3.6.1)63 b(Exp)s(orting)41
6570
 
b(Wisdom)390 4463 y Fo(#include)46 b(<fftw.h>)390 4671
 
6554
b(,)27 b(that)g(is,)g(the)f(iden)m(tit)m(y)150 408 y
 
6555
Fm(Y)20 b Fv([)p Fm(i)279 422 y Fo(1)317 408 y Fm(;)15
 
6556
b(i)388 422 y Fo(2)426 408 y Fm(;)g(:)g(:)g(:)h(;)f(i)658
 
6557
422 y Fl(d)698 408 y Fv(])28 b(=)g Fm(Y)20 b Fv([)p Fm(n)1003
 
6558
422 y Fo(1)1062 408 y Ft(\000)h Fm(i)1185 422 y Fo(1)1223
 
6559
408 y Fm(;)15 b(n)1318 422 y Fo(2)1376 408 y Ft(\000)21
 
6560
b Fm(i)1499 422 y Fo(2)1537 408 y Fm(;)15 b(:)g(:)g(:)h(;)f(n)1793
 
6561
422 y Fl(d)1854 408 y Ft(\000)21 b Fm(i)1977 422 y Fl(d)2016
 
6562
408 y Fv(])2041 375 y Fk(\003)2111 408 y Fv(holds)31
 
6563
b(for)h(all)f(0)e Ft(\024)f Fm(i)2824 422 y Fl(s)2888
 
6564
408 y Fm(<)g(n)3042 422 y Fl(s)3077 408 y Fv(.)46 b(Because)34
 
6565
b(of)e(this)150 518 y(symmetry)-8 b(,)29 b Fm(Y)48 b
 
6566
Fv(is)28 b(stored)g(in)g(the)g(p)s(eculiar)e(w)m(a)m(y)k(describ)s(ed)c
 
6567
(in)h(Section)h(3.5.3)j([Arra)m(y)e(Dimensions)d(for)150
 
6568
628 y(Real)k(Multi-dimensional)d(T)-8 b(ransforms],)30
 
6569
b(page)h(32.)150 761 y Fr(Complex)44 b(to)f(r)-5 b(e)g(al)44
 
6570
b(\(b)-5 b(ackwar)g(d\))45 b(tr)-5 b(ansform.)74 b Fv(Let)42
 
6571
b Fm(X)48 b Fv(b)s(e)41 b(a)g Fm(d)p Fv(-dimensional)e(complex)i(arra)m
 
6572
(y)g(whose)150 870 y(elemen)m(ts)f(are)h Fm(X)7 b Fv([)p
 
6573
Fm(j)837 884 y Fo(1)875 870 y Fm(;)15 b(j)952 884 y Fo(2)990
 
6574
870 y Fm(;)g(:)g(:)g(:)i(;)e(j)1229 884 y Fl(d)1268 870
 
6575
y Fv(],)43 b(where)c(0)j Ft(\024)f Fm(j)1869 884 y Fl(s)1946
 
6576
870 y Fm(<)g(n)2113 884 y Fl(s)2188 870 y Fv(for)e(all)g
 
6577
Fm(s)i Ft(2)g(f)p Fv(1)p Fm(;)15 b Fv(2)p Fm(;)g(:)g(:)g(:)j(;)d(d)p
 
6578
Ft(g)p Fv(.)71 b(The)39 b(arra)m(y)i Fm(X)150 980 y Fv(m)m(ust)33
 
6579
b(b)s(e)f(hermitian,)f(that)i(is,)g(the)g(iden)m(tit)m(y)f
 
6580
Fm(X)7 b Fv([)p Fm(j)1906 994 y Fo(1)1944 980 y Fm(;)15
 
6581
b(j)2021 994 y Fo(2)2059 980 y Fm(;)g(:)g(:)g(:)i(;)e(j)2298
 
6582
994 y Fl(d)2337 980 y Fv(])30 b(=)f Fm(X)7 b Fv([)p Fm(n)2654
 
6583
994 y Fo(1)2713 980 y Ft(\000)21 b Fm(j)2842 994 y Fo(1)2880
 
6584
980 y Fm(;)15 b(n)2975 994 y Fo(2)3034 980 y Ft(\000)21
 
6585
b Fm(j)3163 994 y Fo(2)3201 980 y Fm(;)15 b(:)g(:)g(:)i(;)e(n)3458
 
6586
994 y Fl(d)3518 980 y Ft(\000)21 b Fm(j)3647 994 y Fl(d)3687
 
6587
980 y Fv(])3712 947 y Fk(\003)150 1090 y Fv(m)m(ust)39
 
6588
b(hold)e(for)i(all)f(0)i Ft(\024)f Fm(j)1109 1104 y Fl(s)1184
 
6589
1090 y Fm(<)g(n)1349 1104 y Fl(s)1384 1090 y Fv(.)66
 
6590
b(Moreo)m(v)m(er,)44 b Fm(X)i Fv(m)m(ust)39 b(b)s(e)f(stored)h(in)f
 
6591
(memory)g(in)g(the)h(p)s(eculiar)150 1199 y(w)m(a)m(y)32
 
6592
b(describ)s(ed)d(in)h(Section)h(3.5.3)i([Arra)m(y)e(Dimensions)e(for)i
 
6593
(Real)g(Multi-dimensional)d(T)-8 b(ransforms],)150 1309
 
6594
y(page)31 b(32.)275 1442 y(Let)f Fm(!)494 1456 y Fl(s)554
 
6595
1442 y Fv(=)25 b Fm(e)692 1409 y Fo(2)p Fl(\031)766 1364
 
6596
y Fk(p)p 821 1364 85 3 v 45 x(\000)p Fo(1)p Fl(=n)981
 
6597
1417 y Fe(s)1017 1442 y Fv(,)30 b(for)f(all)58 b Fm(s)25
 
6598
b Ft(2)g(f)p Fv(1)p Fm(;)15 b Fv(2)p Fm(;)g(:)g(:)g(:)j(;)d(d)p
 
6599
Ft(g)p Fv(.)42 b(The)29 b(complex)g(to)i(real)e(transform)g(computes)h
 
6600
(a)150 1552 y(real)g(arra)m(y)h Fm(Y)20 b Fv(,)30 b(whose)h(structure)f
 
6601
(is)f(the)h(same)h(as)g(that)g(of)f Fm(X)7 b Fv(,)31
 
6602
b(de\014ned)f(b)m(y)612 1847 y Fm(Y)20 b Fv([)p Fm(i)741
 
6603
1861 y Fo(1)779 1847 y Fm(;)15 b(i)850 1861 y Fo(2)888
 
6604
1847 y Fm(;)g(:)g(:)g(:)i(;)e(i)1121 1861 y Fl(d)1160
 
6605
1847 y Fv(])25 b(=)1306 1740 y Fl(n)1347 1748 y Fd(1)1380
 
6606
1740 y Fk(\000)p Fo(1)1326 1766 y Fg(X)1314 1943 y Fl(j)1341
 
6607
1951 y Fd(1)1374 1943 y Fo(=0)1480 1740 y Fl(n)1521 1748
 
6608
y Fd(2)1553 1740 y Fk(\000)p Fo(1)1499 1766 y Fg(X)1487
 
6609
1943 y Fl(j)1514 1951 y Fd(2)1547 1943 y Fo(=0)1654 1847
 
6610
y Ft(\001)15 b(\001)g(\001)1775 1740 y Fl(n)1816 1749
 
6611
y Fe(d)1851 1740 y Fk(\000)p Fo(1)1795 1766 y Fg(X)1782
 
6612
1943 y Fl(j)1809 1952 y Fe(d)1844 1943 y Fo(=0)1951 1847
 
6613
y Fm(X)7 b Fv([)p Fm(j)2095 1861 y Fo(1)2133 1847 y Fm(;)15
 
6614
b(j)2210 1861 y Fo(2)2248 1847 y Fm(;)g(:)g(:)g(:)i(;)e(j)2487
 
6615
1861 y Fl(d)2526 1847 y Fv(])p Fm(!)2611 1804 y Fl(i)2634
 
6616
1812 y Fd(1)2667 1804 y Fl(j)2694 1812 y Fd(1)2608 1866
 
6617
y Fo(1)2731 1847 y Fm(!)2791 1804 y Fl(i)2814 1812 y
 
6618
Fd(2)2847 1804 y Fl(j)2874 1812 y Fd(2)2788 1866 y Fo(2)2926
 
6619
1847 y Ft(\001)g(\001)g(\001)i Fm(!)3108 1804 y Fl(i)3131
 
6620
1813 y Fe(d)3166 1804 y Fl(j)3193 1813 y Fe(d)3105 1869
 
6621
y Fl(d)3263 1847 y Fm(:)275 2117 y Fv(\(That)30 b Fm(Y)50
 
6622
b Fv(is)30 b(real)g(is)f(not)i(mean)m(t)g(to)g(b)s(e)f(ob)m(vious,)g
 
6623
(although)g(the)g(pro)s(of)g(is)f(easy)-8 b(.\))275 2250
 
6624
y(Computing)26 b(the)i(forw)m(ard)f(transform)g(follo)m(w)m(ed)h(b)m(y)
 
6625
g(the)g(bac)m(kw)m(ard)g(transform)f(will)f(m)m(ultiply)f(the)150
 
6626
2359 y(arra)m(y)31 b(b)m(y)514 2295 y Fg(Q)593 2316 y
 
6627
Fl(d)593 2382 y(s)p Fo(=1)727 2359 y Fm(n)782 2373 y
 
6628
Fl(d)821 2359 y Fv(.)150 2613 y Fu(3.6)68 b(Wisdom)45
 
6629
b(Reference)150 2891 y Fh(3.6.1)63 b(Exp)s(orting)41
 
6630
b(Wisdom)390 3053 y Fp(#include)46 b(<fftw.h>)390 3261
6571
6631
y(void)h(fftw_export_wisdom\(void)41 b(\(*emitter\)\(char)j(c,)j(void)g
6572
 
(*\),)f(void)h(*data\);)390 4775 y(void)g(fftw_export_wisdom_to_f)o
6573
 
(ile\()o(FIL)o(E)42 b(*output_file\);)390 4878 y(char)47
 
6632
(*\),)f(void)h(*data\);)390 3365 y(void)g(fftw_export_wisdom_to_f)o
 
6633
(ile\()o(FIL)o(E)42 b(*output_file\);)390 3468 y(char)47
6574
6634
b(*fftw_export_wisdom_to_)o(stri)o(ng\()o(void)o(\);)275
6575
 
5011 y Fu(These)35 b(functions)g(allo)m(w)g(y)m(ou)i(to)f(exp)s(ort)g
6576
 
(all)f(curren)m(tly)g(accum)m(ulated)i Fo(wisdom)d Fu(in)g(a)j(form)e
6577
 
(from)150 5121 y(whic)m(h)g(it)g(can)i(b)s(e)e(later)h(imp)s(orted)f
 
6635
3602 y Fv(These)35 b(functions)g(allo)m(w)g(y)m(ou)i(to)f(exp)s(ort)g
 
6636
(all)f(curren)m(tly)g(accum)m(ulated)i Fp(wisdom)d Fv(in)g(a)j(form)e
 
6637
(from)150 3711 y(whic)m(h)g(it)g(can)i(b)s(e)e(later)h(imp)s(orted)f
6578
6638
(and)g(restored,)j(ev)m(en)f(during)c(a)k(separate)g(run)d(of)j(the)f
6579
 
(program.)150 5230 y(\(See)c(Section)e(2.6)i([W)-8 b(ords)32
 
6639
(program.)150 3821 y(\(See)c(Section)e(2.6)i([W)-8 b(ords)32
6580
6640
b(of)f(Wisdom],)g(page)g(13.\))44 b(The)31 b(curren)m(t)f(store)i(of)f
6581
 
Fo(wisdom)e Fu(is)h(not)h(a\013ected)150 5340 y(b)m(y)f(calling)f(an)m
6582
 
(y)i(of)f(these)h(routines.)p eop
6583
 
%%Page: 35 37
6584
 
35 36 bop 150 -116 a Fu(Chapter)30 b(3:)41 b(FFTW)31
6585
 
b(Reference)2354 b(35)275 299 y Fo(fftw_export_wisdom)34
6586
 
b Fu(exp)s(orts)39 b(the)h Fo(wisdom)e Fu(to)i(an)m(y)g(output)f
6587
 
(medium,)i(as)e(sp)s(eci\014ed)f(b)m(y)i(the)150 408
6588
 
y(callbac)m(k)33 b(function)f Fo(emitter)p Fu(.)47 b
6589
 
Fo(emitter)32 b Fu(is)g(a)h Fo(putc)p Fu(-lik)m(e)f(function)g(that)i
6590
 
(writes)e(the)i(c)m(haracter)g Fo(c)f Fu(to)150 518 y(some)27
6591
 
b(output;)g(its)f(second)g(parameter)g(is)g(the)g Fo(data)f
6592
 
Fu(p)s(oin)m(ter)g(passed)h(to)h Fo(fftw_export_wisdom)p
6593
 
Fu(.)34 b(F)-8 b(or)150 628 y(con)m(v)m(enience,)32 b(the)e(follo)m
6594
 
(wing)f(t)m(w)m(o)j(\\wrapp)s(er")d(routines)h(are)g(pro)m(vided:)275
6595
 
762 y Fo(fftw_export_wisdom_to_f)o(ile)k Fu(writes)40
6596
 
b(the)g Fo(wisdom)f Fu(to)j(the)f(curren)m(t)f(p)s(osition)f(in)g
6597
 
Fo(output_)150 872 y(file)p Fu(,)28 b(whic)m(h)g(should)f(b)s(e)i(op)s
 
6641
Fp(wisdom)e Fv(is)h(not)h(a\013ected)150 3930 y(b)m(y)f(calling)f(an)m
 
6642
(y)i(of)f(these)h(routines.)275 4064 y Fp(fftw_export_wisdom)j
 
6643
Fv(exp)s(orts)39 b(the)h Fp(wisdom)e Fv(to)i(an)m(y)g(output)f(medium,)
 
6644
i(as)e(sp)s(eci\014ed)f(b)m(y)i(the)150 4173 y(callbac)m(k)33
 
6645
b(function)f Fp(emitter)p Fv(.)47 b Fp(emitter)32 b Fv(is)g(a)h
 
6646
Fp(putc)p Fv(-lik)m(e)f(function)g(that)i(writes)e(the)i(c)m(haracter)g
 
6647
Fp(c)f Fv(to)150 4283 y(some)27 b(output;)g(its)f(second)g(parameter)g
 
6648
(is)g(the)g Fp(data)f Fv(p)s(oin)m(ter)g(passed)h(to)h
 
6649
Fp(fftw_export_wisdom)p Fv(.)34 b(F)-8 b(or)150 4392
 
6650
y(con)m(v)m(enience,)32 b(the)e(follo)m(wing)f(t)m(w)m(o)j(\\wrapp)s
 
6651
(er")d(routines)h(are)g(pro)m(vided:)275 4526 y Fp
 
6652
(fftw_export_wisdom_to_f)o(ile)k Fv(writes)40 b(the)g
 
6653
Fp(wisdom)f Fv(to)j(the)f(curren)m(t)f(p)s(osition)f(in)g
 
6654
Fp(output_)150 4635 y(file)p Fv(,)28 b(whic)m(h)g(should)f(b)s(e)i(op)s
6598
6655
(en)f(with)g(write)g(p)s(ermission.)37 b(Up)s(on)28 b(exit,)i(the)f
6599
 
(\014le)f(remains)g(op)s(en)g(and)h(is)150 981 y(p)s(ositioned)f(at)j
6600
 
(the)g(end)f(of)g(the)h Fo(wisdom)e Fu(data.)275 1116
6601
 
y Fo(fftw_export_wisdom_to_s)o(tri)o(ng)22 b Fu(returns)27
6602
 
b(a)h(p)s(oin)m(ter)f(to)i(a)f Fo(NULL)p Fu(-terminated)f(string)g
6603
 
(holding)150 1225 y(the)37 b Fo(wisdom)e Fu(data.)60
 
6656
(\014le)f(remains)g(op)s(en)g(and)h(is)150 4745 y(p)s(ositioned)f(at)j
 
6657
(the)g(end)f(of)g(the)h Fp(wisdom)e Fv(data.)275 4878
 
6658
y Fp(fftw_export_wisdom_to_s)o(tri)o(ng)22 b Fv(returns)27
 
6659
b(a)h(p)s(oin)m(ter)f(to)i(a)f Fp(NULL)p Fv(-terminated)f(string)g
 
6660
(holding)150 4988 y(the)37 b Fp(wisdom)e Fv(data.)60
6604
6661
b(This)34 b(string)i(is)f(dynamically)g(allo)s(cated,)j(and)e(it)g(is)g
6605
 
(the)g(resp)s(onsibilit)m(y)d(of)k(the)150 1335 y(caller)30
6606
 
b(to)h(deallo)s(cate)g(it)e(with)h Fo(fftw_free)d Fu(when)j(it)g(is)f
6607
 
(no)h(longer)g(needed.)275 1469 y(All)24 b(of)j(these)g(routines)e(exp)
 
6662
(the)g(resp)s(onsibilit)m(y)d(of)k(the)150 5097 y(caller)30
 
6663
b(to)h(deallo)s(cate)g(it)e(with)h Fp(fftw_free)d Fv(when)j(it)g(is)f
 
6664
(no)h(longer)g(needed.)275 5230 y(All)24 b(of)j(these)g(routines)e(exp)
6608
6665
s(ort)h(the)g(wisdom)f(in)g(the)i(same)f(format,)i(whic)m(h)d(w)m(e)i
6609
 
(will)d(not)i(do)s(cumen)m(t)150 1579 y(here)k(except)i(to)f(sa)m(y)g
 
6666
(will)d(not)i(do)s(cumen)m(t)150 5340 y(here)k(except)i(to)f(sa)m(y)g
6610
6667
(that)g(it)f(is)f(LISP-lik)m(e)g(ASCI)s(I)g(text)i(that)g(is)f
6611
 
(insensitiv)m(e)e(to)j(white)f(space.)150 1803 y Fi(3.6.2)63
6612
 
b(Imp)s(orting)41 b(Wisdom)390 1990 y Fo(#include)46
6613
 
b(<fftw.h>)390 2198 y(fftw_status)f(fftw_import_wisdom\(int)c
6614
 
(\(*get_input\)\(void)j(*\),)i(void)h(*data\);)390 2301
6615
 
y(fftw_status)e(fftw_import_wisdom_from)o(_fil)o(e\(FI)o(LE)c
6616
 
(*input_file\);)390 2405 y(fftw_status)k(fftw_import_wisdom_from)o
6617
 
(_str)o(ing\()o(con)o(st)d(char)k(*input_string\);)275
6618
 
2540 y Fu(These)32 b(functions)f(imp)s(ort)g Fo(wisdom)g
6619
 
Fu(in)m(to)i(a)g(program)f(from)h(data)g(stored)g(b)m(y)f(the)h
6620
 
Fo(fftw_export_)150 2649 y(wisdom)39 b Fu(functions)g(ab)s(o)m(v)m(e.)
6621
 
72 b(\(See)41 b(Section)g(2.6)g([W)-8 b(ords)41 b(of)g(Wisdom],)i(page)
6622
 
e(13.\))73 b(The)40 b(imp)s(orted)150 2759 y Fo(wisdom)29
6623
 
b Fu(supplemen)m(ts)h(rather)g(than)h(replaces)g(an)m(y)g
6624
 
Fo(wisdom)f Fu(already)g(accum)m(ulated)i(b)m(y)e(the)i(running)150
6625
 
2868 y(program)40 b(\(except)i(when)d(there)h(is)g(con\015icting)f
6626
 
Fo(wisdom)p Fu(,)i(in)e(whic)m(h)g(case)i(the)g(existing)e(wisdom)g(is)
6627
 
150 2978 y(replaced\).)275 3112 y Fo(fftw_import_wisdom)25
6628
 
b Fu(imp)s(orts)k Fo(wisdom)g Fu(from)h(an)m(y)h(input)d(medium,)h(as)i
6629
 
(sp)s(eci\014ed)e(b)m(y)i(the)f(call-)150 3222 y(bac)m(k)25
6630
 
b(function)d Fo(get_input)p Fu(.)36 b Fo(get_input)22
6631
 
b Fu(is)h(a)h Fo(getc)p Fu(-lik)m(e)e(function)h(that)h(returns)f(the)h
6632
 
(next)g(c)m(haracter)150 3332 y(in)30 b(the)h(input;)f(its)g(parameter)
6633
 
i(is)e(the)h Fo(data)f Fu(p)s(oin)m(ter)g(passed)h(to)h
6634
 
Fo(fftw_import_wisdom)p Fu(.)38 b(If)30 b(the)i(end)150
6635
 
3441 y(of)41 b(the)g(input)e(data)i(is)f(reac)m(hed)h(\(whic)m(h)f
6636
 
(should)f(nev)m(er)i(happ)s(en)e(for)i(v)-5 b(alid)39
6637
 
b(data\),)44 b(it)d(ma)m(y)g(return)150 3551 y(either)25
6638
 
b Fo(NULL)g Fu(\(ASCI)s(I)g(0\))i(or)f Fo(EOF)f Fu(\(as)h(de\014ned)f
6639
 
(in)g Fo(<stdio.h>)p Fu(\).)37 b(F)-8 b(or)27 b(con)m(v)m(enience,)h
6640
 
(the)e(follo)m(wing)f(t)m(w)m(o)150 3660 y(\\wrapp)s(er")k(routines)h
6641
 
(are)g(pro)m(vided:)275 3795 y Fo(fftw_import_wisdom_from)o(_fi)o(le)40
6642
 
b Fu(reads)46 b Fo(wisdom)f Fu(from)h(the)g(curren)m(t)g(p)s(osition)f
6643
 
(in)g Fo(input_)150 3905 y(file)p Fu(,)30 b(whic)m(h)g(should)f(b)s(e)i
6644
 
(op)s(en)f(with)g(read)h(p)s(ermission.)39 b(Up)s(on)30
6645
 
b(exit,)i(the)f(\014le)f(remains)g(op)s(en)g(and)h(is)150
6646
 
4014 y(p)s(ositioned)d(at)j(the)g(end)f(of)g(the)h Fo(wisdom)e
6647
 
Fu(data.)275 4149 y Fo(fftw_import_wisdom_from)o(_st)o(ring)52
6648
 
b Fu(reads)59 b Fo(wisdom)f Fu(from)h(the)g Fo(NULL)p
6649
 
Fu(-terminated)f(string)150 4258 y Fo(input_string)p
6650
 
Fu(.)275 4393 y(The)28 b(return)f(v)-5 b(alue)28 b(of)h(these)g
6651
 
(routines)e(is)h Fo(FFTW_SUCCESS)d Fu(if)i(the)i(wisdom)e(w)m(as)i
6652
 
(read)f(successfully)-8 b(,)150 4502 y(and)36 b Fo(FFTW_FAILURE)d
6653
 
Fu(otherwise.)58 b(Note)38 b(that,)g(in)d(all)h(of)g(these)h
6654
 
(functions,)g(an)m(y)f(data)h(in)e(the)i(input)150 4612
6655
 
y(stream)29 b(past)g(the)g(end)f(of)h(the)g Fo(wisdom)e
6656
 
Fu(data)i(is)f(simply)f(ignored)g(\(it)i(is)f(not)h(ev)m(en)g(read)g
6657
 
(if)f(the)h Fo(wisdom)150 4721 y Fu(data)i(is)e(w)m(ell-formed\).)150
6658
 
4946 y Fi(3.6.3)63 b(F)-10 b(orgetting)41 b(Wisdom)390
6659
 
5132 y Fo(#include)46 b(<fftw.h>)390 5340 y(void)h
6660
 
(fftw_forget_wisdom\(void)o(\);)p eop
6661
 
%%Page: 36 38
6662
 
36 37 bop 150 -116 a Fu(36)3232 b(FFTW)275 299 y(Calling)39
6663
 
b Fo(fftw_forget_wisdom)d Fu(causes)42 b(all)e(accum)m(ulated)i
6664
 
Fo(wisdom)e Fu(to)i(b)s(e)f(discarded)f(and)h(its)150
6665
 
408 y(asso)s(ciated)30 b(memory)f(to)h(b)s(e)e(freed.)40
6666
 
b(\(New)30 b Fo(wisdom)d Fu(can)j(still)d(b)s(e)i(gathered)g(subsequen)
6667
 
m(tly)-8 b(,)29 b(ho)m(w)m(ev)m(er.\))150 666 y Ft(3.7)68
6668
 
b(Memory)45 b(Allo)t(cator)h(Reference)390 853 y Fo(#include)g
6669
 
(<fftw.h>)390 1060 y(void)h(*\(*fftw_malloc_hook\))42
6670
 
b(\(size_t)k(n\);)390 1164 y(void)h(\(*fftw_free_hook\))c(\(void)j
6671
 
(*p\);)275 1298 y Fu(Whenev)m(er)21 b(it)e(has)i(to)g(allo)s(cate)f
 
6668
(insensitiv)m(e)e(to)j(white)f(space.)p eop end
 
6669
%%Page: 35 37
 
6670
TeXDict begin 35 36 bop 150 -116 a Fv(Chapter)30 b(3:)41
 
6671
b(FFTW)31 b(Reference)2354 b(35)150 299 y Fh(3.6.2)63
 
6672
b(Imp)s(orting)41 b(Wisdom)390 461 y Fp(#include)46 b(<fftw.h>)390
 
6673
668 y(fftw_status)f(fftw_import_wisdom\(int)c(\(*get_input\)\(void)j
 
6674
(*\),)i(void)h(*data\);)390 772 y(fftw_status)e
 
6675
(fftw_import_wisdom_from)o(_fil)o(e\(FI)o(LE)c(*input_file\);)390
 
6676
876 y(fftw_status)k(fftw_import_wisdom_from)o(_str)o(ing\()o(con)o(st)d
 
6677
(char)k(*input_string\);)275 1011 y Fv(These)32 b(functions)f(imp)s
 
6678
(ort)g Fp(wisdom)g Fv(in)m(to)i(a)g(program)f(from)h(data)g(stored)g(b)
 
6679
m(y)f(the)h Fp(fftw_export_)150 1121 y(wisdom)39 b Fv(functions)g(ab)s
 
6680
(o)m(v)m(e.)72 b(\(See)41 b(Section)g(2.6)g([W)-8 b(ords)41
 
6681
b(of)g(Wisdom],)i(page)e(13.\))73 b(The)40 b(imp)s(orted)150
 
6682
1230 y Fp(wisdom)29 b Fv(supplemen)m(ts)h(rather)g(than)h(replaces)g
 
6683
(an)m(y)g Fp(wisdom)f Fv(already)g(accum)m(ulated)i(b)m(y)e(the)i
 
6684
(running)150 1340 y(program)40 b(\(except)i(when)d(there)h(is)g
 
6685
(con\015icting)f Fp(wisdom)p Fv(,)i(in)e(whic)m(h)g(case)i(the)g
 
6686
(existing)e(wisdom)g(is)150 1449 y(replaced\).)275 1584
 
6687
y Fp(fftw_import_wisdom)25 b Fv(imp)s(orts)k Fp(wisdom)g
 
6688
Fv(from)h(an)m(y)h(input)d(medium,)h(as)i(sp)s(eci\014ed)e(b)m(y)i(the)
 
6689
f(call-)150 1694 y(bac)m(k)25 b(function)d Fp(get_input)p
 
6690
Fv(.)36 b Fp(get_input)22 b Fv(is)h(a)h Fp(getc)p Fv(-lik)m(e)e
 
6691
(function)h(that)h(returns)f(the)h(next)g(c)m(haracter)150
 
6692
1804 y(in)30 b(the)h(input;)f(its)g(parameter)i(is)e(the)h
 
6693
Fp(data)f Fv(p)s(oin)m(ter)g(passed)h(to)h Fp(fftw_import_wisdom)p
 
6694
Fv(.)38 b(If)30 b(the)i(end)150 1913 y(of)41 b(the)g(input)e(data)i(is)
 
6695
f(reac)m(hed)h(\(whic)m(h)f(should)f(nev)m(er)i(happ)s(en)e(for)i(v)-5
 
6696
b(alid)39 b(data\),)44 b(it)d(ma)m(y)g(return)150 2023
 
6697
y(either)25 b Fp(NULL)g Fv(\(ASCI)s(I)g(0\))i(or)f Fp(EOF)f
 
6698
Fv(\(as)h(de\014ned)f(in)g Fp(<stdio.h>)p Fv(\).)37 b(F)-8
 
6699
b(or)27 b(con)m(v)m(enience,)h(the)e(follo)m(wing)f(t)m(w)m(o)150
 
6700
2132 y(\\wrapp)s(er")k(routines)h(are)g(pro)m(vided:)275
 
6701
2267 y Fp(fftw_import_wisdom_from)o(_fi)o(le)40 b Fv(reads)46
 
6702
b Fp(wisdom)f Fv(from)h(the)g(curren)m(t)g(p)s(osition)f(in)g
 
6703
Fp(input_)150 2377 y(file)p Fv(,)30 b(whic)m(h)g(should)f(b)s(e)i(op)s
 
6704
(en)f(with)g(read)h(p)s(ermission.)39 b(Up)s(on)30 b(exit,)i(the)f
 
6705
(\014le)f(remains)g(op)s(en)g(and)h(is)150 2487 y(p)s(ositioned)d(at)j
 
6706
(the)g(end)f(of)g(the)h Fp(wisdom)e Fv(data.)275 2622
 
6707
y Fp(fftw_import_wisdom_from)o(_st)o(ring)52 b Fv(reads)59
 
6708
b Fp(wisdom)f Fv(from)h(the)g Fp(NULL)p Fv(-terminated)f(string)150
 
6709
2731 y Fp(input_string)p Fv(.)275 2866 y(The)28 b(return)f(v)-5
 
6710
b(alue)28 b(of)h(these)g(routines)e(is)h Fp(FFTW_SUCCESS)d
 
6711
Fv(if)i(the)i(wisdom)e(w)m(as)i(read)f(successfully)-8
 
6712
b(,)150 2976 y(and)36 b Fp(FFTW_FAILURE)d Fv(otherwise.)58
 
6713
b(Note)38 b(that,)g(in)d(all)h(of)g(these)h(functions,)g(an)m(y)f(data)
 
6714
h(in)e(the)i(input)150 3085 y(stream)29 b(past)g(the)g(end)f(of)h(the)g
 
6715
Fp(wisdom)e Fv(data)i(is)f(simply)f(ignored)g(\(it)i(is)f(not)h(ev)m
 
6716
(en)g(read)g(if)f(the)h Fp(wisdom)150 3195 y Fv(data)i(is)e(w)m
 
6717
(ell-formed\).)150 3420 y Fh(3.6.3)63 b(F)-10 b(orgetting)41
 
6718
b(Wisdom)390 3582 y Fp(#include)46 b(<fftw.h>)390 3790
 
6719
y(void)h(fftw_forget_wisdom\(void)o(\);)275 3925 y Fv(Calling)39
 
6720
b Fp(fftw_forget_wisdom)d Fv(causes)42 b(all)e(accum)m(ulated)i
 
6721
Fp(wisdom)e Fv(to)i(b)s(e)f(discarded)f(and)h(its)150
 
6722
4035 y(asso)s(ciated)30 b(memory)f(to)h(b)s(e)e(freed.)40
 
6723
b(\(New)30 b Fp(wisdom)d Fv(can)j(still)d(b)s(e)i(gathered)g(subsequen)
 
6724
m(tly)-8 b(,)29 b(ho)m(w)m(ev)m(er.\))150 4293 y Fu(3.7)68
 
6725
b(Memory)45 b(Allo)t(cator)h(Reference)390 4455 y Fp(#include)g
 
6726
(<fftw.h>)390 4663 y(void)h(*\(*fftw_malloc_hook\))42
 
6727
b(\(size_t)k(n\);)390 4767 y(void)h(\(*fftw_free_hook\))c(\(void)j
 
6728
(*p\);)275 4902 y Fv(Whenev)m(er)21 b(it)e(has)i(to)g(allo)s(cate)f
6672
6729
(and)g(release)h(memory)-8 b(,)23 b(FFTW)e(ordinarily)c(calls)i
6673
 
Fo(malloc)g Fu(and)h Fo(free)p Fu(.)150 1408 y(If)25
6674
 
b Fo(malloc)e Fu(fails,)i(FFTW)h(prin)m(ts)e(an)h(error)g(message)h
 
6730
Fp(malloc)g Fv(and)h Fp(free)p Fv(.)150 5011 y(If)25
 
6731
b Fp(malloc)e Fv(fails,)i(FFTW)h(prin)m(ts)e(an)h(error)g(message)h
6675
6732
(and)f(exits.)39 b(This)23 b(b)s(eha)m(vior)h(ma)m(y)i(b)s(e)f
6676
 
(undesirable)150 1518 y(in)i(some)i(applications.)38
 
6733
(undesirable)150 5121 y(in)i(some)i(applications.)38
6677
6734
b(Also,)29 b(sp)s(ecial)e(memory-handling)f(functions)h(ma)m(y)i(b)s(e)
6678
 
e(necessary)i(in)e(certain)150 1627 y(en)m(vironmen)m(ts.)66
 
6735
e(necessary)i(in)e(certain)150 5230 y(en)m(vironmen)m(ts.)66
6679
6736
b(Consequen)m(tly)-8 b(,)41 b(FFTW)f(pro)m(vides)e(means)h(b)m(y)g
6680
6737
(whic)m(h)e(y)m(ou)j(can)f(install)e(y)m(our)i(o)m(wn)150
6681
 
1737 y(memory)k(allo)s(cator)g(and)g(tak)m(e)i(whatev)m(er)e
 
6738
5340 y(memory)k(allo)s(cator)g(and)g(tak)m(e)i(whatev)m(er)e
6682
6739
(error-correcting)h(action)f(y)m(ou)h(\014nd)d(appropriate.)78
6683
 
b(The)150 1846 y(v)-5 b(ariables)31 b Fo(fftw_malloc_hook)d
6684
 
Fu(and)k Fo(fftw_free_hook)d Fu(are)k(p)s(oin)m(ters)e(to)i(functions,)
6685
 
f(and)g(they)h(are)150 1956 y(normally)c Fo(NULL)p Fu(.)42
6686
 
b(If)31 b(y)m(ou)h(set)g(those)f(v)-5 b(ariables)30 b(to)i(p)s(oin)m(t)
6687
 
e(to)i(other)g(functions,)e(then)h(FFTW)h(will)c(use)150
6688
 
2066 y(y)m(our)h(routines)e(instead)h(of)h Fo(malloc)d
6689
 
Fu(and)i Fo(free)p Fu(.)39 b Fo(fftw_malloc_hook)25 b
6690
 
Fu(m)m(ust)j(p)s(oin)m(t)g(to)h(a)g Fo(malloc)p Fu(-lik)m(e)150
6691
 
2175 y(function,)g(and)h Fo(fftw_free_hook)d Fu(m)m(ust)j(p)s(oin)m(t)f
6692
 
(to)i(a)g Fo(free)p Fu(-lik)m(e)e(function.)150 2433
6693
 
y Ft(3.8)68 b(Thread)45 b(safet)l(y)275 2625 y Fu(Users)30
6694
 
b(writing)e(m)m(ulti-threaded)h(programs)h(m)m(ust)h(concern)f
6695
 
(themselv)m(es)h(with)e(the)i Fp(thread)f(safet)m(y)150
6696
 
2735 y Fu(of)d(the)g(libraries)d(they)k(use|that)f(is,)g(whether)f(it)h
6697
 
(is)f(safe)h(to)h(call)e(routines)g(in)g(parallel)f(from)h(m)m(ultiple)
6698
 
150 2844 y(threads.)39 b(FFTW)29 b(can)e(b)s(e)g(used)g(in)f(suc)m(h)i
6699
 
(an)f(en)m(vironmen)m(t,)h(but)f(some)h(care)g(m)m(ust)f(b)s(e)g(tak)m
6700
 
(en)i(b)s(ecause)150 2954 y(certain)21 b(parts)f(of)h(FFTW)g(use)g
6701
 
(priv)-5 b(ate)20 b(global)g(v)-5 b(ariables)19 b(to)j(share)e(data)h
6702
 
(b)s(et)m(w)m(een)h(calls.)36 b(In)20 b(particular,)150
6703
 
3064 y(the)42 b(plan-creation)f(functions)f(share)i(trigonometric)g
6704
 
(tables)f(and)g(accum)m(ulated)h Fo(wisdom)p Fu(.)74
6705
 
b(\(Users)150 3173 y(should)37 b(note)j(that)f(these)h(commen)m(ts)g
6706
 
(only)e(apply)g(to)h(programs)g(using)f(shared-memory)g(threads.)150
6707
 
3283 y(P)m(arallelism)k(using)g(MPI)h(or)h(fork)m(ed)g(pro)s(cesses)f
6708
 
(in)m(v)m(olv)m(es)h(a)g(separate)h(address-space)f(and)f(global)150
6709
 
3392 y(v)-5 b(ariables)29 b(for)h(eac)m(h)i(pro)s(cess,)e(and)g(is)f
 
6740
b(The)p eop end
 
6741
%%Page: 36 38
 
6742
TeXDict begin 36 37 bop 150 -116 a Fv(36)3232 b(FFTW)150
 
6743
299 y(v)-5 b(ariables)31 b Fp(fftw_malloc_hook)d Fv(and)k
 
6744
Fp(fftw_free_hook)d Fv(are)k(p)s(oin)m(ters)e(to)i(functions,)f(and)g
 
6745
(they)h(are)150 408 y(normally)c Fp(NULL)p Fv(.)42 b(If)31
 
6746
b(y)m(ou)h(set)g(those)f(v)-5 b(ariables)30 b(to)i(p)s(oin)m(t)e(to)i
 
6747
(other)g(functions,)e(then)h(FFTW)h(will)c(use)150 518
 
6748
y(y)m(our)h(routines)e(instead)h(of)h Fp(malloc)d Fv(and)i
 
6749
Fp(free)p Fv(.)39 b Fp(fftw_malloc_hook)25 b Fv(m)m(ust)j(p)s(oin)m(t)g
 
6750
(to)h(a)g Fp(malloc)p Fv(-lik)m(e)150 628 y(function,)g(and)h
 
6751
Fp(fftw_free_hook)d Fv(m)m(ust)j(p)s(oin)m(t)f(to)i(a)g
 
6752
Fp(free)p Fv(-lik)m(e)e(function.)150 885 y Fu(3.8)68
 
6753
b(Thread)45 b(safet)l(y)275 1078 y Fv(Users)30 b(writing)e(m)m
 
6754
(ulti-threaded)h(programs)h(m)m(ust)h(concern)f(themselv)m(es)h(with)e
 
6755
(the)i Fq(thread)f(safet)m(y)150 1187 y Fv(of)d(the)g(libraries)d(they)
 
6756
k(use|that)f(is,)g(whether)f(it)h(is)f(safe)h(to)h(call)e(routines)g
 
6757
(in)g(parallel)f(from)h(m)m(ultiple)150 1297 y(threads.)39
 
6758
b(FFTW)29 b(can)e(b)s(e)g(used)g(in)f(suc)m(h)i(an)f(en)m(vironmen)m
 
6759
(t,)h(but)f(some)h(care)g(m)m(ust)f(b)s(e)g(tak)m(en)i(b)s(ecause)150
 
6760
1406 y(certain)21 b(parts)f(of)h(FFTW)g(use)g(priv)-5
 
6761
b(ate)20 b(global)g(v)-5 b(ariables)19 b(to)j(share)e(data)h(b)s(et)m
 
6762
(w)m(een)h(calls.)36 b(In)20 b(particular,)150 1516 y(the)42
 
6763
b(plan-creation)f(functions)f(share)i(trigonometric)g(tables)f(and)g
 
6764
(accum)m(ulated)h Fp(wisdom)p Fv(.)74 b(\(Users)150 1626
 
6765
y(should)37 b(note)j(that)f(these)h(commen)m(ts)g(only)e(apply)g(to)h
 
6766
(programs)g(using)f(shared-memory)g(threads.)150 1735
 
6767
y(P)m(arallelism)k(using)g(MPI)h(or)h(fork)m(ed)g(pro)s(cesses)f(in)m
 
6768
(v)m(olv)m(es)h(a)g(separate)h(address-space)f(and)f(global)150
 
6769
1845 y(v)-5 b(ariables)29 b(for)h(eac)m(h)i(pro)s(cess,)e(and)g(is)f
6710
6770
(not)i(susceptible)d(to)j(problems)e(of)i(this)e(sort.\))275
6711
 
3527 y(The)c(cen)m(tral)i(restriction)e(of)h(FFTW)h(is)e(that)h(it)g
 
6771
1979 y(The)c(cen)m(tral)i(restriction)e(of)h(FFTW)h(is)e(that)h(it)g
6712
6772
(is)f(not)h(safe)h(to)g(create)g(m)m(ultiple)d(plans)h(in)g(parallel.)
6713
 
150 3636 y(Y)-8 b(ou)33 b(m)m(ust)e(either)h(create)i(all)c(of)j(y)m
 
6773
150 2089 y(Y)-8 b(ou)33 b(m)m(ust)e(either)h(create)i(all)c(of)j(y)m
6714
6774
(our)f(plans)e(from)i(a)g(single)f(thread,)h(or)g(instead)g(use)f(a)i
6715
 
(semaphore,)150 3746 y(m)m(utex,)h(or)f(other)g(mec)m(hanism)f(to)i
 
6775
(semaphore,)150 2198 y(m)m(utex,)h(or)f(other)g(mec)m(hanism)f(to)i
6716
6776
(ensure)e(that)i(di\013eren)m(t)e(threads)g(don't)h(attempt)h(to)g
6717
 
(create)g(plans)150 3856 y(at)41 b(the)f(same)h(time.)70
 
6777
(create)g(plans)150 2308 y(at)41 b(the)f(same)h(time.)70
6718
6778
b(The)40 b(same)h(restriction)e(also)h(holds)f(for)h(destruction)f(of)i
6719
 
(plans)e(and)g(imp)s(ort-)150 3965 y(ing/forgetting)31
6720
 
b Fo(wisdom)p Fu(.)39 b(Once)30 b(created,)i(a)e(plan)f(ma)m(y)i
6721
 
(safely)f(b)s(e)g(used)g(in)f(an)m(y)h(thread.)275 4100
 
6779
(plans)e(and)g(imp)s(ort-)150 2418 y(ing/forgetting)31
 
6780
b Fp(wisdom)p Fv(.)39 b(Once)30 b(created,)i(a)e(plan)f(ma)m(y)i
 
6781
(safely)f(b)s(e)g(used)g(in)f(an)m(y)h(thread.)275 2552
6722
6782
y(The)24 b(actual)h(transform)g(routines)e(in)h(FFTW)i(\()p
6723
 
Fo(fftw_one)p Fu(,)e(etcetera\))k(are)d(re-en)m(tran)m(t)i(and)d
6724
 
(thread-)150 4209 y(safe,)i(so)d(it)g(is)g(\014ne)g(to)h(call)f(them)g
 
6783
Fp(fftw_one)p Fv(,)e(etcetera\))k(are)d(re-en)m(tran)m(t)i(and)d
 
6784
(thread-)150 2662 y(safe,)i(so)d(it)g(is)g(\014ne)g(to)h(call)f(them)g
6725
6785
(sim)m(ultaneously)f(from)h(m)m(ultiple)e(threads.)38
6726
 
b(Another)24 b(question)e(arises,)150 4319 y(ho)m(w)m(ev)m(er|is)33
6727
 
b(it)g(safe)g(to)h(use)e(the)i Fq(same)h(plan)41 b Fu(for)33
 
6786
b(Another)24 b(question)e(arises,)150 2771 y(ho)m(w)m(ev)m(er|is)33
 
6787
b(it)g(safe)g(to)h(use)e(the)i Fr(same)h(plan)41 b Fv(for)33
6728
6788
b(m)m(ultiple)e(transforms)h(in)f(parallel?)46 b(\(It)34
6729
 
b(w)m(ould)e(b)s(e)150 4428 y(unsafe)e(if,)h(for)f(example,)i(the)f
 
6789
b(w)m(ould)e(b)s(e)150 2881 y(unsafe)e(if,)h(for)f(example,)i(the)f
6730
6790
(plan)e(w)m(ere)j(mo)s(di\014ed)d(in)g(some)j(w)m(a)m(y)f(b)m(y)g(the)g
6731
 
(transform.\))43 b(W)-8 b(e)32 b(address)150 4538 y(this)40
 
6791
(transform.\))43 b(W)-8 b(e)32 b(address)150 2990 y(this)40
6732
6792
b(question)g(b)m(y)h(de\014ning)e(an)i(additional)e(planner)g(\015ag,)
6733
 
44 b Fo(FFTW_THREADSAFE)p Fu(.)69 b(When)41 b(included)150
6734
 
4648 y(in)d(the)i(\015ags)f(for)h(an)m(y)g(of)f(the)h(plan-creation)f
6735
 
(routines,)i Fo(FFTW_THREADSAFE)35 b Fu(guaran)m(tees)41
6736
 
b(that)f(the)150 4757 y(resulting)28 b(plan)i(will)d(b)s(e)j(read-only)
 
6793
44 b Fp(FFTW_THREADSAFE)p Fv(.)69 b(When)41 b(included)150
 
6794
3100 y(in)d(the)i(\015ags)f(for)h(an)m(y)g(of)f(the)h(plan-creation)f
 
6795
(routines,)i Fp(FFTW_THREADSAFE)35 b Fv(guaran)m(tees)41
 
6796
b(that)f(the)150 3210 y(resulting)28 b(plan)i(will)d(b)s(e)j(read-only)
6737
6797
g(and)g(safe)g(to)h(use)f(in)g(parallel)e(b)m(y)i(m)m(ultiple)e
6738
 
(threads.)p eop
 
6798
(threads.)p eop end
6739
6799
%%Page: 37 39
6740
 
37 38 bop 150 -116 a Fu(Chapter)30 b(4:)41 b(P)m(arallel)29
6741
 
b(FFTW)2435 b(37)150 299 y Fr(4)80 b(P)l(arallel)57 b(FFTW)275
6742
 
569 y Fu(In)46 b(this)g(c)m(hapter)i(w)m(e)f(discuss)f(the)h(use)g(of)g
6743
 
(FFTW)h(in)e(a)h(parallel)f(en)m(vironmen)m(t,)51 b(do)s(cumen)m(t-)150
6744
 
679 y(ing)41 b(the)h(di\013eren)m(t)f(parallel)f(libraries)f(that)k(w)m
6745
 
(e)f(ha)m(v)m(e)h(pro)m(vided.)74 b(\(Users)42 b(calling)f(FFTW)h(from)
6746
 
g(a)150 789 y(m)m(ulti-threaded)e(program)i(should)e(also)i(consult)f
6747
 
(Section)h(3.8)h([Thread)e(safet)m(y],)46 b(page)d(36.\))76
6748
 
b(The)150 898 y(FFTW)35 b(pac)m(k)-5 b(age)36 b(curren)m(tly)d(con)m
6749
 
(tains)h(three)h(parallel)d(transform)h(implemen)m(tations)g(that)i
6750
 
(lev)m(erage)150 1008 y(the)c(unipro)s(cessor)d(FFTW)j(co)s(de:)225
6751
 
1154 y Fs(\017)60 b Fu(The)38 b(\014rst)h(set)g(of)g(routines)f
6752
 
(utilizes)f(shared-memory)h(threads)h(for)f(parallel)f(one-)j(and)e(m)m
6753
 
(ulti-)330 1264 y(dimensional)24 b(transforms)h(of)i(b)s(oth)e(real)h
6754
 
(and)g(complex)g(data.)40 b(An)m(y)27 b(program)f(using)f(FFTW)i(can)
6755
 
330 1374 y(b)s(e)33 b(trivially)e(mo)s(di\014ed)g(to)k(use)e(the)h(m)m
6756
 
(ulti-threaded)e(routines.)49 b(This)32 b(co)s(de)i(can)g(use)f(an)m(y)
6757
 
h(com-)330 1483 y(mon)28 b(threads)g(implemen)m(tation,)f(including)e
6758
 
(POSIX)j(threads.)39 b(\(POSIX)28 b(threads)g(are)h(a)m(v)-5
 
6800
TeXDict begin 37 38 bop 150 -116 a Fv(Chapter)30 b(4:)41
 
6801
b(P)m(arallel)29 b(FFTW)2435 b(37)150 299 y Fs(4)80 b(P)l(arallel)57
 
6802
b(FFTW)275 569 y Fv(In)46 b(this)g(c)m(hapter)i(w)m(e)f(discuss)f(the)h
 
6803
(use)g(of)g(FFTW)h(in)e(a)h(parallel)f(en)m(vironmen)m(t,)51
 
6804
b(do)s(cumen)m(t-)150 679 y(ing)41 b(the)h(di\013eren)m(t)f(parallel)f
 
6805
(libraries)f(that)k(w)m(e)f(ha)m(v)m(e)h(pro)m(vided.)74
 
6806
b(\(Users)42 b(calling)f(FFTW)h(from)g(a)150 789 y(m)m(ulti-threaded)e
 
6807
(program)i(should)e(also)i(consult)f(Section)h(3.8)h([Thread)e(safet)m
 
6808
(y],)46 b(page)d(36.\))76 b(The)150 898 y(FFTW)35 b(pac)m(k)-5
 
6809
b(age)36 b(curren)m(tly)d(con)m(tains)h(three)h(parallel)d(transform)h
 
6810
(implemen)m(tations)g(that)i(lev)m(erage)150 1008 y(the)c(unipro)s
 
6811
(cessor)d(FFTW)j(co)s(de:)225 1154 y Ft(\017)60 b Fv(The)38
 
6812
b(\014rst)h(set)g(of)g(routines)f(utilizes)f(shared-memory)h(threads)h
 
6813
(for)f(parallel)f(one-)j(and)e(m)m(ulti-)330 1264 y(dimensional)24
 
6814
b(transforms)h(of)i(b)s(oth)e(real)h(and)g(complex)g(data.)40
 
6815
b(An)m(y)27 b(program)f(using)f(FFTW)i(can)330 1374 y(b)s(e)33
 
6816
b(trivially)e(mo)s(di\014ed)g(to)k(use)e(the)h(m)m(ulti-threaded)e
 
6817
(routines.)49 b(This)32 b(co)s(de)i(can)g(use)f(an)m(y)h(com-)330
 
6818
1483 y(mon)28 b(threads)g(implemen)m(tation,)f(including)e(POSIX)j
 
6819
(threads.)39 b(\(POSIX)28 b(threads)g(are)h(a)m(v)-5
6759
6820
b(ailable)330 1593 y(on)36 b(most)g(Unix)f(v)-5 b(arian)m(ts,)37
6760
6821
b(including)c(Lin)m(ux.\))56 b(These)36 b(routines)f(are)h(lo)s(cated)g
6761
 
(in)f(the)h Fo(threads)330 1702 y Fu(directory)-8 b(,)31
 
6822
(in)f(the)h Fp(threads)330 1702 y Fv(directory)-8 b(,)31
6762
6823
b(and)e(are)i(do)s(cumen)m(ted)f(in)f(Section)h(4.1)i([Multi-threaded)d
6763
 
(FFTW],)j(page)f(37.)225 1843 y Fs(\017)60 b Fu(The)43
6764
 
b Fo(mpi)g Fu(directory)g(con)m(tains)h(m)m(ulti-dimensional)39
 
6824
(FFTW],)j(page)f(37.)225 1843 y Ft(\017)60 b Fv(The)43
 
6825
b Fp(mpi)g Fv(directory)g(con)m(tains)h(m)m(ulti-dimensional)39
6765
6826
b(transforms)k(of)h(real)f(and)g(complex)g(data)330 1952
6766
6827
y(for)32 b(parallel)f(mac)m(hines)h(supp)s(orting)e(MPI.)i(It)h(also)f
6767
6828
(includes)e(parallel)h(one-dimensional)f(trans-)330 2062
6771
6832
(from)f(w)m(orkstation)h(clusters)g(to)g(massiv)m(ely-)330
6772
6833
2281 y(parallel)h(sup)s(ercomputers.)63 b(More)39 b(information)d(on)i
6773
6834
(MPI)g(can)h(b)s(e)e(found)g(at)i(the)g(MPI)f(home)330
6774
 
2391 y(page)d(\()p Fo(http://www.mcs.anl.gov/mpi)o Fu(\).)48
 
6835
2391 y(page)d(\()p Fp(http://www.mcs.anl.gov/mpi)o Fv(\).)48
6775
6836
b(The)33 b(FFTW)j(MPI)e(routines)f(are)i(do)s(cumen)m(ted)f(in)330
6776
6837
2500 y(Section)c(4.2)i([MPI)e(FFTW],)i(page)f(40.)225
6777
 
2641 y Fs(\017)60 b Fu(W)-8 b(e)28 b(also)e(ha)m(v)m(e)i(an)e(exp)s
 
6838
2641 y Ft(\017)60 b Fv(W)-8 b(e)28 b(also)e(ha)m(v)m(e)i(an)e(exp)s
6778
6839
(erimen)m(tal)f(parallel)g(implemen)m(tation)g(written)h(in)f(Cilk,)g
6779
6840
(a)i(C-lik)m(e)f(parallel)330 2751 y(language)21 b(dev)m(elop)s(ed)f
6780
6841
(at)h(MIT)g(and)f(curren)m(tly)f(a)m(v)-5 b(ailable)20
6781
6842
b(for)h(sev)m(eral)g(SMP)f(platforms.)36 b(F)-8 b(or)22
6782
6843
b(more)330 2860 y(information)27 b(on)h(Cilk)e(see)j(the)f(Cilk)f(home)
6783
 
h(page)h(\()p Fo(http://supertech.lcs.mit.e)o(du/)o(cilk)o
6784
 
Fu(\).)330 2970 y(The)34 b(FFTW)h(Cilk)d(co)s(de)j(can)g(b)s(e)e(found)
6785
 
g(in)g(the)i Fo(cilk)e Fu(directory)-8 b(,)36 b(with)d(parallelized)f
 
6844
h(page)h(\()p Fp(http://supertech.lcs.mit.e)o(du/)o(cilk)o
 
6845
Fv(\).)330 2970 y(The)34 b(FFTW)h(Cilk)d(co)s(de)j(can)g(b)s(e)e(found)
 
6846
g(in)g(the)i Fp(cilk)e Fv(directory)-8 b(,)36 b(with)d(parallelized)f
6786
6847
(one-)j(and)330 3079 y(m)m(ulti-dimensional)h(transforms)j(of)h
6787
6848
(complex)g(data.)69 b(The)40 b(Cilk)e(FFTW)i(routines)f(are)h(do)s(cu-)
6788
 
330 3189 y(men)m(ted)31 b(in)e Fo(cilk/README)p Fu(.)150
6789
 
3477 y Ft(4.1)68 b(Multi-threaded)46 b(FFTW)275 3681
6790
 
y Fu(In)28 b(this)h(section)h(w)m(e)g(do)s(cumen)m(t)g(the)f(parallel)f
 
6849
330 3189 y(men)m(ted)31 b(in)e Fp(cilk/README)p Fv(.)150
 
6850
3477 y Fu(4.1)68 b(Multi-threaded)46 b(FFTW)275 3681
 
6851
y Fv(In)28 b(this)h(section)h(w)m(e)g(do)s(cumen)m(t)g(the)f(parallel)f
6791
6852
(FFTW)j(routines)d(for)i(shared-memory)f(threads)g(on)150
6792
6853
3791 y(SMP)k(hardw)m(are.)49 b(These)33 b(routines,)h(whic)m(h)e(supp)s
6793
6854
(ort)f(parallel)h(one-)i(and)f(m)m(ulti-dimensional)c(trans-)150
6800
6861
(an)h(extra)g(parameter.)150 4229 y(An)m(y)30 b(program)g(that)g(uses)f
6801
6862
(the)h(unipro)s(cessor)e(FFTW)j(can)f(b)s(e)f(trivially)e(mo)s
6802
6863
(di\014ed)h(to)j(use)e(the)h(m)m(ulti-)150 4339 y(threaded)g(FFTW.)150
6803
 
4587 y Fi(4.1.1)63 b(Installation)40 b(and)h(Supp)s(orted)i(Hardw)m
6804
 
(are/Soft)m(w)m(are)275 4792 y Fu(All)38 b(of)h(the)h(FFTW)g(threads)f
6805
 
(co)s(de)g(is)g(lo)s(cated)h(in)e(the)h Fo(threads)f
6806
 
Fu(sub)s(directory)f(of)j(the)f(FFTW)150 4902 y(pac)m(k)-5
 
6864
4587 y Fh(4.1.1)63 b(Installation)40 b(and)h(Supp)s(orted)i(Hardw)m
 
6865
(are/Soft)m(w)m(are)275 4792 y Fv(All)38 b(of)h(the)h(FFTW)g(threads)f
 
6866
(co)s(de)g(is)g(lo)s(cated)h(in)e(the)h Fp(threads)f
 
6867
Fv(sub)s(directory)f(of)j(the)f(FFTW)150 4902 y(pac)m(k)-5
6807
6868
b(age.)82 b(On)43 b(Unix)g(systems,)k(the)d(FFTW)g(threads)f(libraries)
6808
6869
e(and)i(header)g(\014les)g(can)h(b)s(e)f(auto-)150 5011
6809
6870
y(matically)32 b(con\014gured,)i(compiled,)f(and)g(installed)e(along)i
6810
6871
(with)f(the)i(unipro)s(cessor)d(FFTW)j(libraries)150
6811
 
5121 y(simply)22 b(b)m(y)j(including)c Fo(--enable-threads)g
6812
 
Fu(in)i(the)i(\015ags)g(to)h(the)f Fo(configure)d Fu(script)i(\(see)h
 
6872
5121 y(simply)22 b(b)m(y)j(including)c Fp(--enable-threads)g
 
6873
Fv(in)i(the)i(\015ags)g(to)h(the)f Fp(configure)d Fv(script)i(\(see)h
6813
6874
(Section)g(6.1)150 5230 y([Installation)34 b(on)i(Unix],)g(page)g
6814
6875
(55\).)58 b(\(Note)37 b(also)f(that)g(the)g(threads)f(routines,)h(when)
6815
6876
e(enabled,)j(are)150 5340 y(automatically)30 b(tested)h(b)m(y)g(the)f
6816
 
(`)p Fo(make)47 b(check)p Fu(')29 b(self-tests.\))p eop
 
6877
(`)p Fp(make)47 b(check)p Fv(')29 b(self-tests.\))p eop
 
6878
end
6817
6879
%%Page: 38 40
6818
 
38 39 bop 150 -116 a Fu(38)3232 b(FFTW)275 299 y(The)25
6819
 
b(threads)h(routines)f(require)g(y)m(our)h(op)s(erating)g(system)g(to)h
6820
 
(ha)m(v)m(e)g(some)g(sort)f(of)g(shared-memory)150 408
6821
 
y(threads)d(supp)s(ort.)37 b(Sp)s(eci\014cally)-8 b(,)23
6822
 
b(the)h(FFTW)h(threads)e(pac)m(k)-5 b(age)26 b(w)m(orks)e(with)e(POSIX)
6823
 
h(threads)h(\(a)m(v)-5 b(ail-)150 518 y(able)27 b(on)g(most)h(Unix)f(v)
6824
 
-5 b(arian)m(ts,)28 b(including)c(Lin)m(ux\),)j(Solaris)f(threads,)i
6825
 
(BeOS)f(\()p Fo(http://www.be.com)p Fu(\))150 628 y(threads)i(\(tested)
6826
 
i(on)f(BeOS)g(DR8.2\),)h(Mac)m(h)g(C)f(threads)f(\(rep)s(orted)g(to)i
6827
 
(w)m(ork)f(b)m(y)f(users\),)h(and)f(Win32)150 737 y(threads)45
6828
 
b(\(rep)s(orted)f(to)i(w)m(ork)f(b)m(y)g(users\).)83
6829
 
b(\(There)45 b(is)f(also)h(un)m(tested)g(co)s(de)g(to)h(use)f(MacOS)g
6830
 
(MP)150 847 y(threads.\))39 b(If)26 b(y)m(ou)h(ha)m(v)m(e)g(a)g
6831
 
(shared-memory)f(mac)m(hine)g(that)h(uses)e(a)i(di\013eren)m(t)f
6832
 
(threads)g(API,)g(it)g(should)150 956 y(b)s(e)34 b(a)i(simple)d(matter)
6833
 
j(of)f(programming)f(to)h(include)e(supp)s(ort)h(for)g(it;)k(see)d(the)
6834
 
g(\014le)g Fo(fftw_threads-)150 1066 y(int.h)29 b Fu(for)h(more)h
6835
 
(detail.)275 1198 y(SMP)i(hardw)m(are)h(is)f(not)i(required,)e
6836
 
(although)h(of)g(course)h(y)m(ou)f(need)g(m)m(ultiple)e(pro)s(cessors)i
6837
 
(to)h(get)150 1308 y(an)m(y)c(b)s(ene\014t)e(from)h(the)h(m)m
6838
 
(ultithreaded)d(transforms.)150 1525 y Fi(4.1.2)63 b(Usage)41
6839
 
b(of)g(Multi-threaded)g(FFTW)275 1716 y Fu(Here,)28 b(it)e(is)f
6840
 
(assumed)h(that)h(the)g(reader)f(is)g(already)g(familiar)e(with)h(the)i
6841
 
(usage)g(of)f(the)h(unipro)s(cessor)150 1825 y(FFTW)39
6842
 
b(routines,)h(describ)s(ed)d(elsewhere)h(in)f(this)h(man)m(ual.)65
6843
 
b(W)-8 b(e)39 b(only)f(describ)s(e)f(what)i(one)g(has)f(to)150
6844
 
1935 y(c)m(hange)32 b(in)d(order)g(to)j(use)e(the)g(m)m(ulti-threaded)f
6845
 
(routines.)275 2067 y(First,)i(instead)f(of)h(including)d
6846
 
Fo(<fftw.h>)h Fu(or)i Fo(<rfftw.h>)p Fu(,)e(y)m(ou)j(should)d(include)g
6847
 
(the)i(\014les)g Fo(<fftw_)150 2177 y(threads.h>)d Fu(or)i
6848
 
Fo(<rfftw_threads.h>)p Fu(,)c(resp)s(ectiv)m(ely)-8 b(.)275
6849
 
2309 y(Second,)30 b(b)s(efore)g(calling)f(an)m(y)h(FFTW)i(routines,)d
6850
 
(y)m(ou)i(should)d(call)i(the)g(function:)390 2435 y
6851
 
Fo(int)47 b(fftw_threads_init\(void\);)275 2567 y Fu(This)38
6852
 
b(function,)j(whic)m(h)e(should)e(only)i(b)s(e)h(called)f(once)i
6853
 
(\(probably)d(in)h(y)m(our)g Fo(main\(\))g Fu(function\),)150
6854
 
2677 y(p)s(erforms)g(an)m(y)i(one-time)g(initialization)c(required)i
 
6880
TeXDict begin 38 39 bop 150 -116 a Fv(38)3232 b(FFTW)275
 
6881
299 y(The)25 b(threads)h(routines)f(require)g(y)m(our)h(op)s(erating)g
 
6882
(system)g(to)h(ha)m(v)m(e)g(some)g(sort)f(of)g(shared-memory)150
 
6883
408 y(threads)d(supp)s(ort.)37 b(Sp)s(eci\014cally)-8
 
6884
b(,)23 b(the)h(FFTW)h(threads)e(pac)m(k)-5 b(age)26 b(w)m(orks)e(with)e
 
6885
(POSIX)h(threads)h(\(a)m(v)-5 b(ail-)150 518 y(able)27
 
6886
b(on)g(most)h(Unix)f(v)-5 b(arian)m(ts,)28 b(including)c(Lin)m(ux\),)j
 
6887
(Solaris)f(threads,)i(BeOS)f(\()p Fp(http://www.be.com)p
 
6888
Fv(\))150 628 y(threads)i(\(tested)i(on)f(BeOS)g(DR8.2\),)h(Mac)m(h)g
 
6889
(C)f(threads)f(\(rep)s(orted)g(to)i(w)m(ork)f(b)m(y)f(users\),)h(and)f
 
6890
(Win32)150 737 y(threads)45 b(\(rep)s(orted)f(to)i(w)m(ork)f(b)m(y)g
 
6891
(users\).)83 b(\(There)45 b(is)f(also)h(un)m(tested)g(co)s(de)g(to)h
 
6892
(use)f(MacOS)g(MP)150 847 y(threads.\))40 b(W)-8 b(e)31
 
6893
b(also)e(supp)s(ort)e(using)h(Op)s(enMP)f(\()p Fp
 
6894
(http://www.openmp.org)p Fv(\))d(or)29 b(SGI)g(MP)g(compiler)150
 
6895
956 y(directiv)m(es)c(to)h(launc)m(h)f(threads,)h(enabled)f(b)m(y)g
 
6896
(using)f Fp(--with-openmp)e Fv(or)j Fp(--with-sgimp)e
 
6897
Fv(in)h(addition)150 1066 y(to)33 b Fp(--enable-threads)p
 
6898
Fv(.)42 b(This)30 b(is)h(esp)s(ecially)g(useful)f(if)h(y)m(ou)i(are)g
 
6899
(emplo)m(ying)e(that)i(sort)f(of)g(directiv)m(e)150 1176
 
6900
y(in)k(y)m(our)i(o)m(wn)f(co)s(de,)j(in)c(order)h(to)h(minimize)d
 
6901
(con\015icts.)62 b(If)37 b(y)m(ou)h(ha)m(v)m(e)g(a)g(shared-memory)f
 
6902
(mac)m(hine)150 1285 y(that)32 b(uses)f(a)g(di\013eren)m(t)g(threads)g
 
6903
(API,)h(it)e(should)g(b)s(e)g(a)i(simple)d(matter)k(of)e(programming)f
 
6904
(to)i(include)150 1395 y(supp)s(ort)d(for)h(it;)g(see)h(the)f(\014le)g
 
6905
Fp(fftw_threads-int.h)25 b Fv(for)30 b(more)h(detail.)275
 
6906
1535 y(SMP)i(hardw)m(are)h(is)f(not)i(required,)e(although)h(of)g
 
6907
(course)h(y)m(ou)f(need)g(m)m(ultiple)e(pro)s(cessors)i(to)h(get)150
 
6908
1645 y(an)m(y)c(b)s(ene\014t)e(from)h(the)h(m)m(ultithreaded)d
 
6909
(transforms.)150 1881 y Fh(4.1.2)63 b(Usage)41 b(of)g(Multi-threaded)g
 
6910
(FFTW)275 2080 y Fv(Here,)28 b(it)e(is)f(assumed)h(that)h(the)g(reader)
 
6911
f(is)g(already)g(familiar)e(with)h(the)i(usage)g(of)f(the)h(unipro)s
 
6912
(cessor)150 2190 y(FFTW)39 b(routines,)h(describ)s(ed)d(elsewhere)h(in)
 
6913
f(this)h(man)m(ual.)65 b(W)-8 b(e)39 b(only)f(describ)s(e)f(what)i(one)
 
6914
g(has)f(to)150 2299 y(c)m(hange)32 b(in)d(order)g(to)j(use)e(the)g(m)m
 
6915
(ulti-threaded)f(routines.)275 2440 y(First,)i(instead)f(of)h
 
6916
(including)d Fp(<fftw.h>)h Fv(or)i Fp(<rfftw.h>)p Fv(,)e(y)m(ou)j
 
6917
(should)d(include)g(the)i(\014les)g Fp(<fftw_)150 2550
 
6918
y(threads.h>)d Fv(or)i Fp(<rfftw_threads.h>)p Fv(,)c(resp)s(ectiv)m
 
6919
(ely)-8 b(.)275 2690 y(Second,)30 b(b)s(efore)g(calling)f(an)m(y)h
 
6920
(FFTW)i(routines,)d(y)m(ou)i(should)d(call)i(the)g(function:)390
 
6921
2825 y Fp(int)47 b(fftw_threads_init\(void\);)275 2966
 
6922
y Fv(This)38 b(function,)j(whic)m(h)e(should)e(only)i(b)s(e)h(called)f
 
6923
(once)i(\(probably)d(in)h(y)m(our)g Fp(main\(\))g Fv(function\),)150
 
6924
3075 y(p)s(erforms)g(an)m(y)i(one-time)g(initialization)c(required)i
6855
6925
(to)i(use)g(threads)f(on)g(y)m(our)h(system.)71 b(It)41
6856
 
b(returns)150 2787 y(zero)36 b(if)e(successful,)h(and)f(a)h(non-zero)h
 
6926
b(returns)150 3185 y(zero)36 b(if)e(successful,)h(and)f(a)h(non-zero)h
6857
6927
(v)-5 b(alue)34 b(if)g(there)h(w)m(as)g(an)g(error)g(\(in)f(whic)m(h)f
6858
 
(case,)38 b(something)c(is)150 2896 y(seriously)28 b(wrong)i(and)g(y)m
 
6928
(case,)38 b(something)c(is)150 3294 y(seriously)28 b(wrong)i(and)g(y)m
6859
6929
(ou)h(should)d(probably)h(exit)h(the)h(program\).)275
6860
 
3028 y(Third,)38 b(when)g(y)m(ou)h(w)m(an)m(t)h(to)f(actually)g
 
6930
3435 y(Third,)38 b(when)g(y)m(ou)h(w)m(an)m(t)h(to)f(actually)g
6861
6931
(compute)g(the)g(transform,)h(y)m(ou)f(should)e(use)i(one)g(of)g(the)
6862
 
150 3138 y(follo)m(wing)29 b(transform)h(routines)f(instead)g(of)i(the)
6863
 
f(ordinary)f(FFTW)i(functions:)390 3264 y Fo(fftw_threads\(nthreads,)42
6864
 
b(plan,)k(howmany,)g(in,)h(istride,)1010 3368 y(idist,)g(out,)f
6865
 
(ostride,)g(odist\);)390 3576 y(fftw_threads_one\(nthread)o(s,)c(plan,)
6866
 
k(in,)h(out\);)390 3783 y(fftwnd_threads\(nthreads,)41
6867
 
b(plan,)47 b(howmany,)e(in,)i(istride,)1106 3887 y(idist,)f(out,)h
6868
 
(ostride,)e(odist\);)390 4095 y(fftwnd_threads_one\(nthre)o(ads,)c
6869
 
(plan,)46 b(in,)h(out\);)390 4302 y(rfftw_threads\(nthreads,)41
6870
 
b(plan,)47 b(howmany,)e(in,)i(istride,)1058 4406 y(idist,)f(out,)h
6871
 
(ostride,)e(odist\);)390 4614 y(rfftw_threads_one\(nthrea)o(ds,)c
6872
 
(plan,)47 b(in,)g(out\);)390 4821 y(rfftwnd_threads_real_to_)o(comp)o
 
6932
150 3545 y(follo)m(wing)29 b(transform)h(routines)f(instead)g(of)i(the)
 
6933
f(ordinary)f(FFTW)i(functions:)390 3680 y Fp(fftw_threads\(nthreads,)42
 
6934
b(plan,)k(howmany,)g(in,)h(istride,)1010 3783 y(idist,)g(out,)f
 
6935
(ostride,)g(odist\);)390 3991 y(fftw_threads_one\(nthread)o(s,)c(plan,)
 
6936
k(in,)h(out\);)390 4198 y(fftwnd_threads\(nthreads,)41
 
6937
b(plan,)47 b(howmany,)e(in,)i(istride,)1106 4302 y(idist,)f(out,)h
 
6938
(ostride,)e(odist\);)390 4510 y(fftwnd_threads_one\(nthre)o(ads,)c
 
6939
(plan,)46 b(in,)h(out\);)390 4717 y(rfftw_threads\(nthreads,)41
 
6940
b(plan,)47 b(howmany,)e(in,)i(istride,)1058 4821 y(idist,)f(out,)h
 
6941
(ostride,)e(odist\);)390 5029 y(rfftw_threads_one\(nthrea)o(ds,)c
 
6942
(plan,)47 b(in,)g(out\);)390 5236 y(rfftwnd_threads_real_to_)o(comp)o
6873
6943
(lex\()o(nth)o(read)o(s,)42 b(plan,)k(howmany,)f(in,)1917
6874
 
4925 y(istride,)h(idist,)g(out,)h(ostride,)e(odist\);)390
6875
 
5132 y(rfftwnd_threads_one_real)o(_to_)o(comp)o(lex)o(\(nth)o(read)o
6876
 
(s,)c(plan,)47 b(in,)g(out\);)390 5340 y(rfftwnd_threads_complex_)o
6877
 
(to_r)o(eal\()o(nth)o(read)o(s,)42 b(plan,)k(howmany,)f(in,)p
6878
 
eop
 
6944
5340 y(istride,)h(idist,)g(out,)h(ostride,)e(odist\);)p
 
6945
eop end
6879
6946
%%Page: 39 41
6880
 
39 40 bop 150 -116 a Fu(Chapter)30 b(4:)41 b(P)m(arallel)29
6881
 
b(FFTW)2435 b(39)1917 299 y Fo(istride,)46 b(idist,)g(out,)h(ostride,)e
6882
 
(odist\);)390 506 y(rfftwnd_threads_one_real)o(_to_)o(comp)o(lex)o
6883
 
(\(nth)o(read)o(s,)c(plan,)47 b(in,)g(out\);)390 714
6884
 
y(rfftwnd_threads_one_comp)o(lex_)o(to_r)o(eal)o(\(nth)o(read)o(s,)41
6885
 
b(plan,)47 b(in,)g(out\);)275 851 y Fu(All)27 b(of)h(these)h(routines)f
6886
 
(tak)m(e)i(exactly)f(the)g(same)g(argumen)m(ts)g(and)f(ha)m(v)m(e)i
6887
 
(exactly)f(the)g(same)g(e\013ects)150 961 y(as)j(their)f(unipro)s
6888
 
(cessor)e(coun)m(terparts)k(\(i.e.)45 b(without)31 b(the)h(`)p
6889
 
Fo(_threads)p Fu('\))e Fq(exc)-5 b(ept)41 b Fu(that)32
6890
 
b(they)g(tak)m(e)i(one)150 1070 y(extra)i(parameter,)h
6891
 
Fo(nthreads)c Fu(\(of)j(t)m(yp)s(e)f Fo(int)p Fu(\),)h(b)s(efore)f(the)
6892
 
g(normal)g(parameters.)3084 1037 y Fn(1)3176 1070 y Fu(The)g
6893
 
Fo(nthreads)150 1180 y Fu(parameter)21 b(sp)s(eci\014es)e(the)i(n)m(um)
6894
 
m(b)s(er)e(of)i(threads)g(of)f(execution)h(to)g(use)g(when)e(p)s
6895
 
(erforming)g(the)i(transform)150 1289 y(\(actually)-8
6896
 
b(,)31 b(the)g(maxim)m(um)e(n)m(um)m(b)s(er)g(of)h(threads\).)275
6897
 
1427 y(F)-8 b(or)31 b(example,)f(to)i(parallelize)c(a)j(single)e
6898
 
(one-dimensional)g(transform)g(of)i(complex)f(data,)i(instead)150
6899
 
1536 y(of)53 b(calling)e(the)i(unipro)s(cessor)e Fo(fftw_one\(plan,)27
6900
 
b(in,)i(out\))p Fu(,)58 b(y)m(ou)53 b(w)m(ould)e(call)i
6901
 
Fo(fftw_threads_)150 1646 y(one\(nthreads,)27 b(plan,)i(in,)g(out\))p
6902
 
Fu(.)47 b(P)m(assing)33 b(an)f Fo(nthreads)f Fu(of)i
6903
 
Fo(1)f Fu(means)h(to)h(use)e(only)g(one)h(thread)150
6904
 
1755 y(\(the)51 b(main)d(thread\),)56 b(and)49 b(is)g(equiv)-5
 
6947
TeXDict begin 39 40 bop 150 -116 a Fv(Chapter)30 b(4:)41
 
6948
b(P)m(arallel)29 b(FFTW)2435 b(39)390 403 y Fp
 
6949
(rfftwnd_threads_one_real)o(_to_)o(comp)o(lex)o(\(nth)o(read)o(s,)41
 
6950
b(plan,)47 b(in,)g(out\);)390 610 y(rfftwnd_threads_complex_)o(to_r)o
 
6951
(eal\()o(nth)o(read)o(s,)42 b(plan,)k(howmany,)f(in,)1917
 
6952
714 y(istride,)h(idist,)g(out,)h(ostride,)e(odist\);)390
 
6953
922 y(rfftwnd_threads_one_real)o(_to_)o(comp)o(lex)o(\(nth)o(read)o(s,)
 
6954
c(plan,)47 b(in,)g(out\);)390 1129 y(rfftwnd_threads_one_comp)o(lex_)o
 
6955
(to_r)o(eal)o(\(nth)o(read)o(s,)41 b(plan,)47 b(in,)g(out\);)275
 
6956
1263 y Fv(All)27 b(of)h(these)h(routines)f(tak)m(e)i(exactly)f(the)g
 
6957
(same)g(argumen)m(ts)g(and)f(ha)m(v)m(e)i(exactly)f(the)g(same)g
 
6958
(e\013ects)150 1373 y(as)j(their)f(unipro)s(cessor)e(coun)m(terparts)k
 
6959
(\(i.e.)45 b(without)31 b(the)h(`)p Fp(_threads)p Fv('\))e
 
6960
Fr(exc)-5 b(ept)41 b Fv(that)32 b(they)g(tak)m(e)i(one)150
 
6961
1482 y(extra)i(parameter,)h Fp(nthreads)c Fv(\(of)j(t)m(yp)s(e)f
 
6962
Fp(int)p Fv(\),)h(b)s(efore)f(the)g(normal)g(parameters.)3084
 
6963
1449 y Fo(1)3176 1482 y Fv(The)g Fp(nthreads)150 1592
 
6964
y Fv(parameter)21 b(sp)s(eci\014es)e(the)i(n)m(um)m(b)s(er)e(of)i
 
6965
(threads)g(of)f(execution)h(to)g(use)g(when)e(p)s(erforming)g(the)i
 
6966
(transform)150 1701 y(\(actually)-8 b(,)31 b(the)g(maxim)m(um)e(n)m(um)
 
6967
m(b)s(er)g(of)h(threads\).)275 1835 y(F)-8 b(or)31 b(example,)f(to)i
 
6968
(parallelize)c(a)j(single)e(one-dimensional)g(transform)g(of)i(complex)
 
6969
f(data,)i(instead)150 1945 y(of)53 b(calling)e(the)i(unipro)s(cessor)e
 
6970
Fp(fftw_one\(plan,)27 b(in,)i(out\))p Fv(,)58 b(y)m(ou)53
 
6971
b(w)m(ould)e(call)i Fp(fftw_threads_)150 2055 y(one\(nthreads,)27
 
6972
b(plan,)i(in,)g(out\))p Fv(.)47 b(P)m(assing)33 b(an)f
 
6973
Fp(nthreads)f Fv(of)i Fp(1)f Fv(means)h(to)h(use)e(only)g(one)h(thread)
 
6974
150 2164 y(\(the)51 b(main)d(thread\),)56 b(and)49 b(is)g(equiv)-5
6905
6975
b(alen)m(t)50 b(to)g(calling)f(the)h(unipro)s(cessor)e(routine.)98
6906
 
b(P)m(assing)50 b(an)150 1865 y Fo(nthreads)32 b Fu(of)j
6907
 
Fo(2)g Fu(means)f(that)h(the)g(transform)f(is)g(p)s(oten)m(tially)f
 
6976
b(P)m(assing)50 b(an)150 2274 y Fp(nthreads)32 b Fv(of)j
 
6977
Fp(2)g Fv(means)f(that)h(the)g(transform)f(is)g(p)s(oten)m(tially)f
6908
6978
(parallelized)g(o)m(v)m(er)j(t)m(w)m(o)g(threads)e(\(and)150
6909
 
1975 y(t)m(w)m(o)e(pro)s(cessors,)e(if)f(y)m(ou)i(ha)m(v)m(e)g(them\),)
6910
 
g(and)f(so)h(on.)275 2112 y(These)36 b(are)i(the)f(only)f(c)m(hanges)j
 
6979
2383 y(t)m(w)m(o)e(pro)s(cessors,)e(if)f(y)m(ou)i(ha)m(v)m(e)g(them\),)
 
6980
g(and)f(so)h(on.)275 2517 y(These)36 b(are)i(the)f(only)f(c)m(hanges)j
6911
6981
(y)m(ou)e(need)g(to)h(mak)m(e)g(to)g(y)m(our)f(source)g(co)s(de.)61
6912
 
b(Calls)36 b(to)i(all)e(other)150 2221 y(FFTW)46 b(routines)e(\(plan)g
 
6982
b(Calls)36 b(to)i(all)e(other)150 2627 y(FFTW)46 b(routines)e(\(plan)g
6913
6983
(creation,)49 b(destruction,)f(wisdom,)f(etcetera\))h(are)e(not)f
6914
 
(parallelized)e(and)150 2331 y(remain)32 b(the)h(same.)49
 
6984
(parallelized)e(and)150 2736 y(remain)32 b(the)h(same.)49
6915
6985
b(\(The)33 b(same)h(plans)d(and)h(wisdom)g(are)h(used)g(b)m(y)f(b)s
6916
 
(oth)h(unipro)s(cessor)e(and)h(m)m(ulti-)150 2441 y(threaded)37
 
6986
(oth)h(unipro)s(cessor)e(and)h(m)m(ulti-)150 2846 y(threaded)37
6917
6987
b(transforms.\))63 b(Y)-8 b(our)38 b(arra)m(ys)g(are)g(allo)s(cated)g
6918
6988
(and)f(formatted)i(in)d(the)i(same)g(w)m(a)m(y)-8 b(,)42
6919
 
b(and)37 b(so)150 2550 y(on.)275 2687 y(Programs)28 b(using)f(the)i
 
6989
b(and)37 b(so)150 2956 y(on.)275 3090 y(Programs)28 b(using)f(the)i
6920
6990
(parallel)d(complex)j(transforms)e(should)g(b)s(e)h(link)m(ed)f(with)g
6921
 
Fo(-lfftw_threads)150 2797 y(-lfftw)i(-lm)37 b Fu(on)h(Unix.)63
 
6991
Fp(-lfftw_threads)150 3199 y(-lfftw)i(-lm)37 b Fv(on)h(Unix.)63
6922
6992
b(Programs)38 b(using)f(the)h(parallel)f(real)h(transforms)f(should)f
6923
 
(b)s(e)i(link)m(ed)e(with)150 2906 y Fo(-lrfftw_threads)26
6924
 
b(-lfftw_threads)h(-lrfftw)h(-lfftw)h(-lm)p Fu(.)55 b(Y)-8
6925
 
b(ou)36 b(will)c(also)k(need)f(to)h(link)d(with)150 3016
 
6993
(b)s(e)i(link)m(ed)e(with)150 3309 y Fp(-lrfftw_threads)26
 
6994
b(-lfftw_threads)h(-lrfftw)h(-lfftw)h(-lm)p Fv(.)55 b(Y)-8
 
6995
b(ou)36 b(will)c(also)k(need)f(to)h(link)d(with)150 3418
6926
6996
y(whatev)m(er)e(library)d(is)i(resp)s(onsible)d(for)j(threads)g(on)g(y)
6927
 
m(our)h(system)f(\(e.g.)42 b Fo(-lpthread)28 b Fu(on)i(Lin)m(ux\).)150
6928
 
3245 y Fi(4.1.3)63 b(Ho)m(w)40 b(Man)m(y)h(Threads)g(to)g(Use?)275
6929
 
3441 y Fu(There)31 b(is)h(a)h(fair)e(amoun)m(t)i(of)g(o)m(v)m(erhead)g
 
6997
m(our)h(system)f(\(e.g.)42 b Fp(-lpthread)28 b Fv(on)i(Lin)m(ux\).)150
 
6998
3641 y Fh(4.1.3)63 b(Ho)m(w)40 b(Man)m(y)h(Threads)g(to)g(Use?)275
 
6999
3833 y Fv(There)31 b(is)h(a)h(fair)e(amoun)m(t)i(of)g(o)m(v)m(erhead)g
6930
7000
(in)m(v)m(olv)m(ed)f(in)g(spa)m(wning)f(and)g(sync)m(hronizing)g
6931
 
(threads,)i(so)150 3550 y(the)d(optimal)f(n)m(um)m(b)s(er)g(of)h
 
7001
(threads,)i(so)150 3943 y(the)d(optimal)f(n)m(um)m(b)s(er)g(of)h
6932
7002
(threads)f(to)i(use)e(dep)s(ends)f(up)s(on)h(the)h(size)g(of)g(the)g
6933
 
(transform)f(as)h(w)m(ell)f(as)h(on)150 3660 y(the)h(n)m(um)m(b)s(er)e
6934
 
(of)h(pro)s(cessors)g(y)m(ou)h(ha)m(v)m(e.)275 3797 y(As)d(a)g(general)
 
7003
(transform)f(as)h(w)m(ell)f(as)h(on)150 4052 y(the)h(n)m(um)m(b)s(er)e
 
7004
(of)h(pro)s(cessors)g(y)m(ou)h(ha)m(v)m(e.)275 4186 y(As)d(a)g(general)
6935
7005
h(rule,)e(y)m(ou)i(don't)f(w)m(an)m(t)h(to)g(use)f(more)h(threads)f
6936
7006
(than)g(y)m(ou)g(ha)m(v)m(e)i(pro)s(cessors.)39 b(\(Using)150
6937
 
3907 y(more)c(threads)g(will)d(w)m(ork,)37 b(but)d(there)h(will)d(b)s
 
7007
4296 y(more)c(threads)g(will)d(w)m(ork,)37 b(but)d(there)h(will)d(b)s
6938
7008
(e)j(extra)g(o)m(v)m(erhead)i(with)c(no)i(b)s(ene\014t.\))54
6939
 
b(In)34 b(fact,)k(if)c(the)150 4016 y(problem)29 b(size)h(is)f(to)s(o)j
 
7009
b(In)34 b(fact,)k(if)c(the)150 4405 y(problem)29 b(size)h(is)f(to)s(o)j
6940
7010
(small,)d(y)m(ou)h(ma)m(y)h(w)m(an)m(t)h(to)f(use)f(few)m(er)g(threads)
6941
 
g(than)g(y)m(ou)h(ha)m(v)m(e)h(pro)s(cessors.)275 4153
 
7011
g(than)g(y)m(ou)h(ha)m(v)m(e)h(pro)s(cessors.)275 4539
6942
7012
y(Y)-8 b(ou)42 b(will)d(ha)m(v)m(e)k(to)f(exp)s(erimen)m(t)f(with)g(y)m
6943
7013
(our)g(system)h(to)h(see)f(what)g(lev)m(el)f(of)h(parallelization)e(is)
6944
 
150 4263 y(b)s(est)31 b(for)f(y)m(our)h(problem)f(size.)42
 
7014
150 4649 y(b)s(est)31 b(for)f(y)m(our)h(problem)f(size.)42
6945
7015
b(Useful)30 b(to)s(ols)h(to)g(help)f(y)m(ou)h(do)g(this)f(are)h(the)g
6946
 
(test)h(programs)f(that)h(are)150 4373 y(automatically)i(compiled)e
6947
 
(along)i(with)f(the)h(threads)g(libraries,)e Fo(fftw_threads_test)d
6948
 
Fu(and)34 b Fo(rfftw_)150 4482 y(threads_test)25 b Fu(\(in)i(the)h
6949
 
Fo(threads)f Fu(sub)s(directory\).)38 b(These)28 b(tak)m(e)h(the)g
6950
 
(same)f(argumen)m(ts)h(as)f(the)h(other)150 4592 y(FFTW)c(test)f
6951
 
(programs)g(\(see)h Fo(tests/README)p Fu(\),)d(except)j(that)g(they)f
6952
 
(also)g(tak)m(e)h(the)g(n)m(um)m(b)s(er)d(of)i(threads)150
6953
 
4701 y(to)31 b(use)f(as)h(a)g(\014rst)e(argumen)m(t,)i(and)f(rep)s(ort)
6954
 
g(the)g(parallel)f(sp)s(eedup)f(in)h(sp)s(eed)h(tests.)41
6955
 
b(F)-8 b(or)32 b(example,)p 150 4802 1200 4 v 192 4869
6956
 
a Fn(1)275 4902 y Fu(There)g(is)f(one)i(exception:)46
6957
 
b(when)32 b(p)s(erforming)e(one-dimensional)h(in-place)g(transforms,)i
6958
 
(the)g Fo(out)275 5011 y Fu(parameter)g(is)f(alw)m(a)m(ys)h(ignored)f
6959
 
(b)m(y)g(the)h(m)m(ulti-threaded)f(routines,)g(instead)g(of)h(b)s(eing)
6960
 
f(used)g(as)h(a)275 5121 y(w)m(orkspace)28 b(if)e(it)i(is)e(non-)p
6961
 
Fo(NULL)h Fu(as)g(in)g(the)h(unipro)s(cessor)d(routines.)39
6962
 
b(The)27 b(m)m(ulti-threaded)f(routines)275 5230 y(alw)m(a)m(ys)41
6963
 
b(allo)s(cate)g(their)f(o)m(wn)h(w)m(orkspace)h(\(the)f(size)g(of)g
6964
 
(whic)m(h)f(dep)s(ends)f(up)s(on)g(the)i(n)m(um)m(b)s(er)f(of)275
6965
 
5340 y(threads\).)p eop
 
7016
(test)h(programs)f(that)h(are)150 4759 y(automatically)i(compiled)e
 
7017
(along)i(with)f(the)h(threads)g(libraries,)e Fp(fftw_threads_test)d
 
7018
Fv(and)34 b Fp(rfftw_)150 4868 y(threads_test)25 b Fv(\(in)i(the)h
 
7019
Fp(threads)f Fv(sub)s(directory\).)38 b(These)28 b(tak)m(e)h(the)g
 
7020
(same)f(argumen)m(ts)h(as)f(the)h(other)p 150 4967 1200
 
7021
4 v 199 5034 a Fo(1)275 5066 y Fj(There)35 b(is)g(one)g(exception:)52
 
7022
b(when)35 b(p)r(erforming)g(one-dimensional)g(in-place)g(transforms,)i
 
7023
(the)e Fi(out)g Fj(parameter)f(is)275 5157 y(alw)n(a)n(ys)26
 
7024
b(ignored)g(b)n(y)e(the)h(m)n(ulti-threaded)e(routines,)j(instead)g(of)
 
7025
g(b)r(eing)f(used)g(as)h(a)g(w)n(orkspace)g(if)g(it)f(is)h(non-)p
 
7026
Fi(NULL)f Fj(as)275 5249 y(in)f(the)g(unipro)r(cessor)i(routines.)35
 
7027
b(The)25 b(m)n(ulti-threaded)d(routines)j(alw)n(a)n(ys)h(allo)r(cate)h
 
7028
(their)d(o)n(wn)h(w)n(orkspace)h(\(the)e(size)275 5340
 
7029
y(of)i(whic)n(h)g(dep)r(ends)f(up)r(on)g(the)g(n)n(um)n(b)r(er)f(of)i
 
7030
(threads\).)p eop end
6966
7031
%%Page: 40 42
6967
 
40 41 bop 150 -116 a Fu(40)3232 b(FFTW)390 299 y Fo(fftw_threads_test)
6968
 
43 b(2)48 b(-s)f(128x128)275 452 y Fu(will)28 b(b)s(enc)m(hmark)j
6969
 
(complex)g(128x128)j(transforms)d(using)f(t)m(w)m(o)i(threads)f(and)g
6970
 
(rep)s(ort)g(the)h(sp)s(eedup)150 561 y(relativ)m(e)e(to)i(the)e
6971
 
(unipro)s(cessor)e(transform.)275 714 y(F)-8 b(or)40
6972
 
b(instance,)i(on)d(a)h(4-pro)s(cessor)g(200MHz)i(P)m(en)m(tium)d(Pro)h
6973
 
(system)g(running)c(Lin)m(ux)i(2.2.0,)44 b(w)m(e)150
6974
 
823 y(found)32 b(that)j(the)f Fo(")p Fu(crosso)m(v)m(er)p
6975
 
Fo(")h Fu(p)s(oin)m(t)d(at)j(whic)m(h)d(2)i(threads)g(b)s(ecame)g(b)s
6976
 
(ene\014cial)e(for)h(complex)h(trans-)150 933 y(forms)c(w)m(as)h(ab)s
6977
 
(out)f(4k)g(p)s(oin)m(ts,)g(while)e(4)j(threads)f(b)s(ecame)h(b)s
6978
 
(ene\014cial)d(at)j(8k)g(p)s(oin)m(ts.)150 1194 y Fi(4.1.4)63
6979
 
b(Using)42 b(Multi-threaded)f(FFTW)h(in)f(a)f(Multi-threaded)i(Program)
6980
 
275 1404 y Fu(It)29 b(is)f(p)s(erfectly)h(p)s(ossible)d(to)k(use)f(the)
6981
 
h(m)m(ulti-threaded)e(FFTW)i(routines)e(from)h(a)g(m)m(ulti-threaded)
6982
 
150 1514 y(program)c(\(e.g.)41 b(ha)m(v)m(e)27 b(m)m(ultiple)c(threads)
6983
 
i(computing)g(m)m(ulti-threaded)f(transforms)h(sim)m(ultaneously\).)150
6984
 
1624 y(If)k(y)m(ou)g(ha)m(v)m(e)i(the)e(pro)s(cessors,)g(more)g(p)s(o)m
 
7032
TeXDict begin 40 41 bop 150 -116 a Fv(40)3232 b(FFTW)150
 
7033
299 y(FFTW)25 b(test)f(programs)g(\(see)h Fp(tests/README)p
 
7034
Fv(\),)d(except)j(that)g(they)f(also)g(tak)m(e)h(the)g(n)m(um)m(b)s(er)
 
7035
d(of)i(threads)150 408 y(to)31 b(use)f(as)h(a)g(\014rst)e(argumen)m(t,)
 
7036
i(and)f(rep)s(ort)g(the)g(parallel)f(sp)s(eedup)f(in)h(sp)s(eed)h
 
7037
(tests.)41 b(F)-8 b(or)32 b(example,)390 542 y Fp(fftw_threads_test)43
 
7038
b(2)48 b(-s)f(128x128)275 681 y Fv(will)28 b(b)s(enc)m(hmark)j(complex)
 
7039
g(128x128)j(transforms)d(using)f(t)m(w)m(o)i(threads)f(and)g(rep)s(ort)
 
7040
g(the)h(sp)s(eedup)150 790 y(relativ)m(e)e(to)i(the)e(unipro)s(cessor)e
 
7041
(transform.)275 929 y(F)-8 b(or)40 b(instance,)i(on)d(a)h(4-pro)s
 
7042
(cessor)g(200MHz)i(P)m(en)m(tium)d(Pro)h(system)g(running)c(Lin)m(ux)i
 
7043
(2.2.0,)44 b(w)m(e)150 1039 y(found)32 b(that)j(the)f
 
7044
Fp(")p Fv(crosso)m(v)m(er)p Fp(")h Fv(p)s(oin)m(t)d(at)j(whic)m(h)d(2)i
 
7045
(threads)g(b)s(ecame)g(b)s(ene\014cial)e(for)h(complex)h(trans-)150
 
7046
1148 y(forms)c(w)m(as)h(ab)s(out)f(4k)g(p)s(oin)m(ts,)g(while)e(4)j
 
7047
(threads)f(b)s(ecame)h(b)s(ene\014cial)d(at)j(8k)g(p)s(oin)m(ts.)150
 
7048
1381 y Fh(4.1.4)63 b(Using)42 b(Multi-threaded)f(FFTW)h(in)f(a)f
 
7049
(Multi-threaded)i(Program)275 1578 y Fv(It)29 b(is)f(p)s(erfectly)h(p)s
 
7050
(ossible)d(to)k(use)f(the)h(m)m(ulti-threaded)e(FFTW)i(routines)e(from)
 
7051
h(a)g(m)m(ulti-threaded)150 1688 y(program)c(\(e.g.)41
 
7052
b(ha)m(v)m(e)27 b(m)m(ultiple)c(threads)i(computing)g(m)m
 
7053
(ulti-threaded)f(transforms)h(sim)m(ultaneously\).)150
 
7054
1798 y(If)k(y)m(ou)g(ha)m(v)m(e)i(the)e(pro)s(cessors,)g(more)g(p)s(o)m
6985
7055
(w)m(er)h(to)g(y)m(ou!)40 b(Ho)m(w)m(ev)m(er,)32 b(the)d(same)h
6986
 
(restrictions)e(apply)g(as)h(for)150 1733 y(the)k(unipro)s(cessor)d
 
7056
(restrictions)e(apply)g(as)h(for)150 1907 y(the)k(unipro)s(cessor)d
6987
7057
(FFTW)j(routines)e(\(see)i(Section)f(3.8)i([Thread)d(safet)m(y],)k
6988
 
(page)e(36\).)48 b(In)31 b(particular,)150 1843 y(y)m(ou)g(should)d
 
7058
(page)e(36\).)48 b(In)31 b(particular,)150 2017 y(y)m(ou)g(should)d
6989
7059
(recall)i(that)h(y)m(ou)f(ma)m(y)h(not)g(create)h(or)e(destro)m(y)h
6990
 
(plans)e(in)g(parallel.)150 2103 y Fi(4.1.5)63 b(Tips)42
6991
 
b(for)e(Optimal)h(Threading)275 2314 y Fu(Not)34 b(all)f(transforms)g
 
7060
(plans)e(in)g(parallel.)150 2250 y Fh(4.1.5)63 b(Tips)42
 
7061
b(for)e(Optimal)h(Threading)275 2447 y Fv(Not)34 b(all)f(transforms)g
6992
7062
(are)h(equally)e(w)m(ell-parallelized)f(b)m(y)i(the)h(m)m
6993
 
(ulti-threaded)e(FFTW)j(routines.)150 2424 y(\(This)20
 
7063
(ulti-threaded)e(FFTW)j(routines.)150 2556 y(\(This)20
6994
7064
b(is)h(merely)g(a)h(consequence)h(of)e(laziness)g(on)h(the)g(part)f(of)
6995
7065
h(the)g(implemen)m(tors,)g(and)f(is)g(not)h(inheren)m(t)150
6996
 
2533 y(to)38 b(the)g(algorithms)f(emplo)m(y)m(ed.\))63
 
7066
2666 y(to)38 b(the)g(algorithms)f(emplo)m(y)m(ed.\))63
6997
7067
b(Mainly)-8 b(,)39 b(the)f(limitations)e(are)i(in)e(the)i(parallel)e
6998
 
(one-dimensional)150 2643 y(transforms.)k(The)30 b(things)f(to)i(a)m(v)
 
7068
(one-dimensional)150 2776 y(transforms.)k(The)30 b(things)f(to)i(a)m(v)
6999
7069
m(oid)g(if)e(y)m(ou)i(w)m(an)m(t)g(optimal)f(parallelization)e(are)j
7000
 
(as)f(follo)m(ws:)150 2904 y Fi(4.1.6)63 b(P)m(arallelization)39
 
7070
(as)f(follo)m(ws:)150 3009 y Fh(4.1.6)63 b(P)m(arallelization)39
7001
7071
b(de\014ciencies)i(in)h(one-dimensional)g(transforms)225
7002
 
3114 y Fs(\017)60 b Fu(Large)24 b(prime)f(factors)h(can)g(sometimes)g
 
7072
3176 y Ft(\017)60 b Fv(Large)24 b(prime)f(factors)h(can)g(sometimes)g
7003
7073
(parallelize)e(p)s(o)s(orly)-8 b(.)37 b(Of)24 b(course,)h(y)m(ou)f
7004
 
(should)e(a)m(v)m(oid)i(these)330 3224 y(an)m(yw)m(a)m(y)32
 
7074
(should)e(a)m(v)m(oid)i(these)330 3286 y(an)m(yw)m(a)m(y)32
7005
7075
b(if)d(y)m(ou)i(w)m(an)m(t)g(high)e(p)s(erformance.)225
7006
 
3368 y Fs(\017)60 b Fu(Single)31 b(in-place)g(transforms)h(don't)g
 
7076
3423 y Ft(\017)60 b Fv(Single)31 b(in-place)g(transforms)h(don't)g
7007
7077
(parallelize)f(completely)-8 b(.)47 b(\(Multiple)30 b(in-place)i
7008
 
(transforms,)330 3477 y(i.e.)40 b Fo(howmany)28 b(>)j(1)p
7009
 
Fu(,)e(are)g(\014ne.\))40 b(Again,)29 b(y)m(ou)h(should)c(a)m(v)m(oid)k
 
7078
(transforms,)330 3532 y(i.e.)40 b Fp(howmany)28 b(>)j(1)p
 
7079
Fv(,)e(are)g(\014ne.\))40 b(Again,)29 b(y)m(ou)h(should)c(a)m(v)m(oid)k
7010
7080
(these)f(in)f(an)m(y)h(case)h(if)e(y)m(ou)h(w)m(an)m(t)h(high)330
7011
 
3587 y(p)s(erformance,)g(as)h(they)f(require)f(transforming)g(to)i(a)g
 
7081
3642 y(p)s(erformance,)g(as)h(they)f(require)f(transforming)g(to)i(a)g
7012
7082
(scratc)m(h)g(arra)m(y)g(and)f(cop)m(ying)g(bac)m(k.)225
7013
 
3730 y Fs(\017)60 b Fu(Single)33 b(real-complex)i(\()p
7014
 
Fo(rfftw)p Fu(\))f(transforms)g(don't)h(parallelize)e(completely)-8
7015
 
b(.)54 b(This)33 b(is)h(unfortu-)330 3840 y(nate,)41
 
7083
3779 y Ft(\017)60 b Fv(Single)33 b(real-complex)i(\()p
 
7084
Fp(rfftw)p Fv(\))f(transforms)g(don't)h(parallelize)e(completely)-8
 
7085
b(.)54 b(This)33 b(is)h(unfortu-)330 3888 y(nate,)41
7016
7086
b(but)d(parallelizing)d(this)i(correctly)i(w)m(ould)e(ha)m(v)m(e)j(in)m
7017
7087
(v)m(olv)m(ed)e(a)g(lot)g(of)h(extra)g(co)s(de)f(\(and)h(a)330
7018
 
3950 y(m)m(uc)m(h)31 b(larger)g(library\).)42 b(Y)-8
 
7088
3998 y(m)m(uc)m(h)31 b(larger)g(library\).)42 b(Y)-8
7019
7089
b(ou)32 b(still)d(get)j(some)g(b)s(ene\014t)e(from)h(additional)e(pro)s
7020
 
(cessors,)j(but)e(if)h(y)m(ou)330 4059 y(ha)m(v)m(e)25
 
7090
(cessors,)j(but)e(if)h(y)m(ou)330 4107 y(ha)m(v)m(e)25
7021
7091
b(a)g(v)m(ery)g(large)f(n)m(um)m(b)s(er)f(of)i(pro)s(cessors)e(y)m(ou)i
7022
7092
(will)c(probably)i(b)s(e)h(b)s(etter)g(o\013)h(using)d(the)j(parallel)
7023
 
330 4169 y(complex)35 b(\()p Fo(fftw)p Fu(\))h(transforms.)55
 
7093
330 4217 y(complex)35 b(\()p Fp(fftw)p Fv(\))h(transforms.)55
7024
7094
b(Note)37 b(that)g(m)m(ulti-dimensional)31 b(real)36
7025
 
b(transforms)f(or)g(m)m(ultiple)330 4278 y(one-dimensional)28
7026
 
b(real)i(transforms)g(are)g(\014ne.)150 4581 y Ft(4.2)68
7027
 
b(MPI)44 b(FFTW)275 4792 y Fu(This)38 b(section)i(describ)s(es)f(the)h
 
7095
b(transforms)f(or)g(m)m(ultiple)330 4326 y(one-dimensional)28
 
7096
b(real)i(transforms)g(are)g(\014ne.)150 4595 y Fu(4.2)68
 
7097
b(MPI)44 b(FFTW)275 4792 y Fv(This)38 b(section)i(describ)s(es)f(the)h
7028
7098
(MPI)g(FFTW)h(routines)e(for)h(distributed-memory)d(\(and)j(shared-)150
7029
7099
4902 y(memory\))h(mac)m(hines)f(supp)s(orting)e(MPI)i(\(Message)j(P)m
7030
7100
(assing)d(In)m(terface\).)73 b(The)40 b(MPI)g(routines)g(are)150
7031
7101
5011 y(signi\014can)m(tly)35 b(di\013eren)m(t)i(from)f(the)h(ordinary)e
7032
7102
(FFTW)j(b)s(ecause)f(the)g(transform)f(data)i(here)f(are)g
7033
 
Fq(dis-)150 5121 y(tribute)-5 b(d)49 b Fu(o)m(v)m(er)41
 
7103
Fr(dis-)150 5121 y(tribute)-5 b(d)49 b Fv(o)m(v)m(er)41
7034
7104
b(m)m(ultiple)c(pro)s(cesses,)k(so)e(that)h(eac)m(h)g(pro)s(cess)f
7035
7105
(gets)h(only)e(a)i(p)s(ortion)e(of)h(the)g(arra)m(y)-8
7036
7106
b(.)150 5230 y(Curren)m(tly)g(,)43 b(m)m(ulti-dimensional)38
7037
7107
b(transforms)i(of)i(b)s(oth)f(real)g(and)g(complex)g(data,)k(as)d(w)m
7038
7108
(ell)f(as)g(one-)150 5340 y(dimensional)28 b(transforms)h(of)i(complex)
7039
 
f(data,)h(are)g(supp)s(orted.)p eop
 
7109
f(data,)h(are)g(supp)s(orted.)p eop end
7040
7110
%%Page: 41 43
7041
 
41 42 bop 150 -116 a Fu(Chapter)30 b(4:)41 b(P)m(arallel)29
7042
 
b(FFTW)2435 b(41)150 299 y Fi(4.2.1)63 b(MPI)41 b(FFTW)g(Installation)
7043
 
275 494 y Fu(The)25 b(FFTW)i(MPI)e(library)f(co)s(de)i(is)f(all)g(lo)s
7044
 
(cated)h(in)e(the)j Fo(mpi)e Fu(sub)s(directo)m(y)f(of)i(the)g(FFTW)h
7045
 
(pac)m(k)-5 b(age)150 604 y(\(along)26 b(with)e(source)h(co)s(de)h(for)
7046
 
f(test)i(programs\).)39 b(On)24 b(Unix)g(systems,)j(the)f(FFTW)g(MPI)f
 
7111
TeXDict begin 41 42 bop 150 -116 a Fv(Chapter)30 b(4:)41
 
7112
b(P)m(arallel)29 b(FFTW)2435 b(41)150 299 y Fh(4.2.1)63
 
7113
b(MPI)41 b(FFTW)g(Installation)275 494 y Fv(The)25 b(FFTW)i(MPI)e
 
7114
(library)f(co)s(de)i(is)f(all)g(lo)s(cated)h(in)e(the)j
 
7115
Fp(mpi)e Fv(sub)s(directo)m(y)f(of)i(the)g(FFTW)h(pac)m(k)-5
 
7116
b(age)150 604 y(\(along)26 b(with)e(source)h(co)s(de)h(for)f(test)i
 
7117
(programs\).)39 b(On)24 b(Unix)g(systems,)j(the)f(FFTW)g(MPI)f
7047
7118
(libraries)e(and)150 713 y(header)j(\014les)f(can)i(b)s(e)e
7048
7119
(automatically)h(con\014gured,)h(compiled,)f(and)f(installed)f(along)j
7049
7120
(with)d(the)j(unipro-)150 823 y(cessor)39 b(FFTW)g(libraries)d(simply)g
7050
 
(b)m(y)i(including)d Fo(--enable-mpi)g Fu(in)i(the)i(\015ags)g(to)g
7051
 
(the)f Fo(configure)150 932 y Fu(script)29 b(\(see)j(Section)e(6.1)h
 
7121
(b)m(y)i(including)d Fp(--enable-mpi)g Fv(in)i(the)i(\015ags)g(to)g
 
7122
(the)f Fp(configure)150 932 y Fv(script)29 b(\(see)j(Section)e(6.1)h
7052
7123
([Installation)e(on)i(Unix],)e(page)i(55\).)275 1069
7053
7124
y(The)j(only)g(requiremen)m(t)g(of)h(the)g(FFTW)g(MPI)g(co)s(de)g(is)f
7054
7125
(that)i(y)m(ou)f(ha)m(v)m(e)h(the)f(standard)f(MPI)h(1.1)150
7055
7126
1179 y(\(or)28 b(later\))g(libraries)d(and)i(header)h(\014les)f
7056
7127
(installed)e(on)j(y)m(our)g(system.)40 b(A)28 b(free)g(implemen)m
7057
7128
(tation)e(of)i(MPI)150 1288 y(is)h(a)m(v)-5 b(ailable)30
7058
 
b(from)g(the)h(MPICH)f(home)g(page)h(\()p Fo
7059
 
(http://www-unix.mcs.anl.gov)o(/mp)o(i/mp)o(ich/)o Fu(\).)275
 
7129
b(from)g(the)h(MPICH)f(home)g(page)h(\()p Fp
 
7130
(http://www-unix.mcs.anl.gov)o(/mp)o(i/mp)o(ich/)o Fv(\).)275
7060
7131
1425 y(Previous)42 b(v)m(ersions)g(of)i(the)f(FFTW)h(MPI)f(routines)f
7061
7132
(ha)m(v)m(e)j(had)d(an)h(unfortunate)g(tendency)g(to)150
7062
7133
1535 y(exp)s(ose)i(bugs)f(in)f(MPI)i(implemen)m(tations.)82
7064
7135
(rewritten,)150 1645 y(and)c(hop)s(efully)e(a)m(v)m(oids)j(some)g(of)g
7065
7136
(the)g(problems.)71 b(If)40 b(y)m(ou)h(run)f(in)m(to)g(di\016culties,)h
7066
7137
(try)g(passing)f(the)150 1754 y(optional)d(w)m(orkspace)i(to)f
7067
 
Fo(\(r\)fftwnd_mpi)c Fu(\(see)39 b(b)s(elo)m(w\),)h(as)e(this)e(allo)m
 
7138
Fp(\(r\)fftwnd_mpi)c Fv(\(see)39 b(b)s(elo)m(w\),)h(as)e(this)e(allo)m
7068
7139
(ws)i(us)f(to)h(use)g(the)g(standard)150 1864 y(\(and)23
7069
 
b(hop)s(efully)d(w)m(ell-tested\))k Fo(MPI_Alltoall)c
7070
 
Fu(primitiv)m(e)h(for)h(comm)m(unications.)38 b(Please)23
7071
 
b(let)g(us)g(kno)m(w)150 1973 y(\()p Fo(fftw@fftw.org)p
7072
 
Fu(\))k(ho)m(w)k(things)e(w)m(ork)i(out.)275 2110 y(Sev)m(eral)36
7073
 
b(test)h(programs)f(are)g(included)e(in)h(the)h Fo(mpi)f
7074
 
Fu(directory)-8 b(.)59 b(The)35 b(ones)i(most)f(useful)f(to)i(y)m(ou)
7075
 
150 2220 y(are)25 b(probably)f(the)h Fo(fftw_mpi_test)d
7076
 
Fu(and)i Fo(rfftw_mpi_test)d Fu(programs,)26 b(whic)m(h)e(are)i(run)d
 
7140
b(hop)s(efully)d(w)m(ell-tested\))k Fp(MPI_Alltoall)c
 
7141
Fv(primitiv)m(e)h(for)h(comm)m(unications.)38 b(Please)23
 
7142
b(let)g(us)g(kno)m(w)150 1973 y(\()p Fp(fftw@fftw.org)p
 
7143
Fv(\))k(ho)m(w)k(things)e(w)m(ork)i(out.)275 2110 y(Sev)m(eral)36
 
7144
b(test)h(programs)f(are)g(included)e(in)h(the)h Fp(mpi)f
 
7145
Fv(directory)-8 b(.)59 b(The)35 b(ones)i(most)f(useful)f(to)i(y)m(ou)
 
7146
150 2220 y(are)25 b(probably)f(the)h Fp(fftw_mpi_test)d
 
7147
Fv(and)i Fp(rfftw_mpi_test)d Fv(programs,)26 b(whic)m(h)e(are)i(run)d
7077
7148
(just)i(lik)m(e)f(an)150 2330 y(ordinary)k(MPI)h(program)g(and)f
7078
7149
(accept)j(the)f(same)f(parameters)h(as)f(the)h(other)f(FFTW)h(test)g
7079
 
(programs)150 2439 y(\(c.f.)48 b Fo(tests/README)p Fu(\).)c(F)-8
7080
 
b(or)33 b(example,)g Fo(mpirun)28 b Fq(...p)-5 b(ar)g(ams...)33
7081
 
b Fo(fftw_mpi_test)27 b(-r)i(0)j Fu(will)e(run)h(non-)150
 
7150
(programs)150 2439 y(\(c.f.)48 b Fp(tests/README)p Fv(\).)c(F)-8
 
7151
b(or)33 b(example,)g Fp(mpirun)28 b Fr(...p)-5 b(ar)g(ams...)33
 
7152
b Fp(fftw_mpi_test)27 b(-r)i(0)j Fv(will)e(run)h(non-)150
7082
7153
2549 y(terminating)h(complex-transform)h(correctness)h(tests)g(of)f
7083
7154
(random)f(dimensions.)47 b(They)32 b(can)i(also)f(do)150
7084
7155
2658 y(p)s(erformance)d(b)s(enc)m(hmarks.)150 2867 y
7085
 
Fi(4.2.2)63 b(Usage)41 b(of)g(MPI)g(FFTW)g(for)g(Complex)f
 
7156
Fh(4.2.2)63 b(Usage)41 b(of)g(MPI)g(FFTW)g(for)g(Complex)f
7086
7157
(Multi-dimensional)464 2991 y(T)-10 b(ransforms)275 3186
7087
 
y Fu(Usage)41 b(of)g(the)g(MPI)f(FFTW)h(routines)f(is)f(similar)f(to)j
 
7158
y Fv(Usage)41 b(of)g(the)g(MPI)f(FFTW)h(routines)f(is)f(similar)f(to)j
7088
7159
(that)h(of)e(the)h(unipro)s(cessor)d(FFTW.)k(W)-8 b(e)150
7089
7160
3296 y(assume)29 b(that)g(the)g(reader)f(already)h(understands)e(the)i
7090
7161
(usage)g(of)g(the)g(unipro)s(cessor)d(FFTW)k(routines,)150
7092
7163
b(Some)31 b(familiarit)m(y)d(with)h(MPI)h(is)g(also)g(helpful.)275
7093
7164
3543 y(A)39 b(t)m(ypical)h(program)f(p)s(erforming)e(a)j(complex)g(t)m
7094
7165
(w)m(o-dimensional)e(MPI)i(transform)f(migh)m(t)g(lo)s(ok)150
7095
 
3652 y(something)30 b(lik)m(e:)390 3783 y Fo(#include)46
 
7166
3652 y(something)30 b(lik)m(e:)390 3783 y Fp(#include)46
7096
7167
b(<fftw_mpi.h>)390 3991 y(int)h(main\(int)e(argc,)i(char)f(**argv\))390
7097
7168
4095 y({)676 4198 y(const)h(int)g(NX)g(=)g(...,)g(NY)g(=)h(...;)676
7098
7169
4302 y(fftwnd_mpi_plan)c(plan;)676 4406 y(fftw_complex)h(*data;)676
7100
7171
(fftw2d_mpi_create_plan\(MPI)o(_COM)o(M_W)o(ORLD)o(,)2108
7101
7172
4925 y(NX,)g(NY,)2108 5029 y(FFTW_FORWARD,)d(FFTW_ESTIMATE\);)676
7102
7173
5236 y(...allocate)h(and)i(initialize)e(data...)p eop
 
7174
end
7103
7175
%%Page: 42 44
7104
 
42 43 bop 150 -116 a Fu(42)3232 b(FFTW)676 299 y Fo(fftwnd_mpi\(p,)45
7105
 
b(1,)i(data,)f(NULL,)g(FFTW_NORMAL_ORDER\);)676 506 y(...)676
7106
 
714 y(fftwnd_mpi_destroy_plan\(pl)o(an\))o(;)676 818
7107
 
y(MPI_Finalize\(\);)390 922 y(})275 1056 y Fu(The)39
7108
 
b(calls)h(to)h Fo(MPI_Init)d Fu(and)i Fo(MPI_Finalize)d
7109
 
Fu(are)k(required)e(in)g(all)g(MPI)h(programs;)46 b(see)41
7110
 
b(the)150 1165 y(MPI)33 b(home)h(page)g(\()p Fo
7111
 
(http://www.mcs.anl.gov/mpi)o(/)p Fu(\))28 b(for)33 b(more)g
 
7176
TeXDict begin 42 43 bop 150 -116 a Fv(42)3232 b(FFTW)676
 
7177
299 y Fp(fftwnd_mpi\(p,)45 b(1,)i(data,)f(NULL,)g(FFTW_NORMAL_ORDER\);)
 
7178
676 506 y(...)676 714 y(fftwnd_mpi_destroy_plan\(pl)o(an\))o(;)676
 
7179
818 y(MPI_Finalize\(\);)390 922 y(})275 1056 y Fv(The)39
 
7180
b(calls)h(to)h Fp(MPI_Init)d Fv(and)i Fp(MPI_Finalize)d
 
7181
Fv(are)k(required)e(in)g(all)g(MPI)h(programs;)46 b(see)41
 
7182
b(the)150 1165 y(MPI)33 b(home)h(page)g(\()p Fp
 
7183
(http://www.mcs.anl.gov/mpi)o(/)p Fv(\))28 b(for)33 b(more)g
7112
7184
(information.)49 b(Note)34 b(that)g(all)f(of)150 1275
7113
7185
y(y)m(our)39 b(pro)s(cesses)f(run)f(the)i(program)g(in)e(parallel,)i
7114
7186
(as)g(a)g(group;)k(there)c(is)e(no)i(explicit)e(launc)m(hing)g(of)150
7115
7187
1384 y(threads/pro)s(cesses)30 b(in)f(an)i(MPI)f(program.)275
7116
7188
1519 y(As)36 b(in)g(the)h(ordinary)e(FFTW,)j(the)f(\014rst)f(thing)g(w)
7117
7189
m(e)i(do)f(is)e(to)j(create)h(a)e(plan)e(\(of)j(t)m(yp)s(e)f
7118
 
Fo(fftwnd_)150 1628 y(mpi_plan)p Fu(\),)29 b(using:)390
7119
 
1757 y Fo(fftwnd_mpi_plan)44 b(fftw2d_mpi_create_plan\()o(MPI_)o(Com)o
 
7190
Fp(fftwnd_)150 1628 y(mpi_plan)p Fv(\),)29 b(using:)390
 
7191
1757 y Fp(fftwnd_mpi_plan)44 b(fftw2d_mpi_create_plan\()o(MPI_)o(Com)o
7120
7192
(m)e(comm,)2251 1860 y(int)47 b(nx,)g(int)g(ny,)2251
7121
7193
1964 y(fftw_direction)d(dir,)j(int)g(flags\);)275 2098
7122
 
y Fu(Except)29 b(for)g(the)h(\014rst)e(argumen)m(t,)j(the)e(parameters)
7123
 
h(are)f(iden)m(tical)g(to)h(those)f(of)h Fo(fftw2d_create_)150
7124
 
2208 y(plan)p Fu(.)39 b(\(There)28 b(are)g(also)g(analogous)g
7125
 
Fo(fftwnd_mpi_create_plan)22 b Fu(and)27 b Fo(fftw3d_mpi_create_plan)
7126
 
150 2317 y Fu(functions.)45 b(T)-8 b(ransforms)31 b(of)h(an)m(y)h(rank)
 
7194
y Fv(Except)29 b(for)g(the)h(\014rst)e(argumen)m(t,)j(the)e(parameters)
 
7195
h(are)f(iden)m(tical)g(to)h(those)f(of)h Fp(fftw2d_create_)150
 
7196
2208 y(plan)p Fv(.)39 b(\(There)28 b(are)g(also)g(analogous)g
 
7197
Fp(fftwnd_mpi_create_plan)22 b Fv(and)27 b Fp(fftw3d_mpi_create_plan)
 
7198
150 2317 y Fv(functions.)45 b(T)-8 b(ransforms)31 b(of)h(an)m(y)h(rank)
7127
7199
f(greater)h(than)f(one)h(are)f(supp)s(orted.\))45 b(The)32
7128
7200
b(\014rst)f(argumen)m(t)i(is)150 2427 y(an)27 b(MPI)g
7129
 
Fp(comm)m(unicator)p Fu(,)h(whic)m(h)e(sp)s(eci\014es)g(the)h(group)g
 
7201
Fq(comm)m(unicator)p Fv(,)h(whic)m(h)e(sp)s(eci\014es)g(the)h(group)g
7130
7202
(of)g(pro)s(cesses)g(that)g(are)h(to)g(b)s(e)e(in)m(v)m(olv)m(ed)h(in)f
7131
7203
(the)150 2537 y(transform;)k(the)g(standard)g(constan)m(t)i
7132
 
Fo(MPI_COMM_WORLD)26 b Fu(indicates)j(all)h(a)m(v)-5
 
7204
Fp(MPI_COMM_WORLD)26 b Fv(indicates)j(all)h(a)m(v)-5
7133
7205
b(ailable)29 b(pro)s(cesses.)275 2671 y(Next,)34 b(one)g(has)e(to)i
7134
7206
(allo)s(cate)g(and)e(initialize)e(the)j(data.)50 b(This)31
7135
7207
b(is)h(somewhat)h(tric)m(ky)-8 b(,)35 b(b)s(ecause)e(the)150
7139
7211
(section)f(\(see)i(Section)e(4.2.3)i([MPI)f(Data)h(La)m(y)m(out],)g
7140
7212
(page)f(43\).)275 3024 y(The)42 b(actual)i(computation)f(of)g(the)g
7141
7213
(transform)g(is)f(p)s(erformed)f(b)m(y)i(the)h(function)d
7142
 
Fo(fftwnd_mpi)p Fu(,)150 3134 y(whic)m(h)29 b(di\013ers)g(somewhat)i
 
7214
Fp(fftwnd_mpi)p Fv(,)150 3134 y(whic)m(h)29 b(di\013ers)g(somewhat)i
7143
7215
(from)f(its)f(unipro)s(cessor)g(equiv)-5 b(alen)m(t)30
7144
 
b(and)f(is)h(describ)s(ed)e(b)m(y:)390 3262 y Fo(void)47
 
7216
b(and)f(is)h(describ)s(ed)e(b)m(y:)390 3262 y Fp(void)47
7145
7217
b(fftwnd_mpi\(fftwnd_mpi_p)o(lan)41 b(p,)1154 3366 y(int)47
7146
7218
b(n_fields,)1154 3470 y(fftw_complex)d(*local_data,)h(fftw_complex)f
7147
7219
(*work,)1154 3573 y(fftwnd_mpi_output_order)d(output_order\);)275
7148
 
3707 y Fu(There)29 b(are)i(sev)m(eral)g(things)e(to)i(notice)g(here:)
7149
 
225 3842 y Fs(\017)60 b Fu(First)40 b(of)g(all,)i(all)d
7150
 
Fo(fftw_mpi)f Fu(transforms)h(are)i(in-place:)59 b(the)40
7151
 
b(output)g(is)f(in)g(the)i Fo(local_data)330 3951 y Fu(parameter,)31
7152
 
b(and)f(there)g(is)g(no)g(need)g(to)h(sp)s(ecify)e Fo(FFTW_IN_PLACE)e
7153
 
Fu(in)i(the)i(plan)e(\015ags.)225 4085 y Fs(\017)60 b
7154
 
Fu(The)32 b(MPI)h(transforms)f(also)h(only)f(supp)s(ort)f(a)j(limited)c
7155
 
(subset)j(of)g(the)g Fo(howmany)p Fu(/)p Fo(stride)p
7156
 
Fu(/)p Fo(dist)330 4195 y Fu(functionalit)m(y)39 b(of)h(the)h(unipro)s
7157
 
(cessor)d(routines:)60 b(the)41 b Fo(n_fields)d Fu(parameter)j(is)e
7158
 
(equiv)-5 b(alen)m(t)40 b(to)330 4305 y Fo(howmany=n_fields)p
7159
 
Fu(,)26 b Fo(stride=n_fields)p Fu(,)h(and)i Fo(dist=1)p
7160
 
Fu(.)39 b(\(Conceptually)-8 b(,)30 b(the)h Fo(n_fields)d
7161
 
Fu(pa-)330 4414 y(rameter)36 b(allo)m(ws)f(y)m(ou)h(to)h(transform)e
 
7220
3707 y Fv(There)29 b(are)i(sev)m(eral)g(things)e(to)i(notice)g(here:)
 
7221
225 3842 y Ft(\017)60 b Fv(First)40 b(of)g(all,)i(all)d
 
7222
Fp(fftw_mpi)f Fv(transforms)h(are)i(in-place:)59 b(the)40
 
7223
b(output)g(is)f(in)g(the)i Fp(local_data)330 3951 y Fv(parameter,)31
 
7224
b(and)f(there)g(is)g(no)g(need)g(to)h(sp)s(ecify)e Fp(FFTW_IN_PLACE)e
 
7225
Fv(in)i(the)i(plan)e(\015ags.)225 4085 y Ft(\017)60 b
 
7226
Fv(The)32 b(MPI)h(transforms)f(also)h(only)f(supp)s(ort)f(a)j(limited)c
 
7227
(subset)j(of)g(the)g Fp(howmany)p Fv(/)p Fp(stride)p
 
7228
Fv(/)p Fp(dist)330 4195 y Fv(functionalit)m(y)39 b(of)h(the)h(unipro)s
 
7229
(cessor)d(routines:)60 b(the)41 b Fp(n_fields)d Fv(parameter)j(is)e
 
7230
(equiv)-5 b(alen)m(t)40 b(to)330 4305 y Fp(howmany=n_fields)p
 
7231
Fv(,)26 b Fp(stride=n_fields)p Fv(,)h(and)i Fp(dist=1)p
 
7232
Fv(.)39 b(\(Conceptually)-8 b(,)30 b(the)h Fp(n_fields)d
 
7233
Fv(pa-)330 4414 y(rameter)36 b(allo)m(ws)f(y)m(ou)h(to)h(transform)e
7162
7234
(an)g(arra)m(y)i(of)f(con)m(tiguous)f(v)m(ectors,)k(eac)m(h)e(with)d
7163
 
(length)i Fo(n_)330 4524 y(fields)p Fu(.\))j Fo(n_fields)28
7164
 
b Fu(is)i Fo(1)g Fu(if)f(y)m(ou)i(are)g(only)e(transforming)g(a)i
7165
 
(single,)e(ordinary)g(arra)m(y)-8 b(.)225 4658 y Fs(\017)60
7166
 
b Fu(The)24 b Fo(work)f Fu(parameter)i(is)f(an)g(optional)g(w)m
7167
 
(orkspace.)40 b(If)24 b(it)g(is)f(not)i Fo(NULL)p Fu(,)g(it)f(should)f
 
7235
(length)i Fp(n_)330 4524 y(fields)p Fv(.\))j Fp(n_fields)28
 
7236
b Fv(is)i Fp(1)g Fv(if)f(y)m(ou)i(are)g(only)e(transforming)g(a)i
 
7237
(single,)e(ordinary)g(arra)m(y)-8 b(.)225 4658 y Ft(\017)60
 
7238
b Fv(The)24 b Fp(work)f Fv(parameter)i(is)f(an)g(optional)g(w)m
 
7239
(orkspace.)40 b(If)24 b(it)g(is)f(not)i Fp(NULL)p Fv(,)g(it)f(should)f
7168
7240
(b)s(e)g(exactly)j(the)330 4767 y(same)31 b(size)f(as)g(the)g
7169
 
Fo(local_data)e Fu(arra)m(y)-8 b(.)41 b(If)30 b(it)g(is)f(pro)m(vided,)
 
7241
Fp(local_data)e Fv(arra)m(y)-8 b(.)41 b(If)30 b(it)g(is)f(pro)m(vided,)
7170
7242
g(FFTW)i(is)e(able)h(to)h(use)f(the)g(built-in)330 4877
7171
 
y Fo(MPI_Alltoall)i Fu(primitiv)m(e)i(for)i(\(often\))g(greater)h
 
7243
y Fp(MPI_Alltoall)i Fv(primitiv)m(e)i(for)i(\(often\))g(greater)h
7172
7244
(e\016ciency)f(at)h(the)f(exp)s(ense)f(of)h(extra)g(storage)330
7173
 
4987 y(space.)225 5121 y Fs(\017)60 b Fu(Finally)-8 b(,)28
 
7245
4987 y(space.)225 5121 y Ft(\017)60 b Fv(Finally)-8 b(,)28
7174
7246
b(the)h(last)f(parameter)i(sp)s(eci\014es)d(whether)h(the)h(output)g
7175
7247
(data)g(has)g(the)g(same)g(ordering)f(as)330 5230 y(the)c(input)e(data)
7176
 
j(\()p Fo(FFTW_NORMAL_ORDER)p Fu(\),)c(or)j(if)f(it)g(is)g(transp)s
7177
 
(osed)g(\()p Fo(FFTW_TRANSPOSED_ORDER)p Fu(\).)330 5340
 
7248
j(\()p Fp(FFTW_NORMAL_ORDER)p Fv(\),)c(or)j(if)f(it)g(is)g(transp)s
 
7249
(osed)g(\()p Fp(FFTW_TRANSPOSED_ORDER)p Fv(\).)330 5340
7178
7250
y(Lea)m(ving)31 b(the)f(data)h(transp)s(osed)f(results)f(in)g
7179
7251
(signi\014can)m(t)g(p)s(erformance)h(impro)m(v)m(emen)m(ts)h(due)e(to)j
7180
 
(a)p eop
 
7252
(a)p eop end
7181
7253
%%Page: 43 45
7182
 
43 44 bop 150 -116 a Fu(Chapter)30 b(4:)41 b(P)m(arallel)29
7183
 
b(FFTW)2435 b(43)330 299 y(sa)m(v)m(ed)41 b(comm)m(unication)f(step)g
7184
 
(\(needed)g(to)h(un-transp)s(ose)e(the)h(data\).)71 b(Sp)s
7185
 
(eci\014cally)-8 b(,)40 b(the)h(\014rst)330 408 y(t)m(w)m(o)32
7186
 
b(dimensions)27 b(of)k(the)g(arra)m(y)g(are)f(transp)s(osed,)g(as)h(is)
7187
 
e(describ)s(ed)g(in)g(more)h(detail)g(b)m(y)g(the)h(next)330
7188
 
518 y(section.)275 694 y(The)f(output)h(of)h Fo(fftwnd_mpi)d
7189
 
Fu(is)h(iden)m(tical)g(to)i(that)g(of)g(the)f(corresp)s(onding)f
7190
 
(unipro)s(cessor)f(trans-)150 803 y(form.)40 b(In)27
7191
 
b(particular,)g(y)m(ou)i(should)d(recall)h(our)h(con)m(v)m(en)m(tions)h
7192
 
(for)f(normalization)e(and)i(the)g(sign)f(of)i(the)150
 
7254
TeXDict begin 43 44 bop 150 -116 a Fv(Chapter)30 b(4:)41
 
7255
b(P)m(arallel)29 b(FFTW)2435 b(43)330 299 y(sa)m(v)m(ed)41
 
7256
b(comm)m(unication)f(step)g(\(needed)g(to)h(un-transp)s(ose)e(the)h
 
7257
(data\).)71 b(Sp)s(eci\014cally)-8 b(,)40 b(the)h(\014rst)330
 
7258
408 y(t)m(w)m(o)32 b(dimensions)27 b(of)k(the)g(arra)m(y)g(are)f
 
7259
(transp)s(osed,)g(as)h(is)e(describ)s(ed)g(in)g(more)h(detail)g(b)m(y)g
 
7260
(the)h(next)330 518 y(section.)275 694 y(The)f(output)h(of)h
 
7261
Fp(fftwnd_mpi)d Fv(is)h(iden)m(tical)g(to)i(that)g(of)g(the)f(corresp)s
 
7262
(onding)f(unipro)s(cessor)f(trans-)150 803 y(form.)40
 
7263
b(In)27 b(particular,)g(y)m(ou)i(should)d(recall)h(our)h(con)m(v)m(en)m
 
7264
(tions)h(for)f(normalization)e(and)i(the)g(sign)f(of)i(the)150
7193
7265
913 y(transform)h(exp)s(onen)m(t.)275 1045 y(The)c(same)i(plan)e(can)i
7194
7266
(b)s(e)f(used)f(to)i(compute)g(man)m(y)f(transforms)g(of)g(the)h(same)g
7195
7267
(size.)39 b(After)28 b(y)m(ou)g(are)150 1154 y(done)i(with)f(it,)h(y)m
7196
7268
(ou)h(should)e(deallo)s(cate)h(it)g(b)m(y)g(calling)f
7197
 
Fo(fftwnd_mpi_destroy_plan)p Fu(.)275 1286 y Fm(Imp)s(ortan)m(t:)40
7198
 
b Fu(The)30 b(FFTW)h(MPI)e(routines)g(m)m(ust)h(b)s(e)g(called)f(in)f
 
7269
Fp(fftwnd_mpi_destroy_plan)p Fv(.)275 1286 y Fn(Imp)s(ortan)m(t:)40
 
7270
b Fv(The)30 b(FFTW)h(MPI)e(routines)g(m)m(ust)h(b)s(e)g(called)f(in)f
7199
7271
(the)j(same)f(order)f(b)m(y)h(all)f(pro)s(cesses)150
7200
7272
1396 y(in)m(v)m(olv)m(ed)22 b(in)g(the)h(transform.)37
7201
7273
b(Y)-8 b(ou)24 b(should)c(assume)j(that)g(they)g(all)f(are)h(blo)s(c)m
7202
7274
(king,)g(as)g(if)f(eac)m(h)i(con)m(tained)150 1505 y(a)31
7203
 
b(call)e(to)j Fo(MPI_Barrier)p Fu(.)275 1637 y(Programs)25
 
7275
b(call)e(to)j Fp(MPI_Barrier)p Fv(.)275 1637 y(Programs)25
7204
7276
b(using)f(the)h(FFTW)h(MPI)g(routines)e(should)f(b)s(e)i(link)m(ed)f
7205
 
(with)g Fo(-lfftw_mpi)j(-lfftw)i(-lm)150 1746 y Fu(on)h(Unix,)g(in)f
 
7277
(with)g Fp(-lfftw_mpi)j(-lfftw)i(-lm)150 1746 y Fv(on)h(Unix,)g(in)f
7206
7278
(addition)f(to)j(whatev)m(er)h(libraries)27 b(are)k(required)d(for)j
7207
 
(MPI.)150 1962 y Fi(4.2.3)63 b(MPI)41 b(Data)f(La)m(y)m(out)275
7208
 
2152 y Fu(The)31 b(transform)f(data)j(used)e(b)m(y)g(the)h(MPI)f(FFTW)i
7209
 
(routines)d(is)h Fp(distributed)t Fu(:)39 b(a)32 b(distinct)e(p)s
 
7279
(MPI.)150 1962 y Fh(4.2.3)63 b(MPI)41 b(Data)f(La)m(y)m(out)275
 
7280
2152 y Fv(The)31 b(transform)f(data)j(used)e(b)m(y)g(the)h(MPI)f(FFTW)i
 
7281
(routines)d(is)h Fq(distributed)t Fv(:)39 b(a)32 b(distinct)e(p)s
7210
7282
(ortion)150 2261 y(of)k(it)f(resides)f(with)g(eac)m(h)j(pro)s(cess)e
7211
7283
(in)m(v)m(olv)m(ed)g(in)g(the)g(transform.)50 b(This)32
7212
7284
b(allo)m(ws)h(the)g(transform)g(to)i(b)s(e)150 2371 y(parallelized,)23
7223
7295
b(consequence)h(of)g(this)e(is)g(that)i(y)m(ou)f(can't)i(tak)m(e)f(adv)
7224
7296
-5 b(an)m(tage)40 b(of)d(more)g(pro)s(cessors)150 3051
7225
7297
y(than)23 b(y)m(ou)h(ha)m(v)m(e)g(ro)m(ws)f(\(e.g.)40
7226
 
b Fo(64x64x64)21 b Fu(matrix)h(can)i(at)g(most)f(use)g(64)h(pro)s
 
7298
b Fp(64x64x64)21 b Fv(matrix)h(can)i(at)g(most)f(use)g(64)h(pro)s
7227
7299
(cessors\).)38 b(This)22 b(isn't)g(usually)150 3160 y(m)m(uc)m(h)34
7228
7300
b(of)h(a)f(limitation,)f(ho)m(w)m(ev)m(er,)k(as)e(eac)m(h)g(pro)s
7229
7301
(cessor)f(needs)g(a)g(fair)g(amoun)m(t)g(of)h(data)g(in)d(order)i(for)
7230
7302
150 3270 y(the)d(parallel-computation)d(b)s(ene\014ts)i(to)h(out)m(w)m
7231
7303
(eigh)m(t)g(the)g(comm)m(unications)f(costs.)275 3401
7232
7304
y(Belo)m(w,)d(the)f(\014rst)g(dimension)d(of)j(the)g(data)h(will)c(b)s
7233
 
(e)j(referred)f(to)i(as)f(`)p Fo(x)p Fu(')g(and)f(the)h(second)h
7234
 
(dimension)150 3511 y(as)k(`)p Fo(y)p Fu('.)275 3643
 
7305
(e)j(referred)f(to)i(as)f(`)p Fp(x)p Fv(')g(and)f(the)h(second)h
 
7306
(dimension)150 3511 y(as)k(`)p Fp(y)p Fv('.)275 3643
7235
7307
y(FFTW)43 b(supplies)e(a)i(routine)f(to)i(tell)f(y)m(ou)g(exactly)h(ho)
7236
7308
m(w)g(m)m(uc)m(h)f(data)h(resides)e(on)h(the)g(curren)m(t)150
7237
 
3752 y(pro)s(cess:)390 3878 y Fo(void)k(fftwnd_mpi_local_sizes\()o
 
7309
3752 y(pro)s(cess:)390 3878 y Fp(void)k(fftwnd_mpi_local_sizes\()o
7238
7310
(fftw)o(nd_)o(mpi_)o(plan)41 b(p,)1726 3982 y(int)47
7239
7311
b(*local_nx,)1726 4086 y(int)g(*local_x_start,)1726 4189
7240
7312
y(int)g(*local_ny_after_transpose)o(,)1726 4293 y(int)g
7241
7313
(*local_y_start_after_tran)o(spos)o(e,)1726 4397 y(int)g
7242
 
(*total_local_size\);)275 4529 y Fu(Giv)m(en)39 b(a)h(plan)e
7243
 
Fo(p)p Fu(,)k(the)e(other)g(parameters)f(of)h(this)f(routine)f(are)i
 
7314
(*total_local_size\);)275 4529 y Fv(Giv)m(en)39 b(a)h(plan)e
 
7315
Fp(p)p Fv(,)k(the)e(other)g(parameters)f(of)h(this)f(routine)f(are)i
7244
7316
(set)g(to)h(v)-5 b(alues)38 b(describing)g(the)150 4638
7245
7317
y(required)29 b(data)i(la)m(y)m(out,)g(describ)s(ed)d(b)s(elo)m(w.)275
7246
 
4770 y Fo(total_local_size)c Fu(is)k(the)i(n)m(um)m(b)s(er)d(of)i
7247
 
Fo(fftw_complex)d Fu(elemen)m(ts)k(that)f(y)m(ou)h(m)m(ust)f(allo)s
 
7318
4770 y Fp(total_local_size)c Fv(is)k(the)i(n)m(um)m(b)s(er)d(of)i
 
7319
Fp(fftw_complex)d Fv(elemen)m(ts)k(that)f(y)m(ou)h(m)m(ust)f(allo)s
7248
7320
(cate)g(for)150 4880 y(y)m(our)e(lo)s(cal)g(data)h(\(and)f(w)m
7249
7321
(orkspace,)i(if)d(y)m(ou)h(c)m(ho)s(ose\).)41 b(\(This)26
7250
7322
b(v)-5 b(alue)27 b(should,)f(of)i(course,)g(b)s(e)e(m)m(ultiplied)150
7251
 
4989 y(b)m(y)k Fo(n_fields)e Fu(if)i(that)h(parameter)g(to)g
7252
 
Fo(fftwnd_mpi)c Fu(is)j(not)g Fo(1)p Fu(.\))275 5121
 
7323
4989 y(b)m(y)k Fp(n_fields)e Fv(if)i(that)h(parameter)g(to)g
 
7324
Fp(fftwnd_mpi)c Fv(is)j(not)g Fp(1)p Fv(.\))275 5121
7253
7325
y(The)e(data)i(on)f(the)g(curren)m(t)g(pro)s(cess)g(has)f
7254
 
Fo(local_nx)f Fu(ro)m(ws,)j(starting)f(at)h(ro)m(w)f
7255
 
Fo(local_x_start)p Fu(.)36 b(If)150 5230 y Fo(fftwnd_mpi)26
7256
 
b Fu(is)i(called)g(with)g Fo(FFTW_TRANSPOSED_ORDER)23
7257
 
b Fu(output,)29 b(then)g Fo(y)f Fu(will)f(b)s(e)h(the)h(\014rst)f
 
7326
Fp(local_nx)f Fv(ro)m(ws,)j(starting)f(at)h(ro)m(w)f
 
7327
Fp(local_x_start)p Fv(.)36 b(If)150 5230 y Fp(fftwnd_mpi)26
 
7328
b Fv(is)i(called)g(with)g Fp(FFTW_TRANSPOSED_ORDER)23
 
7329
b Fv(output,)29 b(then)g Fp(y)f Fv(will)f(b)s(e)h(the)h(\014rst)f
7258
7330
(dimen-)150 5340 y(sion)d(of)i(the)f(output,)h(and)f(the)h(lo)s(cal)e
7259
 
Fo(y)h Fu(exten)m(t)i(will)c(b)s(e)h(giv)m(en)i(b)m(y)f
7260
 
Fo(local_ny_after_transpose)20 b Fu(and)p eop
 
7331
Fp(y)h Fv(exten)m(t)i(will)c(b)s(e)h(giv)m(en)i(b)m(y)f
 
7332
Fp(local_ny_after_transpose)20 b Fv(and)p eop end
7261
7333
%%Page: 44 46
7262
 
44 45 bop 150 -116 a Fu(44)3232 b(FFTW)150 299 y Fo
7263
 
(local_y_start_after_tran)o(spos)o(e)p Fu(.)59 b(Otherwise,)39
7264
 
b(the)g(output)e(has)i(the)f(same)h(dimensions)d(and)150
7265
 
408 y(la)m(y)m(out)31 b(as)g(the)f(input.)275 542 y(F)-8
7266
 
b(or)24 b(instance,)g(supp)s(ose)e(y)m(ou)i(w)m(an)m(t)g(to)g
7267
 
(transform)f(three-dimensional)e(data)j(of)g(size)f Fo(nx)29
7268
 
b(x)i(ny)e(x)h(nz)p Fu(.)150 651 y(Then,)c(the)f(curren)m(t)g(pro)s
 
7334
TeXDict begin 44 45 bop 150 -116 a Fv(44)3232 b(FFTW)150
 
7335
299 y Fp(local_y_start_after_tran)o(spos)o(e)p Fv(.)59
 
7336
b(Otherwise,)39 b(the)g(output)e(has)i(the)f(same)h(dimensions)d(and)
 
7337
150 408 y(la)m(y)m(out)31 b(as)g(the)f(input.)275 542
 
7338
y(F)-8 b(or)24 b(instance,)g(supp)s(ose)e(y)m(ou)i(w)m(an)m(t)g(to)g
 
7339
(transform)f(three-dimensional)e(data)j(of)g(size)f Fp(nx)29
 
7340
b(x)i(ny)e(x)h(nz)p Fv(.)150 651 y(Then,)c(the)f(curren)m(t)g(pro)s
7269
7341
(cess)g(will)e(store)j(a)f(subset)g(of)g(this)g(data,)i(of)e(size)h
7270
 
Fo(local_nx)i(x)i(ny)f(x)h(nz)p Fu(,)c(where)150 761
7271
 
y(the)d Fo(x)g Fu(indices)e(corresp)s(ond)h(to)h(the)g(range)h
7272
 
Fo(local_x_start)19 b Fu(to)24 b Fo(local_x_start+local_nx-)o(1)17
7273
 
b Fu(in)22 b(the)150 870 y(\\real")34 b(\(i.e.)50 b(logical\))33
7274
 
b(arra)m(y)-8 b(.)51 b(If)33 b Fo(fftwnd_mpi)d Fu(is)j(called)f(with)g
7275
 
Fo(FFTW_TRANSPOSED_ORDER)c Fu(output,)150 980 y(then)33
7276
 
b(the)h(result)e(will)f(b)s(e)i(a)g Fo(ny)d(x)g(nx)g(x)g(nz)j
7277
 
Fu(arra)m(y)-8 b(,)35 b(of)f(whic)m(h)e(a)i Fo(local_ny_after_transpos)
7278
 
o(e)25 b(x)30 b(nx)150 1090 y(x)g(nz)h Fu(subset)g(is)g(stored)h(on)g
 
7342
Fp(local_nx)i(x)i(ny)f(x)h(nz)p Fv(,)c(where)150 761
 
7343
y(the)d Fp(x)g Fv(indices)e(corresp)s(ond)h(to)h(the)g(range)h
 
7344
Fp(local_x_start)19 b Fv(to)24 b Fp(local_x_start+local_nx-)o(1)17
 
7345
b Fv(in)22 b(the)150 870 y(\\real")34 b(\(i.e.)50 b(logical\))33
 
7346
b(arra)m(y)-8 b(.)51 b(If)33 b Fp(fftwnd_mpi)d Fv(is)j(called)f(with)g
 
7347
Fp(FFTW_TRANSPOSED_ORDER)c Fv(output,)150 980 y(then)33
 
7348
b(the)h(result)e(will)f(b)s(e)i(a)g Fp(ny)d(x)g(nx)g(x)g(nz)j
 
7349
Fv(arra)m(y)-8 b(,)35 b(of)f(whic)m(h)e(a)i Fp(local_ny_after_transpos)
 
7350
o(e)25 b(x)30 b(nx)150 1090 y(x)g(nz)h Fv(subset)g(is)g(stored)h(on)g
7279
7351
(the)g(curren)m(t)g(pro)s(cess)f(\(corresp)s(onding)f(to)j
7280
 
Fo(y)e Fu(v)-5 b(alues)31 b(starting)h(at)h Fo(local_)150
7281
 
1199 y(y_start_after_transpose)p Fu(\).)275 1332 y(The)c(follo)m(wing)g
 
7352
Fp(y)e Fv(v)-5 b(alues)31 b(starting)h(at)h Fp(local_)150
 
7353
1199 y(y_start_after_transpose)p Fv(\).)275 1332 y(The)c(follo)m(wing)g
7282
7354
(is)h(an)g(example)g(of)g(allo)s(cating)g(suc)m(h)g(a)h
7283
 
(three-dimensional)d(arra)m(y)i(arra)m(y)h(\()p Fo(local_)150
7284
 
1442 y(data)p Fu(\))f(b)s(efore)g(the)g(transform)g(and)g(initializing)
7285
 
c(it)k(to)h(some)g(function)e Fo(f\(x,y,z\))p Fu(:)772
7286
 
1569 y Fo(fftwnd_mpi_local_sizes\(p)o(lan)o(,)42 b(&local_nx,)j
 
7355
(three-dimensional)d(arra)m(y)i(arra)m(y)h(\()p Fp(local_)150
 
7356
1442 y(data)p Fv(\))f(b)s(efore)g(the)g(transform)g(and)g(initializing)
 
7357
c(it)k(to)h(some)g(function)e Fp(f\(x,y,z\))p Fv(:)772
 
7358
1569 y Fp(fftwnd_mpi_local_sizes\(p)o(lan)o(,)42 b(&local_nx,)j
7287
7359
(&local_x_start,)1870 1673 y(&local_ny_after_transpo)o(se,)1870
7288
7360
1777 y(&local_y_start_after_tr)o(ans)o(pose)o(,)1870
7289
7361
1881 y(&total_local_size\);)772 2088 y(local_data)g(=)i
7293
7365
(++y\))1535 2607 y(for)g(\(z)h(=)f(0;)g(z)h(<)f(nz;)g(++z\))1917
7294
7366
2711 y(local_data[\(x*ny)d(+)j(y\)*nz)f(+)i(z])2299 2815
7295
7367
y(=)g(f\(x)e(+)i(local_x_start,)c(y,)j(z\);)275 2948
7296
 
y Fu(Some)30 b(imp)s(ortan)m(t)g(things)f(to)i(remem)m(b)s(er:)225
7297
 
3081 y Fs(\017)60 b Fu(Although)26 b(the)i(lo)s(cal)e(data)i(is)e(of)i
7298
 
(dimensions)c Fo(local_nx)k(x)i(ny)g(x)g(nz)d Fu(in)f(the)h(ab)s(o)m(v)
7299
 
m(e)h(example,)g(do)330 3190 y Fq(not)40 b Fu(allo)s(cate)31
7300
 
b(the)f(arra)m(y)h(to)g(b)s(e)f(of)g(size)h Fo(local_nx*ny*nz)p
7301
 
Fu(.)36 b(Use)31 b Fo(total_local_size)26 b Fu(instead.)225
7302
 
3324 y Fs(\017)60 b Fu(The)32 b(amoun)m(t)g(of)g(data)h(on)f(eac)m(h)h
 
7368
y Fv(Some)30 b(imp)s(ortan)m(t)g(things)f(to)i(remem)m(b)s(er:)225
 
7369
3081 y Ft(\017)60 b Fv(Although)26 b(the)i(lo)s(cal)e(data)i(is)e(of)i
 
7370
(dimensions)c Fp(local_nx)k(x)i(ny)g(x)g(nz)d Fv(in)f(the)h(ab)s(o)m(v)
 
7371
m(e)h(example,)g(do)330 3190 y Fr(not)40 b Fv(allo)s(cate)31
 
7372
b(the)f(arra)m(y)h(to)g(b)s(e)f(of)g(size)h Fp(local_nx*ny*nz)p
 
7373
Fv(.)36 b(Use)31 b Fp(total_local_size)26 b Fv(instead.)225
 
7374
3324 y Ft(\017)60 b Fv(The)32 b(amoun)m(t)g(of)g(data)h(on)f(eac)m(h)h
7303
7375
(pro)s(cess)f(will)d(not)k(necessarily)e(b)s(e)g(the)i(same;)g(in)e
7304
 
(fact,)j Fo(local_)330 3433 y(nx)g Fu(ma)m(y)h(ev)m(en)f(b)s(e)g(zero)h
 
7376
(fact,)j Fp(local_)330 3433 y(nx)g Fv(ma)m(y)h(ev)m(en)f(b)s(e)g(zero)h
7305
7377
(for)f(some)h(pro)s(cesses.)52 b(\(F)-8 b(or)36 b(example,)f(supp)s
7306
 
(ose)e(y)m(ou)h(are)h(doing)e(a)i Fo(6x6)330 3543 y Fu(transform)f(on)g
 
7378
(ose)e(y)m(ou)h(are)h(doing)e(a)i Fp(6x6)330 3543 y Fv(transform)f(on)g
7307
7379
(four)g(pro)s(cessors.)52 b(There)34 b(is)f(no)h(w)m(a)m(y)i(to)f
7308
7380
(e\013ectiv)m(ely)g(use)f(the)h(fourth)e(pro)s(cessor)330
7309
7381
3652 y(in)27 b(a)i(slab)f(decomp)s(osition,)g(so)g(w)m(e)i(lea)m(v)m(e)
7310
7382
g(it)e(empt)m(y)-8 b(.)41 b(Pro)s(of)28 b(left)g(as)h(an)f(exercise)h
7311
 
(for)g(the)f(reader.\))225 3786 y Fs(\017)60 b Fu(All)27
 
7383
(for)g(the)f(reader.\))225 3786 y Ft(\017)60 b Fv(All)27
7312
7384
b(arra)m(ys)i(are,)g(of)f(course,)i(in)d(ro)m(w-ma)5
7313
7385
b(jor)28 b(order)g(\(see)h(Section)g(2.5)g([Multi-dimensional)c(Arra)m
7314
7386
(y)330 3895 y(F)-8 b(ormat],)32 b(page)f(11\).)225 4028
7315
 
y Fs(\017)60 b Fu(If)27 b(y)m(ou)g(w)m(an)m(t)h(to)g(compute)g(the)f
 
7387
y Ft(\017)60 b Fv(If)27 b(y)m(ou)g(w)m(an)m(t)h(to)g(compute)g(the)f
7316
7388
(in)m(v)m(erse)g(transform)g(of)g(the)g(output)g(of)h
7317
 
Fo(fftwnd_mpi)p Fu(,)d(the)i(dimen-)330 4138 y(sions)f(of)i(the)g(in)m
 
7389
Fp(fftwnd_mpi)p Fv(,)d(the)i(dimen-)330 4138 y(sions)f(of)i(the)g(in)m
7318
7390
(v)m(erse)f(transform)g(are)h(giv)m(en)g(b)m(y)f(the)h(dimensions)d(of)
7319
7391
j(the)g(output)f(of)h(the)f(forw)m(ard)330 4247 y(transform.)39
7320
7392
b(F)-8 b(or)27 b(example,)g(if)e(y)m(ou)h(are)h(using)d
7321
 
Fo(FFTW_TRANSPOSED_ORDER)d Fu(output)26 b(in)e(the)j(ab)s(o)m(v)m(e)330
 
7393
Fp(FFTW_TRANSPOSED_ORDER)d Fv(output)26 b(in)e(the)j(ab)s(o)m(v)m(e)330
7322
7394
4357 y(example,)j(then)g(the)h(in)m(v)m(erse)f(plan)f(should)g(b)s(e)g
7323
 
(created)j(with)d(dimensions)f Fo(ny)h(x)h(nx)g(x)g(nz)p
7324
 
Fu(.)225 4490 y Fs(\017)60 b Fu(The)35 b(data)h(la)m(y)m(out)h(only)d
 
7395
(created)j(with)d(dimensions)f Fp(ny)h(x)h(nx)g(x)g(nz)p
 
7396
Fv(.)225 4490 y Ft(\017)60 b Fv(The)35 b(data)h(la)m(y)m(out)h(only)d
7325
7397
(dep)s(ends)g(up)s(on)g(the)h(dimensions)e(of)j(the)f(arra)m(y)-8
7326
7398
b(,)38 b(not)e(on)f(the)h(plan,)g(so)330 4600 y(y)m(ou)28
7327
7399
b(are)h(guaran)m(teed)g(that)f(di\013eren)m(t)g(plans)e(for)i(the)g
7328
7400
(same)g(size)g(\(or)g(in)m(v)m(erse)g(plans\))f(will)e(use)j(the)330
7329
7401
4709 y(same)j(\(consisten)m(t\))g(data)g(la)m(y)m(outs.)150
7330
 
4930 y Fi(4.2.4)63 b(Usage)41 b(of)g(MPI)g(FFTW)g(for)g(Real)f
7331
 
(Multi-dimensional)i(T)-10 b(ransforms)275 5121 y Fu(MPI)31
 
7402
4930 y Fh(4.2.4)63 b(Usage)41 b(of)g(MPI)g(FFTW)g(for)g(Real)f
 
7403
(Multi-dimensional)i(T)-10 b(ransforms)275 5121 y Fv(MPI)31
7332
7404
b(transforms)h(sp)s(ecialized)e(for)h(real)h(data)h(are)f(also)g(a)m(v)
7333
7405
-5 b(ailable,)32 b(similiar)c(to)33 b(the)f(unipro)s(cessor)150
7334
 
5230 y Fo(rfftwnd)37 b Fu(transforms.)66 b(Just)39 b(as)g(in)f(the)h
 
7406
5230 y Fp(rfftwnd)37 b Fv(transforms.)66 b(Just)39 b(as)g(in)f(the)h
7335
7407
(unipro)s(cessor)e(case,)43 b(the)c(real-data)h(MPI)f(functions)e(gain)
7336
7408
150 5340 y(roughly)29 b(a)h(factor)h(of)f(t)m(w)m(o)h(in)e(sp)s(eed)g
7337
7409
(\(and)g(sa)m(v)m(e)j(a)e(factor)h(of)f(t)m(w)m(o)h(in)e(space\))i(at)f
7338
 
(the)g(exp)s(ense)g(of)g(more)p eop
 
7410
(the)g(exp)s(ense)g(of)g(more)p eop end
7339
7411
%%Page: 45 47
7340
 
45 46 bop 150 -116 a Fu(Chapter)30 b(4:)41 b(P)m(arallel)29
7341
 
b(FFTW)2435 b(45)150 299 y(complicated)34 b(data)i(formats)f(in)e(the)i
7342
 
(calling)f(program.)53 b(Before)36 b(reading)e(this)g(section,)i(y)m
7343
 
(ou)f(should)150 408 y(de\014nitely)27 b(understand)g(ho)m(w)i(to)h
7344
 
(call)e(the)h(unipro)s(cessor)e Fo(rfftwnd)g Fu(functions)g(and)h(also)
7345
 
h(the)g(complex)150 518 y(MPI)h(FFTW)h(functions.)275
7346
 
689 y(The)26 b(follo)m(wing)f(is)h(an)h(example)g(of)g(a)g(program)g
7347
 
(using)e Fo(rfftwnd_mpi)p Fu(.)37 b(It)27 b(computes)g(the)g(size)g
7348
 
Fo(nx)i(x)150 799 y(ny)h(x)g(nz)g Fu(transform)g(of)h(a)h(real)e
7349
 
(function)g Fo(f\(x,y,z\))p Fu(,)f(m)m(ultiplies)f(the)j(imaginary)e
7350
 
(part)i(b)m(y)g Fo(2)f Fu(for)h(fun,)150 908 y(then)i(computes)g(the)g
7351
 
(in)m(v)m(erse)g(transform.)48 b(\(W)-8 b(e'll)33 b(also)g(use)f
7352
 
Fo(FFTW_TRANSPOSED_ORDER)c Fu(output)k(for)150 1018 y(the)d(transform,)
7353
 
g(and)g(additionally)d(supply)h(the)j(optional)e(w)m(orkspace)i
7354
 
(parameter)g(to)g Fo(rfftwnd_mpi)p Fu(,)150 1127 y(just)g(to)h(add)f(a)
7355
 
g(little)g(spice.\))390 1293 y Fo(#include)46 b(<rfftw_mpi.h>)390
 
7412
TeXDict begin 45 46 bop 150 -116 a Fv(Chapter)30 b(4:)41
 
7413
b(P)m(arallel)29 b(FFTW)2435 b(45)150 299 y(complicated)34
 
7414
b(data)i(formats)f(in)e(the)i(calling)f(program.)53 b(Before)36
 
7415
b(reading)e(this)g(section,)i(y)m(ou)f(should)150 408
 
7416
y(de\014nitely)27 b(understand)g(ho)m(w)i(to)h(call)e(the)h(unipro)s
 
7417
(cessor)e Fp(rfftwnd)g Fv(functions)g(and)h(also)h(the)g(complex)150
 
7418
518 y(MPI)h(FFTW)h(functions.)275 689 y(The)26 b(follo)m(wing)f(is)h
 
7419
(an)h(example)g(of)g(a)g(program)g(using)e Fp(rfftwnd_mpi)p
 
7420
Fv(.)37 b(It)27 b(computes)g(the)g(size)g Fp(nx)i(x)150
 
7421
799 y(ny)h(x)g(nz)g Fv(transform)g(of)h(a)h(real)e(function)g
 
7422
Fp(f\(x,y,z\))p Fv(,)f(m)m(ultiplies)f(the)j(imaginary)e(part)i(b)m(y)g
 
7423
Fp(2)f Fv(for)h(fun,)150 908 y(then)i(computes)g(the)g(in)m(v)m(erse)g
 
7424
(transform.)48 b(\(W)-8 b(e'll)33 b(also)g(use)f Fp
 
7425
(FFTW_TRANSPOSED_ORDER)c Fv(output)k(for)150 1018 y(the)d(transform,)g
 
7426
(and)g(additionally)d(supply)h(the)j(optional)e(w)m(orkspace)i
 
7427
(parameter)g(to)g Fp(rfftwnd_mpi)p Fv(,)150 1127 y(just)g(to)h(add)f(a)
 
7428
g(little)g(spice.\))390 1293 y Fp(#include)46 b(<rfftw_mpi.h>)390
7356
7429
1500 y(int)h(main\(int)e(argc,)i(char)f(**argv\))390
7357
7430
1604 y({)629 1708 y(const)g(int)h(nx)g(=)h(...,)e(ny)h(=)h(...,)e(nz)i
7358
7431
(=)f(...;)629 1812 y(int)g(local_nx,)e(local_x_start,)f
7381
7454
5132 y(for)g(\(z)g(=)h(0;)f(z)g(<)h(nz;)f(++z\))1774
7382
7455
5236 y(data[\(x*ny)e(+)j(y\))f(*)g(\(2*\(nz/2+1\)\))e(+)i(z])2156
7383
7456
5340 y(=)g(f\(x)g(+)h(local_x_start,)c(y,)j(z\);)p eop
 
7457
end
7384
7458
%%Page: 46 48
7385
 
46 47 bop 150 -116 a Fu(46)3232 b(FFTW)629 403 y Fo(/*)47
7386
 
b(Now,)f(compute)g(the)h(forward)f(transform:)f(*/)629
 
7459
TeXDict begin 46 47 bop 150 -116 a Fv(46)3232 b(FFTW)629
 
7460
403 y Fp(/*)47 b(Now,)f(compute)g(the)h(forward)f(transform:)f(*/)629
7387
7461
506 y(rfftwnd_mpi\(plan,)e(1,)k(data,)f(work,)h
7388
7462
(FFTW_TRANSPOSED_ORDER\);)629 714 y(/*)g(the)g(data)f(is)i(now)e
7389
7463
(complex,)g(so)h(typecast)f(a)h(pointer:)f(*/)629 818
7401
7475
2271 y(free\(data\);)629 2374 y(free\(work\);)629 2478
7402
7476
y(rfftwnd_mpi_destroy_pla)o(n\(pl)o(an\))o(;)629 2582
7403
7477
y(rfftwnd_mpi_destroy_pla)o(n\(ip)o(lan)o(\);)629 2686
7404
 
y(MPI_Finalize\(\);)390 2790 y(})275 2946 y Fu(There's)22
 
7478
y(MPI_Finalize\(\);)390 2790 y(})275 2946 y Fv(There's)22
7405
7479
b(a)i(lot)f(of)g(stu\013)g(in)f(this)g(example,)j(but)d(it's)h(all)f
7406
7480
(just)h(what)g(y)m(ou)g(w)m(ould)f(ha)m(v)m(e)j(guessed,)f(righ)m(t?)
7407
 
150 3056 y(W)-8 b(e)25 b(replaced)f(all)f(the)h Fo(fftwnd_mpi*)d
7408
 
Fu(functions)h(b)m(y)i Fo(rfftwnd_mpi*)p Fu(,)f(but)g(otherwise)g(the)i
 
7481
150 3056 y(W)-8 b(e)25 b(replaced)f(all)f(the)h Fp(fftwnd_mpi*)d
 
7482
Fv(functions)h(b)m(y)i Fp(rfftwnd_mpi*)p Fv(,)f(but)g(otherwise)g(the)i
7409
7483
(parameters)150 3165 y(w)m(ere)32 b(prett)m(y)h(m)m(uc)m(h)e(the)h
7410
7484
(same.)46 b(The)31 b(data)i(la)m(y)m(out)f(distributed)d(among)j(the)g
7411
7485
(pro)s(cesses)g(just)f(lik)m(e)g(for)150 3275 y(the)38
7415
7489
(is)f(for)i(the)g(unipro)s(cessor)d(in-place)i(real)g(transforms)150
7416
7490
3494 y(\(see)31 b(Section)f(3.5.3)j([Arra)m(y)d(Dimensions)f(for)h
7417
7491
(Real)g(Multi-dimensional)d(T)-8 b(ransforms],)30 b(page)h(32\).)42
7418
 
b(In)150 3604 y(particular,)26 b(the)h Fo(z)f Fu(dimension)e(of)j(the)g
 
7492
b(In)150 3604 y(particular,)26 b(the)h Fp(z)f Fv(dimension)e(of)j(the)g
7419
7493
(real)f(input)e(data)k(is)d(padded)h(to)h(a)g(size)g
7420
 
Fo(2*\(nz/2+1\))p Fu(,)d(and)i(after)150 3713 y(the)31
7421
 
b(transform)e(it)h(con)m(tains)h Fo(nz/2+1)d Fu(complex)i(v)-5
 
7494
Fp(2*\(nz/2+1\))p Fv(,)d(and)i(after)150 3713 y(the)31
 
7495
b(transform)e(it)h(con)m(tains)h Fp(nz/2+1)d Fv(complex)i(v)-5
7422
7496
b(alues.)275 3870 y(Some)30 b(other)h(imp)s(ortan)m(t)e(things)g(to)i
7423
7497
(kno)m(w)g(ab)s(out)f(the)h(real)f(MPI)g(transforms:)225
7424
 
4027 y Fs(\017)60 b Fu(As)36 b(for)g(the)g(unipro)s(cessor)e
7425
 
Fo(rfftwnd_create_plan)p Fu(,)e(the)37 b(dimensions)c(passed)i(for)h
7426
 
(the)g Fo(FFTW_)330 4136 y(COMPLEX_TO_REAL)h Fu(plan)j(are)i(those)g
7427
 
(of)g(the)f Fq(r)-5 b(e)g(al)53 b Fu(data.)74 b(In)41
7428
 
b(particular,)i(ev)m(en)f(when)e Fo(FFTW_)330 4246 y(TRANSPOSED_ORDER)
7429
 
32 b Fu(is)j(used)g(as)h(in)f(this)g(case,)k(the)d(dimensions)e(are)i
 
7498
4027 y Ft(\017)60 b Fv(As)36 b(for)g(the)g(unipro)s(cessor)e
 
7499
Fp(rfftwnd_create_plan)p Fv(,)e(the)37 b(dimensions)c(passed)i(for)h
 
7500
(the)g Fp(FFTW_)330 4136 y(COMPLEX_TO_REAL)h Fv(plan)j(are)i(those)g
 
7501
(of)g(the)f Fr(r)-5 b(e)g(al)53 b Fv(data.)74 b(In)41
 
7502
b(particular,)i(ev)m(en)f(when)e Fp(FFTW_)330 4246 y(TRANSPOSED_ORDER)
 
7503
32 b Fv(is)j(used)g(as)h(in)f(this)g(case,)k(the)d(dimensions)e(are)i
7430
7504
(those)h(of)f(the)g(\(un)m(trans-)330 4355 y(p)s(osed\))29
7431
7505
b(real)h(output,)g(not)g(the)g(\(transp)s(osed\))g(complex)g(input.)38
7432
7506
b(\(F)-8 b(or)31 b(the)f(complex)g(MPI)g(trans-)330 4465
7433
7507
y(forms,)g(on)g(the)h(other)f(hand,)g(the)h(dimensions)c(are)k(alw)m(a)
7434
7508
m(ys)g(those)g(of)f(the)h(input)d(arra)m(y)-8 b(.\))225
7435
 
4610 y Fs(\017)60 b Fu(The)24 b(output)h(ordering)e(of)i(the)g
7436
 
(transform)f(\()p Fo(FFTW_TRANSPOSED_ORDER)c Fu(or)k
7437
 
Fo(FFTW_TRANSPOSED_)330 4720 y(ORDER)p Fu(\))39 b Fq(must)50
7438
 
b Fu(b)s(e)40 b(the)h(same)g(for)f(b)s(oth)g(forw)m(ard)g(and)g(bac)m
 
7509
4610 y Ft(\017)60 b Fv(The)24 b(output)h(ordering)e(of)i(the)g
 
7510
(transform)f(\()p Fp(FFTW_TRANSPOSED_ORDER)c Fv(or)k
 
7511
Fp(FFTW_TRANSPOSED_)330 4720 y(ORDER)p Fv(\))39 b Fr(must)50
 
7512
b Fv(b)s(e)40 b(the)h(same)g(for)f(b)s(oth)g(forw)m(ard)g(and)g(bac)m
7439
7513
(kw)m(ard)h(transforms.)70 b(\(This)39 b(is)h(not)330
7440
7514
4830 y(required)29 b(in)g(the)h(complex)g(case.\))225
7441
 
4975 y Fs(\017)60 b Fo(total_local_size)24 b Fu(is)k(the)h(required)d
7442
 
(size)j(in)e Fo(fftw_real)f Fu(v)-5 b(alues,)29 b(not)f
7443
 
Fo(fftw_complex)e Fu(v)-5 b(alues)330 5085 y(as)31 b(it)f(is)f(for)h
7444
 
(the)h(complex)f(transforms.)225 5230 y Fs(\017)60 b
7445
 
Fo(local_ny_after_transpose)20 b Fu(and)26 b Fo
7446
 
(local_y_start_after_tra)o(nspo)o(se)20 b Fu(describ)s(e)25
 
7515
4975 y Ft(\017)60 b Fp(total_local_size)24 b Fv(is)k(the)h(required)d
 
7516
(size)j(in)e Fp(fftw_real)f Fv(v)-5 b(alues,)29 b(not)f
 
7517
Fp(fftw_complex)e Fv(v)-5 b(alues)330 5085 y(as)31 b(it)f(is)f(for)h
 
7518
(the)h(complex)f(transforms.)225 5230 y Ft(\017)60 b
 
7519
Fp(local_ny_after_transpose)20 b Fv(and)26 b Fp
 
7520
(local_y_start_after_tra)o(nspo)o(se)20 b Fv(describ)s(e)25
7447
7521
b(the)i(p)s(or-)330 5340 y(tion)21 b(of)h(the)h(arra)m(y)f(after)g(the)
7448
7522
h(transform;)h(that)e(is,)h(they)f(are)h(indices)d(in)g(the)i(complex)g
7449
 
(arra)m(y)g(for)g(an)p eop
 
7523
(arra)m(y)g(for)g(an)p eop end
7450
7524
%%Page: 47 49
7451
 
47 48 bop 150 -116 a Fu(Chapter)30 b(4:)41 b(P)m(arallel)29
7452
 
b(FFTW)2435 b(47)330 299 y Fo(FFTW_REAL_TO_COMPLEX)19
7453
 
b Fu(transform)k(and)g(in)g(the)h(real)g(arra)m(y)h(for)f(an)f
7454
 
Fo(FFTW_COMPLEX_TO_REAL)330 408 y Fu(transform.)225 544
7455
 
y Fs(\017)60 b Fo(rfftwnd_mpi)21 b Fu(alw)m(a)m(ys)j(exp)s(ects)g
7456
 
Fo(fftw_real*)d Fu(arra)m(y)k(argumen)m(ts,)h(but)d(of)h(course)g
7457
 
(these)g(p)s(oin)m(ters)330 654 y(can)29 b(refer)f(to)h(either)f(real)g
 
7525
TeXDict begin 47 48 bop 150 -116 a Fv(Chapter)30 b(4:)41
 
7526
b(P)m(arallel)29 b(FFTW)2435 b(47)330 299 y Fp(FFTW_REAL_TO_COMPLEX)19
 
7527
b Fv(transform)k(and)g(in)g(the)h(real)g(arra)m(y)h(for)f(an)f
 
7528
Fp(FFTW_COMPLEX_TO_REAL)330 408 y Fv(transform.)225 541
 
7529
y Ft(\017)60 b Fp(rfftwnd_mpi)21 b Fv(alw)m(a)m(ys)j(exp)s(ects)g
 
7530
Fp(fftw_real*)d Fv(arra)m(y)k(argumen)m(ts,)h(but)d(of)h(course)g
 
7531
(these)g(p)s(oin)m(ters)330 651 y(can)29 b(refer)f(to)h(either)f(real)g
7458
7532
(or)h(complex)f(arra)m(ys,)i(dep)s(ending)c(up)s(on)h(whic)m(h)g(side)g
7459
 
(of)i(the)g(transform)330 764 y(y)m(ou)i(are)h(on.)42
 
7533
(of)i(the)g(transform)330 760 y(y)m(ou)i(are)h(on.)42
7460
7534
b(Just)30 b(as)i(for)e(in-place)g(unipro)s(cessor)f(real)i(transforms)f
7461
 
(\(and)h(also)f(in)g(the)h(example)330 873 y(ab)s(o)m(v)m(e\),)f(this)d
 
7535
(\(and)h(also)f(in)g(the)h(example)330 870 y(ab)s(o)m(v)m(e\),)f(this)d
7462
7536
(is)g(most)i(easily)e(handled)f(b)m(y)i(t)m(yp)s(ecasting)g(to)h(a)g
7463
 
(complex)e(p)s(oin)m(ter)h(when)f(handling)330 983 y(the)k(complex)f
7464
 
(data.)225 1119 y Fs(\017)60 b Fu(As)33 b(with)e(the)i(complex)g
7465
 
(transforms,)g(there)g(are)g(also)f Fo(rfftwnd_create_plan)c
7466
 
Fu(and)k Fo(rfftw2d_)330 1228 y(create_plan)27 b Fu(functions,)i(and)h
 
7537
(complex)e(p)s(oin)m(ter)h(when)f(handling)330 979 y(the)k(complex)f
 
7538
(data.)225 1112 y Ft(\017)60 b Fv(As)33 b(with)e(the)i(complex)g
 
7539
(transforms,)g(there)g(are)g(also)f Fp(rfftwnd_create_plan)c
 
7540
Fv(and)k Fp(rfftw2d_)330 1221 y(create_plan)27 b Fv(functions,)i(and)h
7467
7541
(an)m(y)h(rank)f(greater)h(than)g(one)f(is)g(supp)s(orted.)275
7468
 
1392 y(Programs)c(using)e(the)j(MPI)f(FFTW)h(real)f(transforms)f
7469
 
(should)g(link)f(with)h Fo(-lrfftw_mpi)i(-lfftw_)150
7470
 
1502 y(mpi)i(-lrfftw)g(-lfftw)g(-lm)g Fu(on)h(Unix.)150
7471
 
1714 y Fi(4.2.5)63 b(Usage)41 b(of)g(MPI)g(FFTW)g(for)g(Complex)f
7472
 
(One-dimensional)464 1838 y(T)-10 b(ransforms)275 2034
7473
 
y Fu(The)53 b(MPI)h(FFTW)g(also)g(includes)e(routines)g(for)i(parallel)
7474
 
e(one-dimensional)g(transforms)h(of)150 2143 y(complex)40
7475
 
b(data)h(\(only\).)72 b(Although)39 b(the)i(sp)s(eedup)e(is)g
7476
 
(generally)h(w)m(orse)h(than)f(it)g(is)g(for)g(the)h(m)m(ulti-)150
7477
 
2253 y(dimensional)26 b(routines,)994 2220 y Fn(2)1059
7478
 
2253 y Fu(these)j(distributed-memory)c(one-dimensional)h(transforms)i
7479
 
(are)h(esp)s(ecially)150 2363 y(useful)d(for)i(p)s(erforming)e
7480
 
(one-dimensional)g(transforms)h(that)i(don't)f(\014t)f(in)m(to)h(the)h
7481
 
(memory)f(of)g(a)g(single)150 2472 y(mac)m(hine.)275
7482
 
2610 y(The)45 b(usage)i(of)f(these)g(routines)f(is)g(straigh)m(tforw)m
 
7542
1377 y(Programs)c(using)e(the)j(MPI)f(FFTW)h(real)f(transforms)f
 
7543
(should)g(link)f(with)h Fp(-lrfftw_mpi)i(-lfftw_)150
 
7544
1487 y(mpi)i(-lrfftw)g(-lfftw)g(-lm)g Fv(on)h(Unix.)150
 
7545
1687 y Fh(4.2.5)63 b(Usage)41 b(of)g(MPI)g(FFTW)g(for)g(Complex)f
 
7546
(One-dimensional)464 1811 y(T)-10 b(ransforms)275 2002
 
7547
y Fv(The)67 b(MPI)h(FFTW)h(also)f(includes)d(routines)i(for)h(parallel)
 
7548
e(one-dimensional)g(transforms)150 2111 y(of)51 b(complex)g(data)g
 
7549
(\(only\).)103 b(Although)50 b(the)h(sp)s(eedup)e(is)h(generally)g(w)m
 
7550
(orse)i(than)e(it)h(is)f(for)h(the)150 2221 y(m)m(ulti-dimensional)g
 
7551
(routines,)1260 2188 y Fo(2)1358 2221 y Fv(these)56 b
 
7552
(distributed-memory)d(one-dimensional)g(transforms)i(are)150
 
7553
2331 y(esp)s(ecially)28 b(useful)g(for)h(p)s(erforming)f
 
7554
(one-dimensional)f(transforms)i(that)h(don't)g(\014t)g(in)m(to)f(the)h
 
7555
(memory)150 2440 y(of)h(a)f(single)f(mac)m(hine.)275
 
7556
2573 y(The)45 b(usage)i(of)f(these)g(routines)f(is)g(straigh)m(tforw)m
7483
7557
(ard,)50 b(and)c(is)f(similar)e(to)k(that)g(of)f(the)g(m)m(ulti-)150
7484
 
2719 y(dimensional)41 b(MPI)i(transform)g(functions.)79
7485
 
b(Y)-8 b(ou)45 b(\014rst)e(include)e(the)j(header)f Fo(<fftw_mpi.h>)e
7486
 
Fu(and)150 2829 y(then)30 b(create)i(a)f(plan)e(b)m(y)h(calling:)390
7487
 
2960 y Fo(fftw_mpi_plan)44 b(fftw_mpi_create_plan\(MPI_)o(Comm)d(comm,)
7488
 
46 b(int)h(n,)2060 3064 y(fftw_direction)d(dir,)j(int)g(flags\);)275
7489
 
3202 y Fu(The)32 b(last)i(three)f(argumen)m(ts)h(are)f(the)h(same)g(as)
7490
 
f(for)g Fo(fftw_create_plan)c Fu(\(except)35 b(that)f(all)e(MPI)150
7491
 
3311 y(transforms)k(are)h(automatically)g Fo(FFTW_IN_PLACE)p
7492
 
Fu(\).)58 b(The)36 b(\014rst)g(argumen)m(t)i(sp)s(eci\014es)d(the)i
7493
 
(group)g(of)150 3421 y(pro)s(cesses)26 b(y)m(ou)h(are)g(using,)f(and)g
7494
 
(is)f(usually)g Fo(MPI_COMM_WORLD)d Fu(\(all)k(pro)s(cesses\).)39
7495
 
b(A)27 b(plan)e(can)i(b)s(e)f(used)150 3530 y(for)h(man)m(y)f
 
7558
2682 y(dimensional)41 b(MPI)i(transform)g(functions.)79
 
7559
b(Y)-8 b(ou)45 b(\014rst)e(include)e(the)j(header)f Fp(<fftw_mpi.h>)e
 
7560
Fv(and)150 2792 y(then)30 b(create)i(a)f(plan)e(b)m(y)h(calling:)390
 
7561
2919 y Fp(fftw_mpi_plan)44 b(fftw_mpi_create_plan\(MPI_)o(Comm)d(comm,)
 
7562
46 b(int)h(n,)2060 3022 y(fftw_direction)d(dir,)j(int)g(flags\);)275
 
7563
3155 y Fv(The)32 b(last)i(three)f(argumen)m(ts)h(are)f(the)h(same)g(as)
 
7564
f(for)g Fp(fftw_create_plan)c Fv(\(except)35 b(that)f(all)e(MPI)150
 
7565
3265 y(transforms)k(are)h(automatically)g Fp(FFTW_IN_PLACE)p
 
7566
Fv(\).)58 b(The)36 b(\014rst)g(argumen)m(t)i(sp)s(eci\014es)d(the)i
 
7567
(group)g(of)150 3374 y(pro)s(cesses)26 b(y)m(ou)h(are)g(using,)f(and)g
 
7568
(is)f(usually)g Fp(MPI_COMM_WORLD)d Fv(\(all)k(pro)s(cesses\).)39
 
7569
b(A)27 b(plan)e(can)i(b)s(e)f(used)150 3484 y(for)h(man)m(y)f
7496
7570
(transforms)g(of)h(the)g(same)g(size,)h(and)e(is)g(destro)m(y)m(ed)i
7497
7571
(when)d(y)m(ou)i(are)h(done)e(with)f(it)i(b)m(y)f(calling)150
7498
 
3640 y Fo(fftw_mpi_destroy_plan\(pl)o(an\))p Fu(.)275
7499
 
3777 y(If)32 b(y)m(ou)i(don't)f(care)h(ab)s(out)e(the)i(ordering)d(of)j
 
7572
3593 y Fp(fftw_mpi_destroy_plan\(pl)o(an\))p Fv(.)275
 
7573
3726 y(If)32 b(y)m(ou)i(don't)f(care)h(ab)s(out)e(the)i(ordering)d(of)j
7500
7574
(the)f(input)e(or)i(output)g(data)h(of)f(the)g(transform,)h(y)m(ou)150
7501
 
3887 y(can)d(include)d Fo(FFTW_SCRAMBLED_INPUT)d Fu(and/or)30
7502
 
b Fo(FFTW_SCRAMBLED_OUTPUT)24 b Fu(in)29 b(the)i Fo(flags)p
7503
 
Fu(.)39 b(These)150 3997 y(sa)m(v)m(e)34 b(some)e(comm)m(unications)g
 
7575
3835 y(can)d(include)d Fp(FFTW_SCRAMBLED_INPUT)d Fv(and/or)30
 
7576
b Fp(FFTW_SCRAMBLED_OUTPUT)24 b Fv(in)29 b(the)i Fp(flags)p
 
7577
Fv(.)39 b(These)150 3945 y(sa)m(v)m(e)34 b(some)e(comm)m(unications)g
7504
7578
(at)h(the)f(exp)s(ense)g(of)g(ha)m(ving)g(the)g(input)f(and/or)h
7505
 
(output)g(reordered)f(in)150 4106 y(an)e(undo)s(cumen)m(ted)f(w)m(a)m
 
7579
(output)g(reordered)f(in)150 4055 y(an)e(undo)s(cumen)m(ted)f(w)m(a)m
7506
7580
(y)-8 b(.)42 b(F)-8 b(or)29 b(example,)h(if)e(y)m(ou)h(are)h(p)s
7507
7581
(erforming)d(an)i(FFT-based)g(con)m(v)m(olution,)h(y)m(ou)150
7508
 
4216 y(migh)m(t)d(use)f Fo(FFTW_SCRAMBLED_OUTPUT)21 b
7509
 
Fu(for)27 b(the)g(forw)m(ard)f(transform)g(and)g Fo
7510
 
(FFTW_SCRAMBLED_INPUT)150 4325 y Fu(for)k(the)h(in)m(v)m(erse)f
7511
 
(transform.)275 4463 y(The)f(transform)h(itself)f(is)h(computed)g(b)m
7512
 
(y:)390 4594 y Fo(void)47 b(fftw_mpi\(fftw_mpi_plan)41
7513
 
b(p,)48 b(int)e(n_fields,)1058 4698 y(fftw_complex)f(*local_data,)f
7514
 
(fftw_complex)h(*work\);)p 150 4802 1200 4 v 192 4869
7515
 
a Fn(2)275 4902 y Fu(The)35 b(1D)h(transforms)f(require)f(m)m(uc)m(h)i
7516
 
(more)f(comm)m(unication.)56 b(All)34 b(the)i(comm)m(unication)f(in)f
7517
 
(our)275 5011 y(FFT)46 b(routines)g(tak)m(es)i(the)f(form)f(of)g(an)h
7518
 
(all-to-all)f(comm)m(unication:)72 b(the)47 b(m)m(ulti-dimensional)275
7519
 
5121 y(transforms)29 b(require)f(t)m(w)m(o)k(all-to-all)d(comm)m
7520
 
(unications)g(\(or)i(one,)f(if)f(y)m(ou)i(use)e Fo(FFTW_TRANSPOSED_)275
7521
 
5230 y(ORDER)p Fu(\),)c(while)f(the)i(one-dimensional)d(transforms)i
7522
 
(require)g Fq(thr)-5 b(e)g(e)33 b Fu(\(or)26 b(t)m(w)m(o,)j(if)24
7523
 
b(y)m(ou)i(use)g(scram)m(bled)275 5340 y(input)i(or)i(output\).)p
7524
 
eop
 
7582
4164 y(migh)m(t)d(use)f Fp(FFTW_SCRAMBLED_OUTPUT)21 b
 
7583
Fv(for)27 b(the)g(forw)m(ard)f(transform)g(and)g Fp
 
7584
(FFTW_SCRAMBLED_INPUT)150 4274 y Fv(for)k(the)h(in)m(v)m(erse)f
 
7585
(transform.)275 4406 y(The)f(transform)h(itself)f(is)h(computed)g(b)m
 
7586
(y:)390 4533 y Fp(void)47 b(fftw_mpi\(fftw_mpi_plan)41
 
7587
b(p,)48 b(int)e(n_fields,)1058 4637 y(fftw_complex)f(*local_data,)f
 
7588
(fftw_complex)h(*work\);)275 4769 y(n_fields)p Fv(,)c(as)g(in)f
 
7589
Fp(fftwnd_mpi)p Fv(,)h(is)f(equiv)-5 b(alen)m(t)41 b(to)g
 
7590
Fp(howmany=n_fields)p Fv(,)f Fp(stride=n_fields)p Fv(,)150
 
7591
4879 y(and)h Fp(dist=1)p Fv(,)i(and)e(should)f(b)s(e)h
 
7592
Fp(1)g Fv(when)g(y)m(ou)h(are)g(computing)f(the)h(transform)f(of)g(a)h
 
7593
(single)f(arra)m(y)-8 b(.)p 150 4967 1200 4 v 199 5034
 
7594
a Fo(2)275 5066 y Fj(The)33 b(1D)h(transforms)g(require)g(m)n(uc)n(h)e
 
7595
(more)h(comm)n(unication.)57 b(All)34 b(the)f(comm)n(unication)g(in)h
 
7596
(our)f(FFT)i(routines)275 5157 y(tak)n(es)c(the)g(form)g(of)h(an)f
 
7597
(all-to-all)i(comm)n(unication:)44 b(the)31 b(m)n(ulti-dimensional)f
 
7598
(transforms)i(require)f(t)n(w)n(o)g(all-to-all)275 5249
 
7599
y(comm)n(unications)36 b(\(or)h(one,)k(if)d(y)n(ou)e(use)i
 
7600
Fi(FFTW_TRANSPOSED_ORDER)p Fj(\),)44 b(while)39 b(the)d
 
7601
(one-dimensional)i(transforms)275 5340 y(require)25 b
 
7602
Ff(thr)l(e)l(e)33 b Fj(\(or)26 b(t)n(w)n(o,)g(if)h(y)n(ou)e(use)h
 
7603
(scram)n(bled)f(input)f(or)i(output\).)p eop end
7525
7604
%%Page: 48 50
7526
 
48 49 bop 150 -116 a Fu(48)3232 b(FFTW)275 299 y Fo(n_fields)p
7527
 
Fu(,)41 b(as)g(in)f Fo(fftwnd_mpi)p Fu(,)h(is)f(equiv)-5
7528
 
b(alen)m(t)41 b(to)g Fo(howmany=n_fields)p Fu(,)f Fo(stride=n_fields)p
7529
 
Fu(,)150 408 y(and)h Fo(dist=1)p Fu(,)i(and)e(should)f(b)s(e)h
7530
 
Fo(1)g Fu(when)g(y)m(ou)h(are)g(computing)f(the)h(transform)f(of)g(a)h
7531
 
(single)f(arra)m(y)-8 b(.)150 518 y Fo(local_data)25
7532
 
b Fu(con)m(tains)j(the)f(p)s(ortion)g(of)g(the)h(arra)m(y)g(lo)s(cal)f
7533
 
(to)i(the)f(curren)m(t)f(pro)s(cess,)h(describ)s(ed)d(b)s(elo)m(w.)150
7534
 
628 y Fo(work)37 b Fu(is)f(either)h Fo(NULL)g Fu(or)h(an)f(arra)m(y)h
7535
 
(exactly)h(the)f(same)g(size)g(as)f Fo(local_data)p Fu(;)i(in)e(the)h
7536
 
(latter)g(case,)150 737 y(FFTW)h(can)f(use)g(the)g Fo(MPI_Alltoall)d
7537
 
Fu(comm)m(unications)i(primitiv)m(e)f(whic)m(h)h(is)g(\(usually\))g
7538
 
(faster)h(at)150 847 y(the)30 b(exp)s(ense)g(of)g(extra)h(storage.)42
7539
 
b(Up)s(on)29 b(return,)g Fo(local_data)f Fu(con)m(tains)i(the)g(p)s
7540
 
(ortion)f(of)h(the)g(output)150 956 y(lo)s(cal)g(to)h(the)f(curren)m(t)
7541
 
g(pro)s(cess)g(\(see)i(b)s(elo)m(w\).)275 1104 y(T)-8
 
7605
TeXDict begin 48 49 bop 150 -116 a Fv(48)3232 b(FFTW)150
 
7606
299 y Fp(local_data)25 b Fv(con)m(tains)j(the)f(p)s(ortion)g(of)g(the)h
 
7607
(arra)m(y)g(lo)s(cal)f(to)i(the)f(curren)m(t)f(pro)s(cess,)h(describ)s
 
7608
(ed)d(b)s(elo)m(w.)150 408 y Fp(work)37 b Fv(is)f(either)h
 
7609
Fp(NULL)g Fv(or)h(an)f(arra)m(y)h(exactly)h(the)f(same)g(size)g(as)f
 
7610
Fp(local_data)p Fv(;)i(in)e(the)h(latter)g(case,)150
 
7611
518 y(FFTW)h(can)f(use)g(the)g Fp(MPI_Alltoall)d Fv(comm)m(unications)i
 
7612
(primitiv)m(e)f(whic)m(h)h(is)g(\(usually\))g(faster)h(at)150
 
7613
628 y(the)30 b(exp)s(ense)g(of)g(extra)h(storage.)42
 
7614
b(Up)s(on)29 b(return,)g Fp(local_data)f Fv(con)m(tains)i(the)g(p)s
 
7615
(ortion)f(of)h(the)g(output)150 737 y(lo)s(cal)g(to)h(the)f(curren)m(t)
 
7616
g(pro)s(cess)g(\(see)i(b)s(elo)m(w\).)275 870 y(T)-8
7542
7617
b(o)29 b(\014nd)f(out)i(what)f(p)s(ortion)f(of)h(the)h(arra)m(y)f(is)g
7543
7618
(stored)g(lo)s(cal)g(to)h(the)f(curren)m(t)g(pro)s(cess,)h(y)m(ou)f
7544
 
(call)g(the)150 1213 y(follo)m(wing)g(routine:)390 1355
7545
 
y Fo(void)47 b(fftw_mpi_local_sizes\(ff)o(tw_m)o(pi_)o(plan)41
7546
 
b(p,)1631 1459 y(int)47 b(*local_n,)e(int)i(*local_start,)1631
7547
 
1563 y(int)g(*local_n_after_transform)o(,)1631 1666 y(int)g
7548
 
(*local_start_after_trans)o(for)o(m,)1631 1770 y(int)g
7549
 
(*total_local_size\);)275 1918 y(total_local_size)17
7550
 
b Fu(is)j(the)i(n)m(um)m(b)s(er)f(of)g Fo(fftw_complex)e
7551
 
Fu(elemen)m(ts)j(y)m(ou)g(should)d(actually)i(allo)s(cate)150
7552
 
2027 y(for)34 b Fo(local_data)e Fu(\(and)i Fo(work)p
7553
 
Fu(\).)52 b Fo(local_n)33 b Fu(and)h Fo(local_start)d
7554
 
Fu(indicate)i(that)i(the)g(curren)m(t)f(pro)s(cess)150
7555
 
2137 y(stores)23 b Fo(local_n)d Fu(elemen)m(ts)j(corresp)s(onding)e(to)
7556
 
i(the)g(indices)d Fo(local_start)g Fu(to)j Fo(local_start+local_)150
7557
 
2246 y(n-1)35 b Fu(in)f(the)i(\\real")g(arra)m(y)-8 b(.)58
7558
 
b Fq(After)37 b(the)h(tr)-5 b(ansform,)41 b(the)d(pr)-5
 
7619
(call)g(the)150 980 y(follo)m(wing)g(routine:)390 1107
 
7620
y Fp(void)47 b(fftw_mpi_local_sizes\(ff)o(tw_m)o(pi_)o(plan)41
 
7621
b(p,)1631 1211 y(int)47 b(*local_n,)e(int)i(*local_start,)1631
 
7622
1314 y(int)g(*local_n_after_transform)o(,)1631 1418 y(int)g
 
7623
(*local_start_after_trans)o(for)o(m,)1631 1522 y(int)g
 
7624
(*total_local_size\);)275 1655 y(total_local_size)17
 
7625
b Fv(is)j(the)i(n)m(um)m(b)s(er)f(of)g Fp(fftw_complex)e
 
7626
Fv(elemen)m(ts)j(y)m(ou)g(should)d(actually)i(allo)s(cate)150
 
7627
1765 y(for)34 b Fp(local_data)e Fv(\(and)i Fp(work)p
 
7628
Fv(\).)52 b Fp(local_n)33 b Fv(and)h Fp(local_start)d
 
7629
Fv(indicate)i(that)i(the)g(curren)m(t)f(pro)s(cess)150
 
7630
1874 y(stores)23 b Fp(local_n)d Fv(elemen)m(ts)j(corresp)s(onding)e(to)
 
7631
i(the)g(indices)d Fp(local_start)g Fv(to)j Fp(local_start+local_)150
 
7632
1984 y(n-1)35 b Fv(in)f(the)i(\\real")g(arra)m(y)-8 b(.)58
 
7633
b Fr(After)37 b(the)h(tr)-5 b(ansform,)41 b(the)d(pr)-5
7559
7634
b(o)g(c)g(ess)40 b(may)e(stor)-5 b(e)39 b(a)f(di\013er)-5
7560
 
b(ent)38 b(p)-5 b(ortion)40 b(of)150 2356 y(the)f(arr)-5
7561
 
b(ay.)60 b Fu(The)36 b(p)s(ortion)f(of)i(the)g(data)g(stored)f(on)h
 
7635
b(ent)38 b(p)-5 b(ortion)40 b(of)150 2093 y(the)f(arr)-5
 
7636
b(ay.)60 b Fv(The)36 b(p)s(ortion)f(of)i(the)g(data)g(stored)f(on)h
7562
7637
(the)g(pro)s(cess)f(after)h(the)f(transform)g(is)g(giv)m(en)g(b)m(y)150
7563
 
2466 y Fo(local_n_after_transform)23 b Fu(and)29 b Fo
7564
 
(local_start_after_transfor)o(m)p Fu(.)34 b(This)28 b(data)j(is)d
7565
 
(exactly)j(the)150 2575 y(same)g(as)h(a)f(con)m(tiguous)g(segmen)m(t)h
 
7638
2203 y Fp(local_n_after_transform)23 b Fv(and)29 b Fp
 
7639
(local_start_after_transfor)o(m)p Fv(.)34 b(This)28 b(data)j(is)d
 
7640
(exactly)j(the)150 2312 y(same)g(as)h(a)f(con)m(tiguous)g(segmen)m(t)h
7566
7641
(of)f(the)h(corresp)s(onding)d(unipro)s(cessor)f(transform)j(output)f
7567
 
(\(i.e.)43 b(an)150 2685 y(in-order)29 b(sequence)i(of)f(sequen)m(tial)
7568
 
g(frequency)g(bins\).)275 2832 y(Note)35 b(that,)g(if)e(y)m(ou)h
 
7642
(\(i.e.)43 b(an)150 2422 y(in-order)29 b(sequence)i(of)f(sequen)m(tial)
 
7643
g(frequency)g(bins\).)275 2555 y(Note)35 b(that,)g(if)e(y)m(ou)h
7569
7644
(compute)h(b)s(oth)e(a)h(forw)m(ard)f(and)h(a)g(bac)m(kw)m(ard)g
7570
 
(transform)f(of)h(the)h(same)f(size,)150 2942 y(the)39
 
7645
(transform)f(of)h(the)h(same)f(size,)150 2665 y(the)39
7571
7646
b(lo)s(cal)f(sizes)h(are)g(guaran)m(teed)h(to)f(b)s(e)g(consisten)m(t.)
7572
7647
66 b(That)39 b(is,)h(the)f(lo)s(cal)f(size)h(after)g(the)g(forw)m(ard)
7573
 
150 3051 y(transform)28 b(will)f(b)s(e)h(the)h(same)g(as)h(the)f(lo)s
 
7648
150 2774 y(transform)28 b(will)f(b)s(e)h(the)h(same)g(as)h(the)f(lo)s
7574
7649
(cal)f(size)h(b)s(efore)f(the)h(bac)m(kw)m(ard)h(transform,)f(and)f
7575
 
(vice)h(v)m(ersa.)275 3199 y(Programs)c(using)f(the)h(FFTW)h(MPI)g
7576
 
(routines)e(should)f(b)s(e)i(link)m(ed)f(with)g Fo(-lfftw_mpi)j(-lfftw)
7577
 
i(-lm)150 3308 y Fu(on)h(Unix,)g(in)f(addition)f(to)j(whatev)m(er)h
7578
 
(libraries)27 b(are)k(required)d(for)j(MPI.)150 3558
7579
 
y Fi(4.2.6)63 b(MPI)41 b(Tips)275 3764 y Fu(There)27
 
7650
(vice)h(v)m(ersa.)275 2907 y(Programs)c(using)f(the)h(FFTW)h(MPI)g
 
7651
(routines)e(should)f(b)s(e)i(link)m(ed)f(with)g Fp(-lfftw_mpi)j(-lfftw)
 
7652
i(-lm)150 3017 y Fv(on)h(Unix,)g(in)f(addition)f(to)j(whatev)m(er)h
 
7653
(libraries)27 b(are)k(required)d(for)j(MPI.)150 3236
 
7654
y Fh(4.2.6)63 b(MPI)41 b(Tips)275 3427 y Fv(There)27
7580
7655
b(are)h(sev)m(eral)g(things)f(y)m(ou)h(should)e(consider)g(in)h(order)g
7581
7656
(to)h(get)h(the)f(b)s(est)g(p)s(erformance)f(out)h(of)150
7582
 
3874 y(the)j(MPI)f(FFTW)h(routines.)275 4021 y(First,)39
 
7657
3537 y(the)j(MPI)f(FFTW)h(routines.)275 3670 y(First,)39
7583
7658
b(if)f(p)s(ossible,)g(the)g(\014rst)g(and)g(second)g(dimensions)e(of)i
7584
7659
(y)m(our)g(data)h(should)e(b)s(e)h(divisible)c(b)m(y)150
7585
 
4131 y(the)43 b(n)m(um)m(b)s(er)f(of)h(pro)s(cesses)g(y)m(ou)g(are)g
 
7660
3779 y(the)43 b(n)m(um)m(b)s(er)f(of)h(pro)s(cesses)g(y)m(ou)g(are)g
7586
7661
(using.)77 b(\(If)43 b(only)f(one)i(can)f(b)s(e)f(divisible,)h(then)g
7587
 
(y)m(ou)g(should)150 4240 y(c)m(ho)s(ose)29 b(the)f(\014rst)f
 
7662
(y)m(ou)g(should)150 3889 y(c)m(ho)s(ose)29 b(the)f(\014rst)f
7588
7663
(dimension.\))37 b(This)26 b(allo)m(ws)h(the)h(computational)g(load)f
7589
 
(to)i(b)s(e)e(spread)g(ev)m(enly)g(among)150 4350 y(the)32
 
7664
(to)i(b)s(e)e(spread)g(ev)m(enly)g(among)150 3999 y(the)32
7590
7665
b(pro)s(cesses,)h(and)e(also)h(reduces)g(the)g(comm)m(unications)f
7591
7666
(complexit)m(y)h(and)f(o)m(v)m(erhead.)47 b(In)31 b(the)h(one-)150
7592
 
4459 y(dimensional)39 b(transform)h(case,)46 b(the)41
 
7667
4108 y(dimensional)39 b(transform)h(case,)46 b(the)41
7593
7668
b(size)g(of)h(the)g(transform)e(should)g(ideally)f(b)s(e)i(divisible)c
7594
 
(b)m(y)42 b(the)150 4569 y Fq(squar)-5 b(e)38 b Fu(of)30
7595
 
b(the)h(n)m(um)m(b)s(er)e(of)i(pro)s(cessors.)275 4716
 
7669
(b)m(y)42 b(the)150 4218 y Fr(squar)-5 b(e)38 b Fv(of)30
 
7670
b(the)h(n)m(um)m(b)s(er)e(of)i(pro)s(cessors.)275 4351
7596
7671
y(Second,)38 b(y)m(ou)f(should)e(consider)g(using)h(the)h
7597
 
Fo(FFTW_TRANSPOSED_ORDER)30 b Fu(output)37 b(format)g(if)f(it)g(is)150
7598
 
4826 y(not)26 b(to)s(o)g(burdensome.)37 b(The)25 b(sp)s(eed)g(gains)f
 
7672
Fp(FFTW_TRANSPOSED_ORDER)30 b Fv(output)37 b(format)g(if)f(it)g(is)150
 
7673
4460 y(not)26 b(to)s(o)g(burdensome.)37 b(The)25 b(sp)s(eed)g(gains)f
7599
7674
(from)h(comm)m(unications)g(sa)m(vings)g(are)h(usually)d(substan)m
7600
 
(tial.)275 4973 y(Third,)40 b(y)m(ou)h(should)d(consider)h(allo)s
7601
 
(cating)h(a)h(w)m(orkspace)g(for)f Fo(\(r\)fftw\(nd\)_mpi)p
7602
 
Fu(,)f(as)h(this)g(can)150 5083 y(often)31 b(\(but)f(not)g(alw)m(a)m
 
7675
(tial.)275 4593 y(Third,)40 b(y)m(ou)h(should)d(consider)h(allo)s
 
7676
(cating)h(a)h(w)m(orkspace)g(for)f Fp(\(r\)fftw\(nd\)_mpi)p
 
7677
Fv(,)f(as)h(this)g(can)150 4703 y(often)31 b(\(but)f(not)g(alw)m(a)m
7603
7678
(ys\))i(impro)m(v)m(e)e(p)s(erformance)g(\(at)h(the)g(cost)g(of)g
7604
 
(extra)g(storage\).)275 5230 y(F)-8 b(ourth,)50 b(y)m(ou)c(should)e
 
7679
(extra)g(storage\).)275 4836 y(F)-8 b(ourth,)50 b(y)m(ou)c(should)e
7605
7680
(exp)s(erimen)m(t)h(with)g(the)h(b)s(est)g(n)m(um)m(b)s(er)e(of)i(pro)s
7606
 
(cessors)g(to)h(use)e(for)h(y)m(our)150 5340 y(problem.)71
 
7681
(cessors)g(to)h(use)e(for)h(y)m(our)150 4945 y(problem.)71
7607
7682
b(\(There)41 b(comes)h(a)f(p)s(oin)m(t)f(of)i(diminishing)36
7608
 
b(returns,)42 b(when)e(the)i(comm)m(unications)e(costs)p
7609
 
eop
 
7683
b(returns,)42 b(when)e(the)i(comm)m(unications)e(costs)150
 
7684
5055 y(out)m(w)m(eigh)e(the)g(computational)f(b)s(ene\014ts.)1645
 
7685
5022 y Fo(3)1681 5055 y Fv(\))62 b(The)37 b Fp(fftw_mpi_test)d
 
7686
Fv(program)j(can)h(output)f(helpful)p 150 5149 1200 4
 
7687
v 199 5217 a Fo(3)275 5249 y Fj(An)c(FFT)h(is)h(particularly)f(hard)g
 
7688
(on)g(comm)n(unications)e(systems,)k(as)f(it)f(requires)g(an)g
 
7689
Fc(all-to-all)k Fj(comm)n(unication,)275 5340 y(whic)n(h)25
 
7690
b(is)h(more)g(or)g(less)g(the)g(w)n(orst)g(p)r(ossible)h(case.)p
 
7691
eop end
7610
7692
%%Page: 49 51
7611
 
49 50 bop 150 -116 a Fu(Chapter)30 b(4:)41 b(P)m(arallel)29
7612
 
b(FFTW)2435 b(49)150 299 y(out)m(w)m(eigh)38 b(the)g(computational)f(b)
7613
 
s(ene\014ts.)1645 266 y Fn(3)1681 299 y Fu(\))62 b(The)37
7614
 
b Fo(fftw_mpi_test)d Fu(program)j(can)h(output)f(helpful)150
7615
 
408 y(p)s(erformance)23 b(b)s(enc)m(hmarks.)37 b(It)24
7616
 
b(accepts)h(the)e(same)h(parameters)g(as)g(the)g(unipro)s(cessor)d
7617
 
(test)j(programs)150 518 y(\(c.f.)53 b Fo(tests/README)p
7618
 
Fu(\))31 b(and)i(is)g(run)g(lik)m(e)g(an)h(ordinary)f(MPI)h(program.)51
7619
 
b(F)-8 b(or)35 b(example,)g Fo(mpirun)29 b(-np)150 628
7620
 
y(4)h(fftw_mpi_test)d(-s)j(128x128x128)g Fu(will)h(b)s(enc)m(hmark)h(a)
7621
 
i Fo(128x128x128)c Fu(transform)j(on)g(four)g(pro-)150
7622
 
737 y(cessors,)28 b(rep)s(orting)d(timings)g(and)g(parallel)g(sp)s
7623
 
(eedups)f(for)j(all)e(v)-5 b(arian)m(ts)26 b(of)h Fo(fftwnd_mpi)c
7624
 
Fu(\(transp)s(osed,)150 847 y(with)34 b(w)m(orkspace,)k(etcetera\).)60
7625
 
b(\(Note)37 b(also)e(that)i(there)f(is)e(the)i Fo(rfftw_mpi_test)c
7626
 
Fu(program)j(for)h(the)150 956 y(real)30 b(transforms.\))p
7627
 
150 5131 1200 4 v 192 5197 a Fn(3)275 5230 y Fu(An)42
7628
 
b(FFT)h(is)f(particularly)e(hard)i(on)h(comm)m(unications)f(systems,)k
7629
 
(as)d(it)g(requires)e(an)i Fp(all-to-all)275 5340 y Fu(comm)m
7630
 
(unication,)30 b(whic)m(h)f(is)g(more)i(or)f(less)g(the)g(w)m(orst)h(p)
7631
 
s(ossible)d(case.)p eop
 
7693
TeXDict begin 49 50 bop 150 -116 a Fv(Chapter)30 b(4:)41
 
7694
b(P)m(arallel)29 b(FFTW)2435 b(49)150 299 y(p)s(erformance)23
 
7695
b(b)s(enc)m(hmarks.)37 b(It)24 b(accepts)h(the)e(same)h(parameters)g
 
7696
(as)g(the)g(unipro)s(cessor)d(test)j(programs)150 408
 
7697
y(\(c.f.)53 b Fp(tests/README)p Fv(\))31 b(and)i(is)g(run)g(lik)m(e)g
 
7698
(an)h(ordinary)f(MPI)h(program.)51 b(F)-8 b(or)35 b(example,)g
 
7699
Fp(mpirun)29 b(-np)150 518 y(4)h(fftw_mpi_test)d(-s)j(128x128x128)g
 
7700
Fv(will)h(b)s(enc)m(hmark)h(a)i Fp(128x128x128)c Fv(transform)j(on)g
 
7701
(four)g(pro-)150 628 y(cessors,)28 b(rep)s(orting)d(timings)g(and)g
 
7702
(parallel)g(sp)s(eedups)f(for)j(all)e(v)-5 b(arian)m(ts)26
 
7703
b(of)h Fp(fftwnd_mpi)c Fv(\(transp)s(osed,)150 737 y(with)34
 
7704
b(w)m(orkspace,)k(etcetera\).)60 b(\(Note)37 b(also)e(that)i(there)f
 
7705
(is)e(the)i Fp(rfftw_mpi_test)c Fv(program)j(for)h(the)150
 
7706
847 y(real)30 b(transforms.\))p eop end
7632
7707
%%Page: 50 52
7633
 
50 51 bop 150 -116 a Fu(50)3232 b(FFTW)p eop
 
7708
TeXDict begin 50 51 bop 150 -116 a Fv(50)3232 b(FFTW)p
 
7709
eop end
7634
7710
%%Page: 51 53
7635
 
51 52 bop 150 -116 a Fu(Chapter)30 b(5:)41 b(Calling)28
7636
 
b(FFTW)j(from)f(F)-8 b(ortran)1909 b(51)150 299 y Fr(5)80
7637
 
b(Calling)56 b(FFTW)f(from)e(F)-13 b(ortran)275 528 y
7638
 
Fu(The)27 b(standard)g(FFTW)h(libraries)d(include)g(sp)s(ecial)h(wrapp)
7639
 
s(er)g(functions)g(that)j(allo)m(w)e(F)-8 b(ortran)28
7640
 
b(pro-)150 638 y(grams)44 b(to)h(call)f(FFTW)h(subroutines.)79
 
7711
TeXDict begin 51 52 bop 150 -116 a Fv(Chapter)30 b(5:)41
 
7712
b(Calling)28 b(FFTW)j(from)f(F)-8 b(ortran)1909 b(51)150
 
7713
299 y Fs(5)80 b(Calling)56 b(FFTW)f(from)e(F)-13 b(ortran)275
 
7714
503 y Fv(The)27 b(standard)g(FFTW)h(libraries)d(include)g(sp)s(ecial)h
 
7715
(wrapp)s(er)g(functions)g(that)j(allo)m(w)e(F)-8 b(ortran)28
 
7716
b(pro-)150 612 y(grams)44 b(to)h(call)f(FFTW)h(subroutines.)79
7641
7717
b(This)43 b(c)m(hapter)i(describ)s(es)d(ho)m(w)i(those)h(functions)e
7642
 
(ma)m(y)i(b)s(e)150 747 y(emplo)m(y)m(ed)32 b(to)h(use)e(FFTW)i(from)e
 
7718
(ma)m(y)i(b)s(e)150 722 y(emplo)m(y)m(ed)32 b(to)h(use)e(FFTW)i(from)e
7643
7719
(F)-8 b(ortran.)46 b(W)-8 b(e)33 b(assume)f(here)g(that)g(the)g(reader)
7644
 
g(is)f(already)g(familiar)150 857 y(with)e(the)i(usage)g(of)f(FFTW)h
 
7720
g(is)f(already)g(familiar)150 832 y(with)e(the)i(usage)g(of)f(FFTW)h
7645
7721
(in)e(C,)i(as)f(describ)s(ed)e(elsewhere)i(in)f(this)h(man)m(ual.)275
7646
 
991 y(In)h(general,)i(it)e(is)g(not)i(p)s(ossible)c(to)k(call)f(C)f
 
7722
960 y(In)h(general,)i(it)e(is)g(not)i(p)s(ossible)c(to)k(call)f(C)f
7647
7723
(functions)g(directly)g(from)g(F)-8 b(ortran,)34 b(due)d(to)i(F)-8
7648
 
b(ortran's)150 1100 y(inabilit)m(y)33 b(to)k(pass)f(argumen)m(ts)h(b)m
 
7724
b(ortran's)150 1070 y(inabilit)m(y)33 b(to)k(pass)f(argumen)m(ts)h(b)m
7649
7725
(y)f(v)-5 b(alue)35 b(and)h(also)g(b)s(ecause)g(F)-8
7650
7726
b(ortran)37 b(compilers)d(t)m(ypically)h(exp)s(ect)150
7651
 
1210 y(iden)m(ti\014ers)i(to)i(b)s(e)g(mangled)f(someho)m(w)h(for)g
 
7727
1179 y(iden)m(ti\014ers)i(to)i(b)s(e)g(mangled)f(someho)m(w)h(for)g
7652
7728
(linking.)63 b(Ho)m(w)m(ev)m(er,)43 b(if)38 b(C)g(functions)g(are)h
7653
 
(written)f(in)g(a)150 1319 y(sp)s(ecial)28 b(w)m(a)m(y)-8
7654
 
b(,)31 b(they)e Fq(ar)-5 b(e)37 b Fu(callable)28 b(from)h(F)-8
 
7729
(written)f(in)g(a)150 1289 y(sp)s(ecial)28 b(w)m(a)m(y)-8
 
7730
b(,)31 b(they)e Fr(ar)-5 b(e)37 b Fv(callable)28 b(from)h(F)-8
7655
7731
b(ortran,)30 b(and)e(w)m(e)i(ha)m(v)m(e)g(emplo)m(y)m(ed)g(this)e(tec)m
7656
 
(hnique)h(to)g(create)150 1429 y(F)-8 b(ortran-callable)42
 
7732
(hnique)h(to)g(create)150 1398 y(F)-8 b(ortran-callable)42
7657
7733
b(\\wrapp)s(er")f(functions)g(around)g(the)h(main)f(FFTW)h(routines.)75
7658
 
b(These)42 b(wrapp)s(er)150 1538 y(functions)36 b(are)j(included)c(in)h
 
7734
b(These)42 b(wrapp)s(er)150 1508 y(functions)36 b(are)j(included)c(in)h
7659
7735
(the)j(FFTW)f(libraries)d(b)m(y)j(default,)h(unless)d(a)j(F)-8
7660
 
b(ortran)38 b(compiler)f(isn't)150 1648 y(found)29 b(on)h(y)m(our)h
7661
 
(system)f(or)h Fo(--disable-fortran)25 b Fu(is)30 b(included)d(in)i
7662
 
(the)i Fo(configure)d Fu(\015ags.)275 1782 y(As)34 b(a)h(result,)f
 
7736
b(ortran)38 b(compiler)f(isn't)150 1618 y(found)29 b(on)h(y)m(our)h
 
7737
(system)f(or)h Fp(--disable-fortran)25 b Fv(is)30 b(included)d(in)i
 
7738
(the)i Fp(configure)d Fv(\015ags.)275 1746 y(As)34 b(a)h(result,)f
7663
7739
(calling)f(FFTW)i(from)f(F)-8 b(ortran)35 b(requires)e(little)g(more)h
7664
 
(than)h(app)s(ending)c(`)p Fo(_f77)p Fu(')j(to)150 1891
 
7740
(than)h(app)s(ending)c(`)p Fp(_f77)p Fv(')j(to)150 1856
7665
7741
y(the)g(function)f(names)i(and)e(then)h(linking)e(normally)g(with)h
7666
7742
(the)i(FFTW)g(libraries.)49 b(There)34 b(are)h(a)f(few)150
7667
 
2001 y(wrinkles,)28 b(ho)m(w)m(ev)m(er,)k(as)f(w)m(e)g(shall)d(discuss)
7668
 
h(b)s(elo)m(w.)150 2255 y Ft(5.1)68 b(W)-11 b(rapp)t(er)44
7669
 
b(Routines)275 2447 y Fu(All)g(of)j(the)f(unipro)s(cessor)e(and)h(m)m
 
7743
1965 y(wrinkles,)28 b(ho)m(w)m(ev)m(er,)k(as)f(w)m(e)g(shall)d(discuss)
 
7744
h(b)s(elo)m(w.)150 2205 y Fu(5.1)68 b(W)-11 b(rapp)t(er)44
 
7745
b(Routines)275 2391 y Fv(All)g(of)j(the)f(unipro)s(cessor)e(and)h(m)m
7670
7746
(ulti-threaded)g(transform)h(routines)f(ha)m(v)m(e)i(F)-8
7671
 
b(ortran-callable)150 2556 y(wrapp)s(ers,)38 b(except)h(for)f(the)h
 
7747
b(ortran-callable)150 2501 y(wrapp)s(ers,)38 b(except)h(for)f(the)h
7672
7748
(wisdom)d(imp)s(ort/exp)s(ort)h(functions)g(\(since)h(it)f(is)g(not)i
7673
 
(p)s(ossible)c(to)k(ex-)150 2666 y(c)m(hange)c(string)d(and)h(\014le)f
 
7749
(p)s(ossible)c(to)k(ex-)150 2610 y(c)m(hange)c(string)d(and)h(\014le)f
7674
7750
(argumen)m(ts)i(p)s(ortably)e(with)g(F)-8 b(ortran\))35
7675
7751
b(and)e(the)g(sp)s(eci\014c)f(planner)g(routines)150
7676
 
2776 y(\(see)25 b(Section)g(3.2.2)h([Discussion)d(on)h(Sp)s(eci\014c)f
 
7752
2720 y(\(see)25 b(Section)g(3.2.2)h([Discussion)d(on)h(Sp)s(eci\014c)f
7677
7753
(Plans],)i(page)h(20\).)39 b(The)24 b(name)h(of)g(the)f(wrapp)s(er)f
7678
 
(routine)150 2885 y(is)k(the)h(same)g(as)g(that)h(of)e(the)h(corresp)s
7679
 
(onding)e(C)i(routine,)f(but)g(with)g Fo(fftw/fftwnd/rfftw/rfftwn)o(d)
7680
 
150 2995 y Fu(replaced)g(b)m(y)g Fo(fftw_f77/fftwnd_f77/rff)o(tw_)o
7681
 
(f77/)o(rfft)o(wnd)o(_f77)o Fu(.)34 b(F)-8 b(or)28 b(example,)f(in)f(F)
7682
 
-8 b(ortran,)29 b(in-)150 3104 y(stead)41 b(of)f(calling)f
7683
 
Fo(fftw_one)e Fu(y)m(ou)k(w)m(ould)e(call)g Fo(fftw_f77_one)p
7684
 
Fu(.)2464 3071 y Fn(1)2568 3104 y Fu(F)-8 b(or)41 b(the)f(most)h(part,)
7685
 
h(all)d(of)i(the)150 3214 y(argumen)m(ts)31 b(to)g(the)f(functions)f
 
7754
(routine)150 2829 y(is)k(the)h(same)g(as)g(that)h(of)e(the)h(corresp)s
 
7755
(onding)e(C)i(routine,)f(but)g(with)g Fp(fftw/fftwnd/rfftw/rfftwn)o(d)
 
7756
150 2939 y Fv(replaced)g(b)m(y)g Fp(fftw_f77/fftwnd_f77/rff)o(tw_)o
 
7757
(f77/)o(rfft)o(wnd)o(_f77)o Fv(.)34 b(F)-8 b(or)28 b(example,)f(in)f(F)
 
7758
-8 b(ortran,)29 b(in-)150 3049 y(stead)41 b(of)f(calling)f
 
7759
Fp(fftw_one)e Fv(y)m(ou)k(w)m(ould)e(call)g Fp(fftw_f77_one)p
 
7760
Fv(.)2464 3016 y Fo(1)2568 3049 y Fv(F)-8 b(or)41 b(the)f(most)h(part,)
 
7761
h(all)d(of)i(the)150 3158 y(argumen)m(ts)31 b(to)g(the)f(functions)f
7686
7762
(are)i(the)g(same,)g(with)e(the)h(follo)m(wing)f(exceptions:)225
7687
 
3348 y Fs(\017)60 b Fo(plan)22 b Fu(v)-5 b(ariables)23
7688
 
b(\(what)g(w)m(ould)g(b)s(e)g(of)g(t)m(yp)s(e)h Fo(fftw_plan)p
7689
 
Fu(,)f Fo(rfftwnd_plan)p Fu(,)f(etcetera,)27 b(in)22
7690
 
b(C\),)i(m)m(ust)330 3457 y(b)s(e)37 b(declared)h(as)g(a)g(t)m(yp)s(e)g
 
7763
3287 y Ft(\017)60 b Fp(plan)22 b Fv(v)-5 b(ariables)23
 
7764
b(\(what)g(w)m(ould)g(b)s(e)g(of)g(t)m(yp)s(e)h Fp(fftw_plan)p
 
7765
Fv(,)f Fp(rfftwnd_plan)p Fv(,)f(etcetera,)27 b(in)22
 
7766
b(C\),)i(m)m(ust)330 3396 y(b)s(e)37 b(declared)h(as)g(a)g(t)m(yp)s(e)g
7691
7767
(that)g(is)f(the)h(same)h(size)e(as)i(a)f(p)s(oin)m(ter)f(\(address\))g
7692
 
(on)h(y)m(our)g(mac)m(hine.)330 3567 y(\(F)-8 b(ortran)33
 
7768
(on)h(y)m(our)g(mac)m(hine.)330 3506 y(\(F)-8 b(ortran)33
7693
7769
b(has)f(no)g(generic)g(p)s(oin)m(ter)f(t)m(yp)s(e.\))47
7694
 
b(The)32 b(F)-8 b(ortran)33 b Fo(integer)d Fu(t)m(yp)s(e)i(is)f
7695
 
(usually)f(the)j(same)330 3676 y(size)c(as)g(a)h(p)s(oin)m(ter,)f(but)f
 
7770
b(The)32 b(F)-8 b(ortran)33 b Fp(integer)d Fv(t)m(yp)s(e)i(is)f
 
7771
(usually)f(the)j(same)330 3615 y(size)c(as)g(a)h(p)s(oin)m(ter,)f(but)f
7696
7772
(y)m(ou)i(need)f(to)h(b)s(e)e(w)m(ary)h(\(esp)s(ecially)f(on)h(64-bit)h
7697
 
(mac)m(hines\).)40 b(\(Y)-8 b(ou)30 b(could)330 3786
7698
 
y(also)36 b(use)f Fo(integer*4)f Fu(on)i(a)g(32-bit)g(mac)m(hine)g(and)
7699
 
f Fo(integer*8)e Fu(on)j(a)g(64-bit)g(mac)m(hine.\))58
7700
 
b(Ugh.)330 3896 y(\()p Fo(g77)38 b Fu(has)g(a)h(sp)s(ecial)e(t)m(yp)s
7701
 
(e,)k Fo(integer\(kind=7\))p Fu(,)c(that)i(is)e(de\014ned)h(to)h(b)s(e)
7702
 
f(the)h(same)f(size)h(as)g(a)330 4005 y(p)s(oin)m(ter.\))225
7703
 
4139 y Fs(\017)60 b Fu(An)m(y)26 b(function)e(that)i(returns)e(a)i(v)-5
7704
 
b(alue)25 b(\(e.g.)40 b Fo(fftw_create_plan)p Fu(\))22
7705
 
b(is)i(con)m(v)m(erted)j(in)m(to)e(a)h(subrou-)330 4248
 
7773
(mac)m(hines\).)40 b(\(Y)-8 b(ou)30 b(could)330 3725
 
7774
y(also)36 b(use)f Fp(integer*4)f Fv(on)i(a)g(32-bit)g(mac)m(hine)g(and)
 
7775
f Fp(integer*8)e Fv(on)j(a)g(64-bit)g(mac)m(hine.\))58
 
7776
b(Ugh.)330 3835 y(\()p Fp(g77)38 b Fv(has)g(a)h(sp)s(ecial)e(t)m(yp)s
 
7777
(e,)k Fp(integer\(kind=7\))p Fv(,)c(that)i(is)e(de\014ned)h(to)h(b)s(e)
 
7778
f(the)h(same)f(size)h(as)g(a)330 3944 y(p)s(oin)m(ter.\))225
 
7779
4073 y Ft(\017)60 b Fv(An)m(y)26 b(function)e(that)i(returns)e(a)i(v)-5
 
7780
b(alue)25 b(\(e.g.)40 b Fp(fftw_create_plan)p Fv(\))22
 
7781
b(is)i(con)m(v)m(erted)j(in)m(to)e(a)h(subrou-)330 4182
7706
7782
y(tine.)40 b(The)28 b(return)g(v)-5 b(alue)28 b(is)g(con)m(v)m(erted)i
7707
7783
(in)m(to)f(an)f(additional)f(\(\014rst\))i(parameter)g(of)g(the)g
7708
 
(wrapp)s(er)330 4358 y(subroutine.)53 b(\(The)35 b(reason)h(for)f(this)
 
7784
(wrapp)s(er)330 4292 y(subroutine.)53 b(\(The)35 b(reason)h(for)f(this)
7709
7785
f(is)g(that)i(some)g(F)-8 b(ortran)35 b(implemen)m(tations)f(seem)i(to)
7710
 
g(ha)m(v)m(e)330 4467 y(trouble)29 b(with)g(C)h(function)f(return)h(v)
7711
 
-5 b(alues.\))225 4601 y Fs(\017)60 b Fu(When)29 b(p)s(erforming)e
7712
 
(one-dimensional)h Fo(FFTW_IN_PLACE)d Fu(transforms,)k(y)m(ou)h(don't)g
7713
 
(ha)m(v)m(e)g(the)g(op-)330 4711 y(tion)36 b(of)i(passing)d
7714
 
Fo(NULL)h Fu(for)h(the)g Fo(out)f Fu(argumen)m(t)i(\(since)e(there)h
7715
 
(is)f(no)h(w)m(a)m(y)h(to)g(pass)f Fo(NULL)e Fu(from)330
7716
 
4820 y(F)-8 b(ortran\).)69 b(Therefore,)42 b(when)c(p)s(erforming)g
7717
 
(suc)m(h)h(transforms,)i(y)m(ou)f Fq(must)48 b Fu(allo)s(cate)40
7718
 
b(and)f(pass)330 4930 y(a)e(con)m(tiguous)f(scratc)m(h)i(arra)m(y)f(of)
 
7786
g(ha)m(v)m(e)330 4401 y(trouble)29 b(with)g(C)h(function)f(return)h(v)
 
7787
-5 b(alues.\))225 4530 y Ft(\017)60 b Fv(When)29 b(p)s(erforming)e
 
7788
(one-dimensional)h Fp(FFTW_IN_PLACE)d Fv(transforms,)k(y)m(ou)h(don't)g
 
7789
(ha)m(v)m(e)g(the)g(op-)330 4639 y(tion)36 b(of)i(passing)d
 
7790
Fp(NULL)h Fv(for)h(the)g Fp(out)f Fv(argumen)m(t)i(\(since)e(there)h
 
7791
(is)f(no)h(w)m(a)m(y)h(to)g(pass)f Fp(NULL)e Fv(from)330
 
7792
4749 y(F)-8 b(ortran\).)69 b(Therefore,)42 b(when)c(p)s(erforming)g
 
7793
(suc)m(h)h(transforms,)i(y)m(ou)f Fr(must)48 b Fv(allo)s(cate)40
 
7794
b(and)f(pass)330 4859 y(a)e(con)m(tiguous)f(scratc)m(h)i(arra)m(y)f(of)
7719
7795
f(the)h(same)g(size)f(as)h(the)g(transform.)58 b(Note)38
7720
 
b(that)f(for)f(in-place)p 150 5022 1200 4 v 192 5088
7721
 
a Fn(1)275 5121 y Fu(T)-8 b(ec)m(hnically)g(,)34 b(F)-8
7722
 
b(ortran)35 b(77)g(iden)m(ti\014ers)d(are)i(not)g(allo)m(w)m(ed)g(to)h
7723
 
(ha)m(v)m(e)g(more)g(than)e(6)i(c)m(haracters,)h(nor)275
7724
 
5230 y(ma)m(y)23 b(they)f(con)m(tain)h(underscores.)37
7725
 
b(An)m(y)23 b(compiler)e(that)i(enforces)g(this)e(limitation)f(do)s
7726
 
(esn't)i(deserv)m(e)275 5340 y(to)31 b(link)d(to)j(FFTW.)p
7727
 
eop
 
7796
b(that)f(for)f(in-place)330 4968 y(m)m(ulti-dimensional)d(\()p
 
7797
Fp(\(r\)fftwnd)p Fv(\))j(transforms,)j(the)e Fp(out)g
 
7798
Fv(argumen)m(t)g(is)g(ignored,)h(so)g(y)m(ou)g(can)330
 
7799
5078 y(pass)30 b(an)m(ything)g(for)g(that)h(parameter.)p
 
7800
150 5149 1200 4 v 199 5217 a Fo(1)275 5249 y Fj(T)-6
 
7801
b(ec)n(hnically)g(,)24 b(F)-6 b(ortran)24 b(77)h(iden)n(ti\014ers)e
 
7802
(are)i(not)e(allo)n(w)n(ed)j(to)e(ha)n(v)n(e)f(more)g(than)h(6)g(c)n
 
7803
(haracters,)h(nor)f(ma)n(y)f(they)g(con)n(tain)275 5340
 
7804
y(underscores.)34 b(An)n(y)25 b(compiler)g(that)h(enforces)h(this)e
 
7805
(limitation)i(do)r(esn't)f(deserv)n(e)f(to)h(link)f(to)h(FFTW.)p
 
7806
eop end
7728
7807
%%Page: 52 54
7729
 
52 53 bop 150 -116 a Fu(52)3232 b(FFTW)330 299 y(m)m(ulti-dimensional)
7730
 
33 b(\()p Fo(\(r\)fftwnd)p Fu(\))j(transforms,)j(the)e
7731
 
Fo(out)g Fu(argumen)m(t)g(is)g(ignored,)h(so)g(y)m(ou)g(can)330
7732
 
408 y(pass)30 b(an)m(ything)g(for)g(that)h(parameter.)225
7733
 
543 y Fs(\017)60 b Fu(The)37 b(wrapp)s(er)e(routines)h(exp)s(ect)i(m)m
 
7808
TeXDict begin 52 53 bop 150 -116 a Fv(52)3232 b(FFTW)225
 
7809
299 y Ft(\017)60 b Fv(The)37 b(wrapp)s(er)e(routines)h(exp)s(ect)i(m)m
7734
7810
(ulti-dimensional)33 b(arra)m(ys)k(to)h(b)s(e)f(in)f(column-ma)5
7735
 
b(jor)36 b(order,)330 652 y(whic)m(h)c(is)g(the)i(ordinary)d(format)j
 
7811
b(jor)36 b(order,)330 408 y(whic)m(h)c(is)g(the)i(ordinary)d(format)j
7736
7812
(of)f(F)-8 b(ortran)34 b(arra)m(ys.)50 b(They)32 b(do)h(this)g
7737
 
(transparen)m(tly)f(and)h(cost-)330 762 y(lessly)f(simply)g(b)m(y)h
 
7813
(transparen)m(tly)f(and)h(cost-)330 518 y(lessly)f(simply)g(b)m(y)h
7738
7814
(rev)m(ersing)h(the)g(order)f(of)h(the)g(dimensions)d(passed)i(to)i
7739
 
(FFTW,)f(but)f(this)g(has)330 872 y(one)38 b(imp)s(ortan)m(t)f
 
7815
(FFTW,)f(but)f(this)g(has)330 628 y(one)38 b(imp)s(ortan)m(t)f
7740
7816
(consequence)i(for)e(m)m(ulti-dimensional)d(real-complex)k(transforms,)
7741
 
h(discussed)330 981 y(b)s(elo)m(w.)275 1165 y(In)32 b(general,)h(y)m
7742
 
(ou)h(should)d(tak)m(e)j(care)g(to)g(use)e(F)-8 b(ortran)34
 
7817
h(discussed)330 737 y(b)s(elo)m(w.)275 923 y(In)32 b(general,)h(y)m(ou)
 
7818
h(should)d(tak)m(e)j(care)g(to)g(use)e(F)-8 b(ortran)34
7743
7819
b(data)f(t)m(yp)s(es)g(that)h(corresp)s(ond)d(to)j(\(i.e.)49
7744
 
b(are)150 1275 y(the)27 b(same)h(size)f(as\))g(the)h(C)e(t)m(yp)s(es)h
 
7820
b(are)150 1033 y(the)27 b(same)h(size)f(as\))g(the)h(C)e(t)m(yp)s(es)h
7745
7821
(used)g(b)m(y)g(FFTW.)h(If)e(y)m(our)h(C)g(and)f(F)-8
7746
 
b(ortran)28 b(compilers)d(are)j(made)f(b)m(y)150 1385
 
7822
b(ortran)28 b(compilers)d(are)j(made)f(b)m(y)150 1142
7747
7823
y(the)33 b(same)h(v)m(endor,)g(the)f(corresp)s(ondence)f(is)h(usually)d
7748
 
(straigh)m(tforw)m(ard)j(\(i.e.)49 b Fo(integer)31 b
7749
 
Fu(corresp)s(onds)150 1494 y(to)38 b Fo(int)p Fu(,)g
7750
 
Fo(real)e Fu(corresp)s(onds)g(to)h Fo(float)p Fu(,)h(etcetera\).)63
 
7824
(straigh)m(tforw)m(ard)j(\(i.e.)49 b Fp(integer)31 b
 
7825
Fv(corresp)s(onds)150 1252 y(to)38 b Fp(int)p Fv(,)g
 
7826
Fp(real)e Fv(corresp)s(onds)g(to)h Fp(float)p Fv(,)h(etcetera\).)63
7751
7827
b(Suc)m(h)37 b(simple)e(corresp)s(ondences)h(are)i(assumed)150
7752
 
1604 y(in)h(the)h(examples)g(b)s(elo)m(w.)68 b(The)40
 
7828
1362 y(in)h(the)h(examples)g(b)s(elo)m(w.)68 b(The)40
7753
7829
b(examples)g(also)g(assume)f(that)i(FFTW)g(w)m(as)f(compiled)f(in)f
7754
 
(double)150 1713 y(precision)29 b(\(the)h(default\).)150
7755
 
1971 y Ft(5.2)68 b(FFTW)43 b(Constan)l(ts)j(in)f(F)-11
7756
 
b(ortran)275 2163 y Fu(When)44 b(creating)h(plans)e(in)h(FFTW,)h(a)g(n)
 
7830
(double)150 1471 y(precision)29 b(\(the)h(default\).)150
 
7831
1731 y Fu(5.2)68 b(FFTW)43 b(Constan)l(ts)j(in)f(F)-11
 
7832
b(ortran)275 1924 y Fv(When)44 b(creating)h(plans)e(in)h(FFTW,)h(a)g(n)
7757
7833
m(um)m(b)s(er)f(of)h(constan)m(ts)h(are)f(used)f(to)h(sp)s(ecify)e
7758
 
(options,)150 2273 y(suc)m(h)36 b(as)g Fo(FFTW_FORWARD)d
7759
 
Fu(or)j Fo(FFTW_USE_WISDOM)p Fu(.)54 b(The)35 b(same)i(constan)m(ts)g
7760
 
(m)m(ust)f(b)s(e)g(used)f(with)g(the)150 2382 y(wrapp)s(er)28
 
7834
(options,)150 2034 y(suc)m(h)36 b(as)g Fp(FFTW_FORWARD)d
 
7835
Fv(or)j Fp(FFTW_USE_WISDOM)p Fv(.)54 b(The)35 b(same)i(constan)m(ts)g
 
7836
(m)m(ust)f(b)s(e)g(used)f(with)g(the)150 2143 y(wrapp)s(er)28
7761
7837
b(routines,)g(but)h(of)g(course)h(the)f(C)g(header)g(\014les)g(where)g
7762
7838
(the)g(constan)m(ts)i(are)e(de\014ned)f(can't)j(b)s(e)150
7763
 
2492 y(incorp)s(orated)e(directly)g(in)m(to)i(F)-8 b(ortran)31
7764
 
b(co)s(de.)275 2626 y(Instead,)c(w)m(e)g(ha)m(v)m(e)h(placed)e(F)-8
 
7839
2253 y(incorp)s(orated)e(directly)g(in)m(to)i(F)-8 b(ortran)31
 
7840
b(co)s(de.)275 2388 y(Instead,)c(w)m(e)g(ha)m(v)m(e)h(placed)e(F)-8
7765
7841
b(ortran)27 b(equiv)-5 b(alen)m(ts)26 b(of)h(the)g(FFTW)g(constan)m(t)h
7766
 
(de\014nitions)c(in)h(the)i(\014le)150 2736 y Fo(fortran/fftw_f77.i)h
7767
 
Fu(of)33 b(the)h(FFTW)f(pac)m(k)-5 b(age.)51 b(If)33
 
7842
(de\014nitions)c(in)h(the)i(\014le)150 2498 y Fp(fortran/fftw_f77.i)h
 
7843
Fv(of)33 b(the)h(FFTW)f(pac)m(k)-5 b(age.)51 b(If)33
7768
7844
b(y)m(our)g(F)-8 b(ortran)33 b(compiler)f(supp)s(orts)f(a)j(prepro-)150
7769
 
2845 y(cessor,)i(y)m(ou)f(can)g(use)g(that)g(to)g(incorp)s(orate)f
 
7845
2607 y(cessor,)i(y)m(ou)f(can)g(use)g(that)g(to)g(incorp)s(orate)f
7770
7846
(this)g(\014le)f(in)m(to)i(y)m(our)g(co)s(de)f(whenev)m(er)h(y)m(ou)g
7771
 
(need)f(to)i(call)150 2955 y(FFTW.)31 b(Otherwise,)f(y)m(ou)g(will)e
 
7847
(need)f(to)i(call)150 2717 y(FFTW.)31 b(Otherwise,)f(y)m(ou)g(will)e
7772
7848
(ha)m(v)m(e)k(to)f(paste)g(the)f(constan)m(t)i(de\014nitions)c(in)h
7773
 
(directly)-8 b(.)40 b(They)30 b(are:)676 3084 y Fo(integer)46
7774
 
b(FFTW_FORWARD,FFTW_BACKWAR)o(D)676 3187 y(parameter)g
7775
 
(\(FFTW_FORWARD=-1,FFTW_B)o(ACKW)o(ARD)o(=1\))676 3395
 
7849
(directly)-8 b(.)40 b(They)30 b(are:)676 2846 y Fp(integer)46
 
7850
b(FFTW_FORWARD,FFTW_BACKWAR)o(D)676 2950 y(parameter)g
 
7851
(\(FFTW_FORWARD=-1,FFTW_B)o(ACKW)o(ARD)o(=1\))676 3158
7776
7852
y(integer)g(FFTW_REAL_TO_COMPLEX,FFTW)o(_COM)o(PLE)o(X_TO)o(_REA)o(L)
7777
 
676 3499 y(parameter)g(\(FFTW_REAL_TO_COMPLEX=-)o(1,FF)o(TW_)o(COMP)o
7778
 
(LEX_)o(TO_)o(REAL)o(=1\))676 3706 y(integer)g
7779
 
(FFTW_ESTIMATE,FFTW_MEASUR)o(E)676 3810 y(parameter)g
7780
 
(\(FFTW_ESTIMATE=0,FFTW_M)o(EASU)o(RE=)o(1\))676 4018
 
7853
676 3262 y(parameter)g(\(FFTW_REAL_TO_COMPLEX=-)o(1,FF)o(TW_)o(COMP)o
 
7854
(LEX_)o(TO_)o(REAL)o(=1\))676 3469 y(integer)g
 
7855
(FFTW_ESTIMATE,FFTW_MEASUR)o(E)676 3573 y(parameter)g
 
7856
(\(FFTW_ESTIMATE=0,FFTW_M)o(EASU)o(RE=)o(1\))676 3780
7781
7857
y(integer)g(FFTW_OUT_OF_PLACE,FFTW_IN)o(_PLA)o(CE,)o(FFTW)o(_USE)o(_WI)
7782
 
o(SDOM)676 4121 y(parameter)g(\(FFTW_OUT_OF_PLACE=0\))676
7783
 
4225 y(parameter)g(\(FFTW_IN_PLACE=8,FFTW_U)o(SE_W)o(ISD)o(OM=1)o(6\))
7784
 
676 4433 y(integer)g(FFTW_THREADSAFE)676 4537 y(parameter)g
7785
 
(\(FFTW_THREADSAFE=128\))275 4671 y Fu(In)31 b(C,)h(y)m(ou)g(com)m
7786
 
(bine)g(di\013eren)m(t)g(\015ags)g(\(lik)m(e)g Fo(FFTW_USE_WISDOM)c
7787
 
Fu(and)j Fo(FFTW_MEASURE)p Fu(\))e(using)i(the)150 4781
7788
 
y(`)p Fo(|)p Fu(')g(op)s(erator;)f(in)g(F)-8 b(ortran)31
7789
 
b(y)m(ou)f(should)f(just)g(use)i(`)p Fo(+)p Fu('.)150
7790
 
5038 y Ft(5.3)68 b(F)-11 b(ortran)45 b(Examples)275 5230
7791
 
y Fu(In)20 b(C)h(y)m(ou)h(migh)m(t)g(ha)m(v)m(e)g(something)f(lik)m(e)g
 
7858
o(SDOM)676 3884 y(parameter)g(\(FFTW_OUT_OF_PLACE=0\))676
 
7859
3988 y(parameter)g(\(FFTW_IN_PLACE=8,FFTW_U)o(SE_W)o(ISD)o(OM=1)o(6\))
 
7860
676 4196 y(integer)g(FFTW_THREADSAFE)676 4299 y(parameter)g
 
7861
(\(FFTW_THREADSAFE=128\))275 4435 y Fv(In)31 b(C,)h(y)m(ou)g(com)m
 
7862
(bine)g(di\013eren)m(t)g(\015ags)g(\(lik)m(e)g Fp(FFTW_USE_WISDOM)c
 
7863
Fv(and)j Fp(FFTW_MEASURE)p Fv(\))e(using)i(the)150 4544
 
7864
y(`)p Fp(|)p Fv(')g(op)s(erator;)f(in)g(F)-8 b(ortran)31
 
7865
b(y)m(ou)f(should)f(just)g(use)i(`)p Fp(+)p Fv('.)150
 
7866
4804 y Fu(5.3)68 b(F)-11 b(ortran)45 b(Examples)275 4997
 
7867
y Fv(In)20 b(C)h(y)m(ou)h(migh)m(t)g(ha)m(v)m(e)g(something)f(lik)m(e)g
7792
7868
(the)h(follo)m(wing)e(to)i(transform)f(a)h(one-dimensional)d(complex)
7793
 
150 5340 y(arra)m(y:)p eop
 
7869
150 5107 y(arra)m(y:)772 5236 y Fp(fftw_complex)44 b(in[N],)i(*out[N];)
 
7870
772 5340 y(fftw_plan)f(plan;)p eop end
7794
7871
%%Page: 53 55
7795
 
53 54 bop 150 -116 a Fu(Chapter)30 b(5:)41 b(Calling)28
7796
 
b(FFTW)j(from)f(F)-8 b(ortran)1909 b(53)772 299 y Fo(fftw_complex)44
7797
 
b(in[N],)i(*out[N];)772 403 y(fftw_plan)f(plan;)772 610
7798
 
y(plan)h(=)i(fftw_create_plan\(N,FFTW_)o(FORW)o(ARD)o(,FFT)o(W_ES)o
7799
 
(TIM)o(ATE\))o(;)772 714 y(fftw_one\(plan,in,out\);)772
7800
 
818 y(fftw_destroy_plan\(plan\);)275 961 y Fu(In)29 b(F)-8
 
7872
TeXDict begin 53 54 bop 150 -116 a Fv(Chapter)30 b(5:)41
 
7873
b(Calling)28 b(FFTW)j(from)f(F)-8 b(ortran)1909 b(53)772
 
7874
403 y Fp(plan)46 b(=)i(fftw_create_plan\(N,FFTW_)o(FORW)o(ARD)o(,FFT)o
 
7875
(W_ES)o(TIM)o(ATE\))o(;)772 506 y(fftw_one\(plan,in,out\);)772
 
7876
610 y(fftw_destroy_plan\(plan\);)275 749 y Fv(In)29 b(F)-8
7801
7877
b(ortran,)31 b(y)m(ou)g(use)f(the)h(follo)m(wing)e(to)i(accomplish)e
7802
 
(the)h(same)h(thing:)772 1099 y Fo(double)46 b(complex)g(in,)h(out)772
7803
 
1203 y(dimension)e(in\(N\),)h(out\(N\))772 1307 y(integer)g(plan)772
7804
 
1515 y(call)g(fftw_f77_create_plan\(plan,)o(N,FF)o(TW_)o(FORW)o(ARD,)o
7805
 
(FFT)o(W_ES)o(TIMA)o(TE\))772 1618 y(call)g
7806
 
(fftw_f77_one\(plan,in,out\))772 1722 y(call)g
7807
 
(fftw_f77_destroy_plan\(plan)o(\))275 1866 y Fu(Notice)34
 
7878
(the)h(same)h(thing:)772 883 y Fp(double)46 b(complex)g(in,)h(out)772
 
7879
986 y(dimension)e(in\(N\),)h(out\(N\))772 1090 y(integer)g(plan)772
 
7880
1298 y(call)g(fftw_f77_create_plan\(plan,)o(N,FF)o(TW_)o(FORW)o(ARD,)o
 
7881
(FFT)o(W_ES)o(TIMA)o(TE\))772 1402 y(call)g
 
7882
(fftw_f77_one\(plan,in,out\))772 1505 y(call)g
 
7883
(fftw_f77_destroy_plan\(plan)o(\))275 1644 y Fv(Notice)34
7808
7884
b(ho)m(w)g(all)f(routines)f(are)i(called)f(as)h(F)-8
7809
7885
b(ortran)35 b(subroutines,)d(and)i(the)g(plan)e(is)h(returned)f(via)150
7810
 
1975 y(the)43 b(\014rst)e(argumen)m(t)i(to)g Fo(fftw_f77_create_plan)p
7811
 
Fu(.)72 b Fq(Imp)-5 b(ortant:)72 b Fu(these)43 b(examples)f(assume)g
7812
 
(that)150 2085 y Fo(integer)26 b Fu(is)h(the)i(same)f(size)g(as)g(a)h
 
7886
1754 y(the)43 b(\014rst)e(argumen)m(t)i(to)g Fp(fftw_f77_create_plan)p
 
7887
Fv(.)72 b Fr(Imp)-5 b(ortant:)72 b Fv(these)43 b(examples)f(assume)g
 
7888
(that)150 1864 y Fp(integer)26 b Fv(is)h(the)i(same)f(size)g(as)g(a)h
7813
7889
(p)s(oin)m(ter,)f(and)f(ma)m(y)i(need)f(mo)s(di\014cation)e(on)i(a)h
7814
 
(64-bit)f(mac)m(hine.)40 b(See)150 2195 y(Section)28
 
7890
(64-bit)f(mac)m(hine.)40 b(See)150 1973 y(Section)28
7815
7891
b(5.1)h([W)-8 b(rapp)s(er)28 b(Routines],)g(page)h(51,)h(ab)s(o)m(v)m
7816
7892
(e.)41 b(T)-8 b(o)29 b(do)f(the)g(same)h(thing,)f(but)f(using)g(8)h
7817
 
(threads)150 2304 y(in)d(parallel)g(\(see)j(Section)f(4.1)h
 
7893
(threads)150 2083 y(in)d(parallel)g(\(see)j(Section)f(4.1)h
7818
7894
([Multi-threaded)d(FFTW],)j(page)g(37\),)h(y)m(ou)e(w)m(ould)f(simply)e
7819
 
(replace)j(the)150 2414 y(call)j(to)h Fo(fftw_f77_one)c
7820
 
Fu(with:)772 2552 y Fo(call)46 b(fftw_f77_threads_one\(8,pla)o(n,in)o
7821
 
(,ou)o(t\))275 2695 y Fu(T)-8 b(o)30 b(transform)g(a)h
 
7895
(replace)j(the)150 2192 y(call)j(to)h Fp(fftw_f77_one)c
 
7896
Fv(with:)772 2326 y Fp(call)46 b(fftw_f77_threads_one\(8,pla)o(n,in)o
 
7897
(,ou)o(t\))275 2465 y Fv(T)-8 b(o)30 b(transform)g(a)h
7822
7898
(three-dimensional)d(arra)m(y)j(in-place)e(with)g(C,)h(y)m(ou)h(migh)m
7823
 
(t)f(do:)772 2833 y Fo(fftw_complex)44 b(arr[L][M][N];)772
7824
 
2937 y(fftwnd_plan)h(plan;)772 3041 y(int)i(n[3])f(=)i({L,M,N};)772
7825
 
3248 y(plan)e(=)i(fftwnd_create_plan\(3,n,F)o(FTW_)o(FOR)o(WARD)o(,)
7826
 
2013 3352 y(FFTW_ESTIMATE)c(|)j(FFTW_IN_PLACE\);)772
7827
 
3456 y(fftwnd_one\(plan,)c(arr,)k(0\);)772 3560 y
7828
 
(fftwnd_destroy_plan\(plan)o(\);)275 3703 y Fu(In)29
 
7899
(t)f(do:)772 2598 y Fp(fftw_complex)44 b(arr[L][M][N];)772
 
7900
2702 y(fftwnd_plan)h(plan;)772 2806 y(int)i(n[3])f(=)i({L,M,N};)772
 
7901
3013 y(plan)e(=)i(fftwnd_create_plan\(3,n,F)o(FTW_)o(FOR)o(WARD)o(,)
 
7902
2013 3117 y(FFTW_ESTIMATE)c(|)j(FFTW_IN_PLACE\);)772
 
7903
3221 y(fftwnd_one\(plan,)c(arr,)k(0\);)772 3325 y
 
7904
(fftwnd_destroy_plan\(plan)o(\);)275 3464 y Fv(In)29
7829
7905
b(F)-8 b(ortran,)31 b(y)m(ou)g(w)m(ould)e(use)h(this)g(instead:)772
7830
 
3841 y Fo(double)46 b(complex)g(arr)772 3945 y(dimension)f
7831
 
(arr\(L,M,N\))772 4049 y(integer)h(n)772 4153 y(dimension)f(n\(3\))772
7832
 
4256 y(integer)h(plan)772 4464 y(n\(1\))g(=)i(L)772 4568
7833
 
y(n\(2\))e(=)i(M)772 4672 y(n\(3\))e(=)i(N)772 4775 y(call)e
 
7906
3597 y Fp(double)46 b(complex)g(arr)772 3701 y(dimension)f
 
7907
(arr\(L,M,N\))772 3805 y(integer)h(n)772 3909 y(dimension)f(n\(3\))772
 
7908
4012 y(integer)h(plan)772 4220 y(n\(1\))g(=)i(L)772 4324
 
7909
y(n\(2\))e(=)i(M)772 4427 y(n\(3\))e(=)i(N)772 4531 y(call)e
7834
7910
(fftwnd_f77_create_plan\(pla)o(n,3,)o(n,F)o(FTW_)o(FORW)o(ARD)o(,)724
7835
 
4879 y(+)1336 b(FFTW_ESTIMATE)44 b(+)k(FFTW_IN_PLACE\))772
7836
 
4983 y(call)e(fftwnd_f77_one\(plan,)d(arr,)j(0\))772
7837
 
5087 y(call)g(fftwnd_f77_destroy_plan\(pl)o(an\))275
7838
 
5230 y Fu(Instead)29 b(of)h(calling)e Fo(fftwnd_f77_create_plan\()o
7839
 
(plan)o(,3,n)o(,..)o(.\))p Fu(,)c(w)m(e)30 b(could)e(also)i(ha)m(v)m(e)
7840
 
h(called)150 5340 y Fo(fftw3d_f77_create_plan\(p)o(lan,)o(L,M,)o(N,.)o
7841
 
(..\))p Fu(.)p eop
 
7911
4635 y(+)1336 b(FFTW_ESTIMATE)44 b(+)k(FFTW_IN_PLACE\))772
 
7912
4739 y(call)e(fftwnd_f77_one\(plan,)d(arr,)j(0\))772
 
7913
4843 y(call)g(fftwnd_f77_destroy_plan\(pl)o(an\))275
 
7914
4982 y Fv(Instead)29 b(of)h(calling)e Fp(fftwnd_f77_create_plan\()o
 
7915
(plan)o(,3,n)o(,..)o(.\))p Fv(,)c(w)m(e)30 b(could)e(also)i(ha)m(v)m(e)
 
7916
h(called)150 5091 y Fp(fftw3d_f77_create_plan\(p)o(lan,)o(L,M,)o(N,.)o
 
7917
(..\))p Fv(.)275 5230 y(Note)j(that)g(w)m(e)g(pass)f(the)h(arra)m(y)g
 
7918
(dimensions)d(in)h(the)h Fp(")p Fv(natural)p Fp(")f Fv(order;)j(also)e
 
7919
(note)h(that)g(the)g(last)150 5340 y(argumen)m(t)d(to)g
 
7920
Fp(fftwnd_f77)d Fv(is)h(ignored)g(since)h(the)h(transform)e(is)h
 
7921
Fp(FFTW_IN_PLACE)p Fv(.)p eop end
7842
7922
%%Page: 54 56
7843
 
54 55 bop 150 -116 a Fu(54)3232 b(FFTW)275 299 y(Note)34
7844
 
b(that)g(w)m(e)g(pass)f(the)h(arra)m(y)g(dimensions)d(in)h(the)h
7845
 
Fo(")p Fu(natural)p Fo(")f Fu(order;)j(also)e(note)h(that)g(the)g(last)
7846
 
150 408 y(argumen)m(t)d(to)g Fo(fftwnd_f77)d Fu(is)h(ignored)g(since)h
7847
 
(the)h(transform)e(is)h Fo(FFTW_IN_PLACE)p Fu(.)275 543
7848
 
y(T)-8 b(o)30 b(transform)g(a)h(one-dimensional)d(real)i(arra)m(y)h(in)
7849
 
e(F)-8 b(ortran,)31 b(y)m(ou)g(migh)m(t)f(do:)772 672
7850
 
y Fo(double)46 b(precision)f(in,)i(out)772 775 y(dimension)e(in\(N\),)h
7851
 
(out\(N\))772 879 y(integer)g(plan)772 1087 y(call)g
7852
 
(rfftw_f77_create_plan\(plan)o(,N,F)o(FTW)o(_REA)o(L_TO)o(_CO)o(MPLE)o
7853
 
(X,)724 1191 y(+)1288 b(FFTW_ESTIMATE\))772 1294 y(call)46
7854
 
b(rfftw_f77_one\(plan,in,out\))772 1398 y(call)g
7855
 
(rfftw_f77_destroy_plan\(pla)o(n\))275 1533 y Fu(T)-8
 
7923
TeXDict begin 54 55 bop 150 -116 a Fv(54)3232 b(FFTW)275
 
7924
299 y(T)-8 b(o)30 b(transform)g(a)h(one-dimensional)d(real)i(arra)m(y)h
 
7925
(in)e(F)-8 b(ortran,)31 b(y)m(ou)g(migh)m(t)f(do:)772
 
7926
428 y Fp(double)46 b(precision)f(in,)i(out)772 531 y(dimension)e
 
7927
(in\(N\),)h(out\(N\))772 635 y(integer)g(plan)772 843
 
7928
y(call)g(rfftw_f77_create_plan\(plan)o(,N,F)o(FTW)o(_REA)o(L_TO)o(_CO)o
 
7929
(MPLE)o(X,)724 946 y(+)1288 b(FFTW_ESTIMATE\))772 1050
 
7930
y(call)46 b(rfftw_f77_one\(plan,in,out\))772 1154 y(call)g
 
7931
(rfftw_f77_destroy_plan\(pla)o(n\))275 1289 y Fv(T)-8
7856
7932
b(o)30 b(transform)g(a)h(t)m(w)m(o-dimensional)e(real)h(arra)m(y)-8
7857
7933
b(,)32 b(out)e(of)h(place,)f(y)m(ou)h(migh)m(t)f(use)g(the)h(follo)m
7858
 
(wing:)772 1661 y Fo(double)46 b(precision)f(in)772 1765
7859
 
y(double)h(complex)g(out)772 1869 y(dimension)f(in\(M,N\),)h(out\(M/2)f
7860
 
(+)j(1,)f(N\))772 1973 y(integer)f(plan)772 2180 y(call)g
 
7934
(wing:)772 1417 y Fp(double)46 b(precision)f(in)772 1521
 
7935
y(double)h(complex)g(out)772 1625 y(dimension)f(in\(M,N\),)h(out\(M/2)f
 
7936
(+)j(1,)f(N\))772 1729 y(integer)f(plan)772 1936 y(call)g
7861
7937
(rfftw2d_f77_create_plan\(pl)o(an,M)o(,N,)o(FFTW)o(_REA)o(L_T)o(O_CO)o
7862
 
(MPLE)o(X,)724 2284 y(+)1384 b(FFTW_ESTIMATE\))772 2388
 
7938
(MPLE)o(X,)724 2040 y(+)1384 b(FFTW_ESTIMATE\))772 2144
7863
7939
y(call)46 b(rfftwnd_f77_one_real_to_co)o(mple)o(x\(p)o(lan,)41
7864
 
b(in,)47 b(out\))772 2491 y(call)f(rfftwnd_f77_destroy_plan\(p)o(lan\))
7865
 
275 2626 y Fm(Imp)s(ortan)m(t:)40 b Fu(Notice)30 b(that)g(it)f(is)f
7866
 
(the)i Fq(\014rst)38 b Fu(dimension)27 b(of)i(the)h(complex)f(output)g
7867
 
(arra)m(y)h(that)f(is)g(cut)150 2736 y(in)38 b(half)g(in)g(F)-8
 
7940
b(in,)47 b(out\))772 2247 y(call)f(rfftwnd_f77_destroy_plan\(p)o(lan\))
 
7941
275 2382 y Fn(Imp)s(ortan)m(t:)40 b Fv(Notice)30 b(that)g(it)f(is)f
 
7942
(the)i Fr(\014rst)38 b Fv(dimension)27 b(of)i(the)h(complex)f(output)g
 
7943
(arra)m(y)h(that)f(is)g(cut)150 2491 y(in)38 b(half)g(in)g(F)-8
7868
7944
b(ortran,)42 b(rather)d(than)g(the)h(last)f(dimension)e(as)i(in)f(C.)h
7869
 
(This)e(is)i(a)g(consequence)h(of)g(the)150 2845 y(wrapp)s(er)29
 
7945
(This)e(is)i(a)g(consequence)h(of)g(the)150 2601 y(wrapp)s(er)29
7870
7946
b(routines)h(rev)m(ersing)g(the)i(order)e(of)h(the)g(arra)m(y)h
7871
7947
(dimensions)c(passed)j(to)g(FFTW)h(so)f(that)h(the)150
7872
 
2955 y(F)-8 b(ortran)31 b(program)f(can)h(use)f(its)g(ordinary)e
7873
 
(column-ma)5 b(jor)30 b(order.)p eop
 
7948
2711 y(F)-8 b(ortran)31 b(program)f(can)h(use)f(its)g(ordinary)e
 
7949
(column-ma)5 b(jor)30 b(order.)p eop end
7874
7950
%%Page: 55 57
7875
 
55 56 bop 150 -116 a Fu(Chapter)30 b(6:)41 b(Installation)29
7876
 
b(and)g(Customization)1808 b(55)150 299 y Fr(6)80 b(Installation)56
7877
 
b(and)d(Customization)275 515 y Fu(This)32 b(c)m(hapter)j(describ)s(es)
7878
 
e(the)h(installation)f(and)h(customization)g(of)g(FFTW,)i(the)e(latest)
7879
 
h(v)m(ersion)150 625 y(of)c(whic)m(h)e(ma)m(y)i(b)s(e)e(do)m(wnloaded)h
7880
 
(from)g(the)g(FFTW)h(home)g(page)g(\()p Fo(http://www.fftw.org)p
7881
 
Fu(\).)275 756 y(As)e(distributed,)e(FFTW)j(mak)m(es)h(v)m(ery)f(few)f
 
7951
TeXDict begin 55 56 bop 150 -116 a Fv(Chapter)30 b(6:)41
 
7952
b(Installation)29 b(and)g(Customization)1808 b(55)150
 
7953
299 y Fs(6)80 b(Installation)56 b(and)d(Customization)275
 
7954
520 y Fv(This)32 b(c)m(hapter)j(describ)s(es)e(the)h(installation)f
 
7955
(and)h(customization)g(of)g(FFTW,)i(the)e(latest)h(v)m(ersion)150
 
7956
629 y(of)c(whic)m(h)e(ma)m(y)i(b)s(e)e(do)m(wnloaded)h(from)g(the)g
 
7957
(FFTW)h(home)g(page)g(\()p Fp(http://www.fftw.org)p Fv(\).)275
 
7958
761 y(As)e(distributed,)e(FFTW)j(mak)m(es)h(v)m(ery)f(few)f
7882
7959
(assumptions)f(ab)s(out)h(y)m(our)h(system.)40 b(All)29
7883
 
b(y)m(ou)h(need)f(is)150 866 y(an)i(ANSI)g(C)g(compiler)e(\()p
7884
 
Fo(gcc)i Fu(is)f(\014ne,)h(although)f(v)m(endor-pro)m(vided)g
7885
 
(compilers)g(often)i(pro)s(duce)e(faster)150 975 y(co)s(de\).)50
 
7960
b(y)m(ou)h(need)f(is)150 871 y(an)i(ANSI)g(C)g(compiler)e(\()p
 
7961
Fp(gcc)i Fv(is)f(\014ne,)h(although)f(v)m(endor-pro)m(vided)g
 
7962
(compilers)g(often)i(pro)s(duce)e(faster)150 980 y(co)s(de\).)50
7886
7963
b(Ho)m(w)m(ev)m(er,)36 b(installation)31 b(of)i(FFTW)h(is)e(somewhat)i
7887
7964
(simpler)d(if)h(y)m(ou)i(ha)m(v)m(e)g(a)f(Unix)g(or)g(a)g(GNU)150
7888
 
1085 y(system,)46 b(suc)m(h)d(as)g(Lin)m(ux.)77 b(In)42
 
7965
1090 y(system,)46 b(suc)m(h)d(as)g(Lin)m(ux.)77 b(In)42
7889
7966
b(this)g(c)m(hapter,)47 b(w)m(e)d(\014rst)e(describ)s(e)f(the)i
7890
 
(installation)e(of)i(FFTW)h(on)150 1194 y(Unix)33 b(and)h(non-Unix)f
 
7967
(installation)e(of)i(FFTW)h(on)150 1200 y(Unix)33 b(and)h(non-Unix)f
7891
7968
(systems.)53 b(W)-8 b(e)36 b(then)e(describ)s(e)f(ho)m(w)i(y)m(ou)g
7892
 
(can)f(customize)h(FFTW)g(to)h(ac)m(hiev)m(e)150 1304
 
7969
(can)f(customize)h(FFTW)g(to)h(ac)m(hiev)m(e)150 1309
7893
7970
y(b)s(etter)41 b(p)s(erformance.)71 b(Sp)s(eci\014cally)-8
7894
 
b(,)41 b(y)m(ou)g(can)g(I\))g(enable)f Fo(gcc)p Fu(/x86-sp)s(eci\014c)g
7895
 
(hac)m(ks)i(that)f(impro)m(v)m(e)150 1414 y(p)s(erformance)26
 
7971
b(,)41 b(y)m(ou)g(can)g(I\))g(enable)f Fp(gcc)p Fv(/x86-sp)s(eci\014c)g
 
7972
(hac)m(ks)i(that)f(impro)m(v)m(e)150 1419 y(p)s(erformance)26
7896
7973
b(on)g(P)m(en)m(tia)h(and)e(P)m(en)m(tiumPro's;)j(I)s(I\))e(adapt)g
7897
7974
(FFTW)h(to)g(use)f(the)h(high-resolution)d(clo)s(c)m(k)150
7898
 
1523 y(of)i(y)m(our)h(mac)m(hine,)g(if)e(an)m(y;)j(I)s(I)s(I\))d(pro)s
7899
 
(duce)g(co)s(de)i(\()p Fq(c)-5 b(o)g(delets)7 b Fu(\))29
 
7975
1528 y(of)i(y)m(our)h(mac)m(hine,)g(if)e(an)m(y;)j(I)s(I)s(I\))d(pro)s
 
7976
(duce)g(co)s(de)i(\()p Fr(c)-5 b(o)g(delets)7 b Fv(\))29
7900
7977
b(to)e(supp)s(ort)d(fast)j(transforms)e(of)h(sizes)g(that)150
7901
 
1633 y(are)31 b(not)f(supp)s(orted)f(e\016cien)m(tly)h(b)m(y)g(the)h
7902
 
(standard)e(FFTW)i(distribution.)150 1880 y Ft(6.1)68
7903
 
b(Installation)47 b(on)e(Unix)275 2069 y Fu(FFTW)24 b(comes)g(with)e(a)
7904
 
h Fo(configure)e Fu(program)i(in)f(the)i(GNU)g(st)m(yle.)38
7905
 
b(Installation)22 b(can)i(b)s(e)e(as)i(simple)150 2178
7906
 
y(as:)390 2303 y Fo(./configure)390 2407 y(make)390 2511
7907
 
y(make)47 b(install)275 2642 y Fu(This)24 b(will)h(build)e(the)k
 
7978
1638 y(are)31 b(not)f(supp)s(orted)f(e\016cien)m(tly)h(b)m(y)g(the)h
 
7979
(standard)e(FFTW)i(distribution.)150 1887 y Fu(6.1)68
 
7980
b(Installation)47 b(on)e(Unix)275 2077 y Fv(FFTW)24 b(comes)g(with)e(a)
 
7981
h Fp(configure)e Fv(program)i(in)f(the)i(GNU)g(st)m(yle.)38
 
7982
b(Installation)22 b(can)i(b)s(e)e(as)i(simple)150 2187
 
7983
y(as:)390 2313 y Fp(./configure)390 2417 y(make)390 2521
 
7984
y(make)47 b(install)275 2652 y Fv(This)24 b(will)h(build)e(the)k
7908
7985
(unipro)s(cessor)e(complex)h(and)g(real)h(transform)f(libraries)e
7909
 
(along)j(with)e(the)i(test)150 2752 y(programs.)37 b(W)-8
 
7986
(along)j(with)e(the)i(test)150 2762 y(programs.)37 b(W)-8
7910
7987
b(e)22 b(strongly)d(recommend)i(that)g(y)m(ou)f(use)g(GNU)h
7911
 
Fo(make)f Fu(if)f(it)h(is)f(a)m(v)-5 b(ailable;)24 b(on)c(some)h
7912
 
(systems)150 2861 y(it)37 b(is)g(called)g Fo(gmake)p
7913
 
Fu(.)62 b(The)37 b(\\)p Fo(make)30 b(install)p Fu(")36
 
7988
Fp(make)f Fv(if)f(it)h(is)f(a)m(v)-5 b(ailable;)24 b(on)c(some)h
 
7989
(systems)150 2872 y(it)37 b(is)g(called)g Fp(gmake)p
 
7990
Fv(.)62 b(The)37 b(\\)p Fp(make)30 b(install)p Fv(")36
7914
7991
b(command)i(installs)d(the)j(\013t)m(w)g(and)f(r\013t)m(w)h(libraries)d
7915
 
(in)150 2971 y(standard)28 b(places,)i(and)e(t)m(ypically)g(requires)g
 
7992
(in)150 2981 y(standard)28 b(places,)i(and)e(t)m(ypically)g(requires)g
7916
7993
(ro)s(ot)i(privileges)d(\(unless)h(y)m(ou)h(sp)s(ecify)f(a)h
7917
 
(di\013eren)m(t)g(install)150 3080 y(directory)34 b(with)e(the)j
7918
 
Fo(--prefix)c Fu(\015ag)k(to)f Fo(configure)p Fu(\).)50
7919
 
b(Y)-8 b(ou)35 b(can)f(also)g(t)m(yp)s(e)g(\\)p Fo(make)c(check)p
7920
 
Fu(")j(to)i(put)150 3190 y(the)f(FFTW)h(test)f(programs)g(through)f
 
7994
(di\013eren)m(t)g(install)150 3091 y(directory)34 b(with)e(the)j
 
7995
Fp(--prefix)c Fv(\015ag)k(to)f Fp(configure)p Fv(\).)50
 
7996
b(Y)-8 b(ou)35 b(can)f(also)g(t)m(yp)s(e)g(\\)p Fp(make)c(check)p
 
7997
Fv(")j(to)i(put)150 3200 y(the)f(FFTW)h(test)f(programs)g(through)f
7921
7998
(their)g(paces.)51 b(If)34 b(y)m(ou)g(ha)m(v)m(e)h(problems)d(during)g
7922
 
(con\014guration)150 3299 y(or)i(compilation,)g(y)m(ou)h(ma)m(y)g(w)m
7923
 
(an)m(t)g(to)g(run)e(\\)p Fo(make)d(distclean)p Fu(")i(b)s(efore)i
7924
 
(trying)f(again;)k(this)c(ensures)150 3409 y(that)e(y)m(ou)g(don't)f
 
7999
(con\014guration)150 3310 y(or)i(compilation,)g(y)m(ou)h(ma)m(y)g(w)m
 
8000
(an)m(t)g(to)g(run)e(\\)p Fp(make)d(distclean)p Fv(")i(b)s(efore)i
 
8001
(trying)f(again;)k(this)c(ensures)150 3419 y(that)e(y)m(ou)g(don't)f
7925
8002
(ha)m(v)m(e)i(an)m(y)e(stale)h(\014les)e(left)h(o)m(v)m(er)i(from)e
7926
 
(previous)f(compilation)g(attempts.)275 3540 y(The)g
7927
 
Fo(configure)f Fu(script)i(kno)m(ws)g(go)s(o)s(d)g Fo(CFLAGS)f
7928
 
Fu(\(C)h(compiler)f(\015ags\))i(for)f(a)h(few)f(systems.)41
7929
 
b(If)30 b(y)m(our)150 3650 y(system)g(is)f(not)h(kno)m(wn,)g(the)g
7930
 
Fo(configure)d Fu(script)i(will)e(prin)m(t)i(out)h(a)g(w)m(arning.)2858
7931
 
3617 y Fn(1)2935 3650 y Fu(In)f(this)g(case,)i(y)m(ou)g(can)150
7932
 
3759 y(compile)e(FFTW)i(with)e(the)i(command)390 3884
7933
 
y Fo(make)47 b(CFLAGS="<write)d(your)i(CFLAGS)g(here>")275
7934
 
4015 y Fu(If)24 b(y)m(ou)h(do)g(\014nd)e(an)i(optimal)f(set)i(of)f
7935
 
Fo(CFLAGS)e Fu(for)i(y)m(our)f(system,)j(please)e(let)f(us)h(kno)m(w)g
7936
 
(what)f(they)i(are)150 4125 y(\(along)31 b(with)e(the)h(output)g(of)h
7937
 
Fo(config.guess)p Fu(\))c(so)k(that)g(w)m(e)g(can)f(include)f(them)h
7938
 
(in)f(future)g(releases.)275 4256 y(The)38 b Fo(configure)e
7939
 
Fu(program)j(supp)s(orts)e(all)h(the)h(standard)f(\015ags)h(de\014ned)f
7940
 
(b)m(y)h(the)g(GNU)h(Co)s(ding)150 4366 y(Standards;)23
7941
 
b(see)d(the)h Fo(INSTALL)d Fu(\014le)i(in)f(FFTW)h(or)h(the)f(GNU)h(w)m
7942
 
(eb)f(page)h(\()p Fo(http://www.gnu.org/prep/sta)o(ndar)o(ds_)p
7943
 
4282 4388 42 91 v 150 4475 a(toc.html)p Fu(\).)38 b(Note)31
7944
 
b(esp)s(ecially)d Fo(--help)f Fu(to)k(list)d(all)g(\015ags)h(and)g
7945
 
Fo(--enable-shared)c Fu(to)30 b(create)h(shared,)150
7946
 
4585 y(rather)21 b(than)g(static,)j(libraries.)35 b Fo(configure)19
7947
 
b Fu(also)j(accepts)g(a)g(few)f(FFTW-sp)s(eci\014c)g(\015ags,)j
7948
 
(particularly:)225 4716 y Fs(\017)60 b Fo(--enable-float)28
7949
 
b Fu(Pro)s(duces)j(a)i(single-precision)c(v)m(ersion)i(of)h(FFTW)h(\()p
7950
 
Fo(float)p Fu(\))e(instead)h(of)g(the)330 4825 y(default)24
7951
 
b(double-precision)d(\()p Fo(double)p Fu(\).)38 b(See)24
 
8003
(previous)f(compilation)g(attempts.)275 3551 y(The)g
 
8004
Fp(configure)f Fv(script)i(kno)m(ws)g(go)s(o)s(d)g Fp(CFLAGS)f
 
8005
Fv(\(C)h(compiler)f(\015ags\))i(for)f(a)h(few)f(systems.)41
 
8006
b(If)30 b(y)m(our)150 3661 y(system)g(is)f(not)h(kno)m(wn,)g(the)g
 
8007
Fp(configure)d Fv(script)i(will)e(prin)m(t)i(out)h(a)g(w)m(arning.)2858
 
8008
3628 y Fo(1)2935 3661 y Fv(In)f(this)g(case,)i(y)m(ou)g(can)150
 
8009
3771 y(compile)e(FFTW)i(with)e(the)i(command)390 3897
 
8010
y Fp(make)47 b(CFLAGS="<write)d(your)i(CFLAGS)g(here>")275
 
8011
4028 y Fv(If)24 b(y)m(ou)h(do)g(\014nd)e(an)i(optimal)f(set)i(of)f
 
8012
Fp(CFLAGS)e Fv(for)i(y)m(our)f(system,)j(please)e(let)f(us)h(kno)m(w)g
 
8013
(what)f(they)i(are)150 4138 y(\(along)31 b(with)e(the)h(output)g(of)h
 
8014
Fp(config.guess)p Fv(\))c(so)k(that)g(w)m(e)g(can)f(include)f(them)h
 
8015
(in)f(future)g(releases.)275 4270 y(The)95 b Fp(configure)f
 
8016
Fv(program)h(supp)s(orts)f(all)h(the)i(standard)e(\015ags)h(de\014ned)f
 
8017
(b)m(y)h(the)150 4379 y(GNU)67 b(Co)s(ding)e(Standards;)84
 
8018
b(see)67 b(the)g Fp(INSTALL)e Fv(\014le)h(in)f(FFTW)j(or)e(the)h(GNU)h
 
8019
(w)m(eb)e(page)150 4489 y(\()p Fp(http://www.gnu.org/prep/st)o(and)o
 
8020
(ards)o(_toc)o(.ht)o(ml)p Fv(\).)93 b(Note)51 b(esp)s(ecially)c
 
8021
Fp(--help)h Fv(to)j(list)d(all)150 4599 y(\015ags)32
 
8022
b(and)f Fp(--enable-shared)d Fv(to)33 b(create)g(shared,)f(rather)g
 
8023
(than)g(static,)h(libraries.)43 b Fp(configure)29 b Fv(also)150
 
8024
4708 y(accepts)j(a)e(few)h(FFTW-sp)s(eci\014c)f(\015ags,)h
 
8025
(particularly:)225 4840 y Ft(\017)60 b Fp(--enable-float)28
 
8026
b Fv(Pro)s(duces)j(a)i(single-precision)c(v)m(ersion)i(of)h(FFTW)h(\()p
 
8027
Fp(float)p Fv(\))e(instead)h(of)g(the)330 4950 y(default)24
 
8028
b(double-precision)d(\()p Fp(double)p Fv(\).)38 b(See)24
7952
8029
b(Section)g(6.3)i([Installing)c(FFTW)j(in)e(b)s(oth)h(single)f(and)330
7953
 
4935 y(double)29 b(precision],)g(page)i(57.)p 150 5022
7954
 
1200 4 v 192 5088 a Fn(1)275 5121 y Fu(Eac)m(h)e(v)m(ersion)e(of)i
7955
 
Fo(cc)f Fu(seems)g(to)i(ha)m(v)m(e)f(its)f(o)m(wn)g(magic)h(incan)m
7956
 
(tation)f(to)h(get)h(the)f(fastest)g(co)s(de)g(most)275
7957
 
5230 y(of)42 b(the)g(time|y)m(ou'd)f(think)g(that)h(p)s(eople)f(w)m
7958
 
(ould)g(ha)m(v)m(e)i(agreed)g(up)s(on)d(some)i(con)m(v)m(en)m(tion,)47
7959
 
b(e.g.)275 5340 y Fo("-Omax")p Fu(,)28 b(b)m(y)i(no)m(w.)p
7960
 
eop
 
8030
5059 y(double)29 b(precision],)g(page)i(57.)p 150 5149
 
8031
1200 4 v 199 5217 a Fo(1)275 5249 y Fj(Eac)n(h)18 b(v)n(ersion)h(of)g
 
8032
Fi(cc)f Fj(seems)g(to)h(ha)n(v)n(e)f(its)g(o)n(wn)h(magic)f(incan)n
 
8033
(tation)h(to)g(get)f(the)g(fastest)i(co)r(de)f(most)e(of)i(the)f
 
8034
(time|y)n(ou'd)275 5340 y(think)24 b(that)i(p)r(eople)g(w)n(ould)g(ha)n
 
8035
(v)n(e)f(agreed)h(up)r(on)f(some)h(con)n(v)n(en)n(tion,)f(e.g.)35
 
8036
b Fi("-Omax")p Fj(,)28 b(b)n(y)d(no)n(w.)p eop end
7961
8037
%%Page: 56 58
7962
 
56 57 bop 150 -116 a Fu(56)3232 b(FFTW)225 299 y Fs(\017)60
7963
 
b Fo(--enable-type-prefix)34 b Fu(Adds)k(a)i(`)p Fo(d)p
7964
 
Fu(')g(or)g(`)p Fo(s)p Fu(')f(pre\014x)g(to)h(all)e(installed)g
 
8038
TeXDict begin 56 57 bop 150 -116 a Fv(56)3232 b(FFTW)225
 
8039
299 y Ft(\017)60 b Fp(--enable-type-prefix)34 b Fv(Adds)k(a)i(`)p
 
8040
Fp(d)p Fv(')g(or)g(`)p Fp(s)p Fv(')f(pre\014x)g(to)h(all)e(installed)g
7965
8041
(libraries)e(and)j(header)330 408 y(\014les)29 b(to)h(indicate)f(the)h
7966
8042
(\015oating-p)s(oin)m(t)f(precision.)39 b(See)30 b(Section)g(6.3)h
7967
8043
([Installing)d(FFTW)i(in)f(b)s(oth)330 518 y(single)c(and)g(double)f
7968
 
(precision],)i(page)g(57.)40 b(\()p Fo(--enable-type-prefix=<prefi)o
7969
 
(x>)20 b Fu(lets)25 b(y)m(ou)i(add)330 628 y(an)j(arbitrary)f
 
8044
(precision],)i(page)g(57.)40 b(\()p Fp(--enable-type-prefix=<prefi)o
 
8045
(x>)20 b Fv(lets)25 b(y)m(ou)i(add)330 628 y(an)j(arbitrary)f
7970
8046
(pre\014x.\))40 b(By)31 b(default,)f(no)g(pre\014x)f(is)h(used.)225
7971
 
764 y Fs(\017)60 b Fo(--enable-threads)24 b Fu(Enables)j(compilation)g
 
8047
758 y Ft(\017)60 b Fp(--enable-threads)24 b Fv(Enables)j(compilation)g
7972
8048
(and)g(installation)g(of)h(the)g(FFTW)h(threads)f(library)330
7973
 
874 y(\(see)36 b(Section)e(4.1)i([Multi-threaded)e(FFTW],)i(page)f
 
8049
867 y(\(see)36 b(Section)e(4.1)i([Multi-threaded)e(FFTW],)i(page)f
7974
8050
(37\),)i(whic)m(h)d(pro)m(vides)g(a)h(simple)d(in)m(terface)330
7975
 
983 y(to)43 b(parallel)e(transforms)g(for)i(SMP)f(systems.)77
 
8051
977 y(to)43 b(parallel)e(transforms)g(for)i(SMP)f(systems.)77
7976
8052
b(\(By)43 b(default,)i(the)d(threads)g(routines)g(are)h(not)330
7977
 
1093 y(compiled.\))225 1229 y Fs(\017)60 b Fo(--enable-mpi)38
7978
 
b Fu(Enables)h(compilation)h(and)g(installation)f(of)i(the)g(FFTW)h
7979
 
(MPI)e(library)f(\(see)330 1339 y(Section)27 b(4.2)i([MPI)e(FFTW],)i
7980
 
(page)f(40\),)i(whic)m(h)c(pro)m(vides)g(parallel)g(transforms)g(for)i
7981
 
(distributed-)330 1449 y(memory)i(systems)h(with)e(MPI.)h(\(By)h
7982
 
(default,)f(the)h(MPI)f(routines)f(are)i(not)g(compiled.\))225
7983
 
1585 y Fs(\017)60 b Fo(--disable-fortran)19 b Fu(Disables)k(inclusion)e
7984
 
(of)j(F)-8 b(ortran-callable)24 b(wrapp)s(er)e(routines)h(\(see)i
7985
 
(Chap-)330 1695 y(ter)32 b(5)h([Calling)d(FFTW)i(from)g(F)-8
7986
 
b(ortran],)33 b(page)g(51\))g(in)e(the)h(standard)f(FFTW)i(libraries.)
7987
 
42 b(These)330 1804 y(wrapp)s(er)e(routines)g(increase)h(the)h(library)
7988
 
d(size)i(b)m(y)g(only)g(a)g(negligible)e(amoun)m(t,)45
7989
 
b(so)d(they)g(are)330 1914 y(included)30 b(b)m(y)j(default)g(as)g(long)
7990
 
g(as)g(the)h Fo(configure)c Fu(script)i(\014nds)g(a)h(F)-8
7991
 
b(ortran)34 b(compiler)e(on)h(y)m(our)330 2023 y(system.)225
7992
 
2160 y Fs(\017)60 b Fo(--with-gcc)32 b Fu(Enables)h(the)i(use)g(of)g
7993
 
Fo(gcc)p Fu(.)53 b(By)35 b(default,)g(FFTW)h(uses)e(the)h(v)m
7994
 
(endor-supplied)d Fo(cc)330 2269 y Fu(compiler)22 b(if)h(presen)m(t.)39
7995
 
b(Unfortunately)-8 b(,)25 b Fo(gcc)e Fu(pro)s(duces)f(slo)m(w)m(er)i
7996
 
(co)s(de)g(than)g Fo(cc)f Fu(on)h(man)m(y)g(systems.)225
7997
 
2406 y Fs(\017)60 b Fo(--enable-i386-hacks)25 b Fu(See)31
 
8053
1086 y(compiled.\))225 1217 y Ft(\017)60 b Fp(--with-openmp)p
 
8054
Fv(,)44 b Fp(--with-sgimp)d Fv(In)j(conjunction)f(with)g
 
8055
Fp(--enable-threads)p Fv(,)g(causes)i(the)330 1326 y(m)m(ulti-threaded)
 
8056
27 b(FFTW)j(library)c(to)j(use)g(Op)s(enMP)e(or)i(SGI)f(MP)h(compiler)e
 
8057
(directiv)m(es)h(in)f(order)330 1436 y(to)g(induce)e(parallelism,)g
 
8058
(rather)h(than)h(spa)m(wning)e(its)h(o)m(wn)g(threads)g(directly)-8
 
8059
b(.)39 b(\(Useful)26 b(esp)s(ecially)330 1545 y(for)i(programs)f
 
8060
(already)g(emplo)m(ying)g(suc)m(h)g(directiv)m(es,)h(in)f(order)g(to)i
 
8061
(minimize)c(con\015icts)i(b)s(et)m(w)m(een)330 1655 y(di\013eren)m(t)j
 
8062
(parallelization)e(mec)m(hanisms.\))225 1785 y Ft(\017)60
 
8063
b Fp(--enable-mpi)38 b Fv(Enables)h(compilation)h(and)g(installation)f
 
8064
(of)i(the)g(FFTW)h(MPI)e(library)f(\(see)330 1895 y(Section)27
 
8065
b(4.2)i([MPI)e(FFTW],)i(page)f(40\),)i(whic)m(h)c(pro)m(vides)g
 
8066
(parallel)g(transforms)g(for)i(distributed-)330 2004
 
8067
y(memory)i(systems)h(with)e(MPI.)h(\(By)h(default,)f(the)h(MPI)f
 
8068
(routines)f(are)i(not)g(compiled.\))225 2134 y Ft(\017)60
 
8069
b Fp(--disable-fortran)19 b Fv(Disables)k(inclusion)e(of)j(F)-8
 
8070
b(ortran-callable)24 b(wrapp)s(er)e(routines)h(\(see)i(Chap-)330
 
8071
2244 y(ter)32 b(5)h([Calling)d(FFTW)i(from)g(F)-8 b(ortran],)33
 
8072
b(page)g(51\))g(in)e(the)h(standard)f(FFTW)i(libraries.)42
 
8073
b(These)330 2353 y(wrapp)s(er)e(routines)g(increase)h(the)h(library)d
 
8074
(size)i(b)m(y)g(only)g(a)g(negligible)e(amoun)m(t,)45
 
8075
b(so)d(they)g(are)330 2463 y(included)30 b(b)m(y)j(default)g(as)g(long)
 
8076
g(as)g(the)h Fp(configure)c Fv(script)i(\014nds)g(a)h(F)-8
 
8077
b(ortran)34 b(compiler)e(on)h(y)m(our)330 2573 y(system.)225
 
8078
2703 y Ft(\017)60 b Fp(--with-gcc)32 b Fv(Enables)h(the)i(use)g(of)g
 
8079
Fp(gcc)p Fv(.)53 b(By)35 b(default,)g(FFTW)h(uses)e(the)h(v)m
 
8080
(endor-supplied)d Fp(cc)330 2812 y Fv(compiler)22 b(if)h(presen)m(t.)39
 
8081
b(Unfortunately)-8 b(,)25 b Fp(gcc)e Fv(pro)s(duces)f(slo)m(w)m(er)i
 
8082
(co)s(de)g(than)g Fp(cc)f Fv(on)h(man)m(y)g(systems.)225
 
8083
2942 y Ft(\017)60 b Fp(--enable-i386-hacks)25 b Fv(See)31
7998
8084
b(Section)f(6.4)h([gcc)h(and)e(P)m(en)m(tium)g(hac)m(ks],)h(page)g(57,)
7999
 
h(b)s(elo)m(w.)225 2542 y Fs(\017)60 b Fo(--enable-pentium-timer)24
8000
 
b Fu(See)31 b(Section)f(6.4)i([gcc)f(and)f(P)m(en)m(tium)g(hac)m(ks],)i
8001
 
(page)f(57,)g(b)s(elo)m(w.)275 2708 y(T)-8 b(o)22 b(force)h
8002
 
Fo(configure)d Fu(to)k(use)e(a)h(particular)d(C)i(compiler)f(\(instead)
8003
 
h(of)h(the)g(default,)g(usually)d Fo(cc)p Fu(\),)k(set)150
8004
 
2817 y(the)i(en)m(vironmen)m(t)f(v)-5 b(ariable)25 b
8005
 
Fo(CC)g Fu(to)h(the)g(name)g(of)g(the)g(desired)e(compiler)g(b)s(efore)
8006
 
h(running)e Fo(configure)p Fu(;)150 2927 y(y)m(ou)31
 
8085
h(b)s(elo)m(w.)225 3072 y Ft(\017)60 b Fp(--enable-pentium-timer)24
 
8086
b Fv(See)31 b(Section)f(6.4)i([gcc)f(and)f(P)m(en)m(tium)g(hac)m(ks],)i
 
8087
(page)f(57,)g(b)s(elo)m(w.)275 3223 y(T)-8 b(o)22 b(force)h
 
8088
Fp(configure)d Fv(to)k(use)e(a)h(particular)d(C)i(compiler)f(\(instead)
 
8089
h(of)h(the)g(default,)g(usually)d Fp(cc)p Fv(\),)k(set)150
 
8090
3333 y(the)i(en)m(vironmen)m(t)f(v)-5 b(ariable)25 b
 
8091
Fp(CC)g Fv(to)h(the)g(name)g(of)g(the)g(desired)e(compiler)g(b)s(efore)
 
8092
h(running)e Fp(configure)p Fv(;)150 3442 y(y)m(ou)31
8007
8093
b(ma)m(y)g(also)f(need)g(to)h(set)g(the)g(\015ags)f(via)g(the)h(v)-5
8008
 
b(ariable)29 b Fo(CFLAGS)p Fu(.)150 3194 y Ft(6.2)68
8009
 
b(Installation)47 b(on)e(non-Unix)g(Systems)275 3390
8010
 
y Fu(It)22 b(is)f(quite)h(straigh)m(tforw)m(ard)g(to)h(install)d(FFTW)j
 
8094
b(ariable)29 b Fp(CFLAGS)p Fv(.)150 3686 y Fu(6.2)68
 
8095
b(Installation)47 b(on)e(non-Unix)g(Systems)275 3874
 
8096
y Fv(It)22 b(is)f(quite)h(straigh)m(tforw)m(ard)g(to)h(install)d(FFTW)j
8011
8097
(ev)m(en)h(on)e(non-Unix)f(systems)h(lac)m(king)g(the)h(niceties)150
8012
 
3500 y(of)33 b(the)g Fo(configure)e Fu(script.)47 b(The)32
 
8098
3984 y(of)33 b(the)g Fp(configure)e Fv(script.)47 b(The)32
8013
8099
b(FFTW)i(Home)g(P)m(age)g(ma)m(y)g(include)c(some)k(FFTW)f(pac)m(k)-5
8014
 
b(ages)35 b(pre-)150 3610 y(con\014gured)29 b(for)h(particular)f
 
8100
b(ages)35 b(pre-)150 4094 y(con\014gured)29 b(for)h(particular)f
8015
8101
(systems/compilers,)g(and)h(also)g(con)m(tains)h(installation)d(notes)i
8016
 
(sen)m(t)h(in)e(b)m(y)150 3719 y(users.)40 b(All)30 b(y)m(ou)h(really)e
 
8102
(sen)m(t)h(in)e(b)m(y)150 4203 y(users.)40 b(All)30 b(y)m(ou)h(really)e
8017
8103
(need)h(to)i(do,)e(though,)h(is)f(to)h(compile)e(all)h(of)h(the)f
8018
 
Fo(.c)g Fu(\014les)g(in)f(the)i(appropriate)150 3829
 
8104
Fp(.c)g Fv(\014les)g(in)f(the)i(appropriate)150 4313
8019
8105
y(directories)39 b(of)i(the)g(FFTW)g(pac)m(k)-5 b(age.)72
8020
8106
b(\(Y)-8 b(ou)42 b(needn't)e(w)m(orry)g(ab)s(out)g(the)h(man)m(y)f
8021
 
(extraneous)h(\014les)150 3938 y(lying)29 b(around.\))275
8022
 
4077 y(F)-8 b(or)37 b(the)g(complex)f(transforms,)i(compile)e(all)f(of)
8023
 
i(the)g Fo(.c)f Fu(\014les)g(in)f(the)i Fo(fftw)f Fu(directory)g(and)g
8024
 
(link)150 4186 y(them)j(in)m(to)h(a)g(library)-8 b(.)66
 
8107
(extraneous)h(\014les)150 4422 y(lying)29 b(around.\))275
 
8108
4552 y(F)-8 b(or)37 b(the)g(complex)f(transforms,)i(compile)e(all)f(of)
 
8109
i(the)g Fp(.c)f Fv(\014les)g(in)f(the)i Fp(fftw)f Fv(directory)g(and)g
 
8110
(link)150 4662 y(them)j(in)m(to)h(a)g(library)-8 b(.)66
8025
8111
b(Similarly)-8 b(,)38 b(for)h(the)h(real)f(transforms,)i(compile)e(all)
8026
 
f(of)i(the)f Fo(.c)g Fu(\014les)g(in)f(the)150 4296 y
8027
 
Fo(rfftw)31 b Fu(directory)h(in)m(to)h(a)g(library)-8
8028
 
b(.)46 b(Note)34 b(that)f(these)g(sources)g Fo(#include)d
8029
 
Fu(v)-5 b(arious)32 b(\014les)g(in)f(the)i Fo(fftw)150
8030
 
4406 y Fu(and)k Fo(rfftw)f Fu(directories,)j(so)f(y)m(ou)g(ma)m(y)g
8031
 
(need)f(to)h(set)h(up)d(the)i Fo(#include)d Fu(paths)i(for)h(y)m(our)f
8032
 
(compiler)150 4515 y(appropriately)-8 b(.)38 b(Be)27
 
8112
f(of)i(the)f Fp(.c)g Fv(\014les)g(in)f(the)150 4772 y
 
8113
Fp(rfftw)31 b Fv(directory)h(in)m(to)h(a)g(library)-8
 
8114
b(.)46 b(Note)34 b(that)f(these)g(sources)g Fp(#include)d
 
8115
Fv(v)-5 b(arious)32 b(\014les)g(in)f(the)i Fp(fftw)150
 
8116
4881 y Fv(and)k Fp(rfftw)f Fv(directories,)j(so)f(y)m(ou)g(ma)m(y)g
 
8117
(need)f(to)h(set)h(up)d(the)i Fp(#include)d Fv(paths)i(for)h(y)m(our)f
 
8118
(compiler)150 4991 y(appropriately)-8 b(.)38 b(Be)27
8033
8119
b(sure)f(to)i(enable)e(the)g(highest-p)s(ossible)e(lev)m(el)i(of)h
8034
 
(optimization)e(in)h(y)m(our)g(compiler.)275 4654 y(By)i(default,)h
 
8120
(optimization)e(in)h(y)m(our)g(compiler.)275 5121 y(By)i(default,)h
8035
8121
(FFTW)g(is)f(compiled)f(for)h(double-precision)e(transforms.)39
8036
 
b(T)-8 b(o)29 b(w)m(ork)g(in)e(single)h(preci-)150 4763
8037
 
y(sion)j(rather)i(than)f(double)e(precision,)i Fo(#define)e
8038
 
Fu(the)j(sym)m(b)s(ol)e Fo(FFTW_ENABLE_FLOAT)c Fu(in)k
8039
 
Fo(fftw.h)g Fu(\(in)150 4873 y(the)g Fo(fftw)e Fu(directory\))h(and)g
8040
 
(\(re\)compile)g(FFTW.)275 5011 y(These)36 b(libraries)e(should)h(b)s
8041
 
(e)h(link)m(ed)f(with)g(an)m(y)i(program)g(that)g(uses)f(the)h(corresp)
8042
 
s(onding)e(trans-)150 5121 y(forms.)44 b(The)31 b(required)f(header)h
8043
 
(\014les,)g Fo(fftw.h)f Fu(and)h Fo(rfftw.h)p Fu(,)f(are)i(lo)s(cated)g
8044
 
(in)e(the)i Fo(fftw)e Fu(and)h Fo(rfftw)150 5230 y Fu(directories)h
8045
 
(resp)s(ectiv)m(ely;)j(y)m(ou)e(ma)m(y)h(w)m(an)m(t)h(to)f(put)e(them)i
8046
 
(with)e(the)h(libraries,)f(or)h(wherev)m(er)g(header)150
8047
 
5340 y(\014les)c(normally)g(go)i(on)f(y)m(our)h(system.)p
8048
 
eop
 
8122
b(T)-8 b(o)29 b(w)m(ork)g(in)e(single)h(preci-)150 5230
 
8123
y(sion)j(rather)i(than)f(double)e(precision,)i Fp(#define)e
 
8124
Fv(the)j(sym)m(b)s(ol)e Fp(FFTW_ENABLE_FLOAT)c Fv(in)k
 
8125
Fp(fftw.h)g Fv(\(in)150 5340 y(the)g Fp(fftw)e Fv(directory\))h(and)g
 
8126
(\(re\)compile)g(FFTW.)p eop end
8049
8127
%%Page: 57 59
8050
 
57 58 bop 150 -116 a Fu(Chapter)30 b(6:)41 b(Installation)29
8051
 
b(and)g(Customization)1808 b(57)275 299 y(FFTW)44 b(includes)d(test)k
8052
 
(programs,)i Fo(fftw_test)41 b Fu(and)i Fo(rfftw_test)p
8053
 
Fu(,)i(in)d(the)i Fo(tests)f Fu(directory)-8 b(.)150
8054
 
408 y(These)33 b(are)g(compiled)f(and)g(link)m(ed)f(lik)m(e)i(an)m(y)g
 
8128
TeXDict begin 57 58 bop 150 -116 a Fv(Chapter)30 b(6:)41
 
8129
b(Installation)29 b(and)g(Customization)1808 b(57)275
 
8130
299 y(These)36 b(libraries)e(should)h(b)s(e)h(link)m(ed)f(with)g(an)m
 
8131
(y)i(program)g(that)g(uses)f(the)h(corresp)s(onding)e(trans-)150
 
8132
408 y(forms.)44 b(The)31 b(required)f(header)h(\014les,)g
 
8133
Fp(fftw.h)f Fv(and)h Fp(rfftw.h)p Fv(,)f(are)i(lo)s(cated)g(in)e(the)i
 
8134
Fp(fftw)e Fv(and)h Fp(rfftw)150 518 y Fv(directories)h(resp)s(ectiv)m
 
8135
(ely;)j(y)m(ou)e(ma)m(y)h(w)m(an)m(t)h(to)f(put)e(them)i(with)e(the)h
 
8136
(libraries,)f(or)h(wherev)m(er)g(header)150 628 y(\014les)c(normally)g
 
8137
(go)i(on)f(y)m(our)h(system.)275 771 y(FFTW)44 b(includes)d(test)k
 
8138
(programs,)i Fp(fftw_test)41 b Fv(and)i Fp(rfftw_test)p
 
8139
Fv(,)i(in)d(the)i Fp(tests)f Fv(directory)-8 b(.)150
 
8140
881 y(These)33 b(are)g(compiled)f(and)g(link)m(ed)f(lik)m(e)i(an)m(y)g
8055
8141
(program)g(using)e(FFTW,)j(except)g(that)g(they)f(use)g(addi-)150
8056
 
518 y(tional)g(header)h(\014les)f(lo)s(cated)h(in)f(the)h
8057
 
Fo(fftw)f Fu(and)g Fo(rfftw)g Fu(directories,)h(so)g(y)m(ou)h(will)c
8058
 
(need)j(to)g(set)h(y)m(our)150 628 y(compiler)41 b Fo(#include)e
8059
 
Fu(paths)j(appropriately)-8 b(.)74 b Fo(fftw_test)40
8060
 
b Fu(is)h(compiled)f(from)i Fo(fftw_test.c)d Fu(and)150
8061
 
737 y Fo(test_main.c)p Fu(,)25 b(while)h Fo(rfftw_test)f
8062
 
Fu(is)i(compiled)f(from)i Fo(rfftw_test.c)c Fu(and)j
8063
 
Fo(test_main.c)p Fu(.)37 b(When)150 847 y(y)m(ou)i(run)f(these)h
 
8142
991 y(tional)g(header)h(\014les)f(lo)s(cated)h(in)f(the)h
 
8143
Fp(fftw)f Fv(and)g Fp(rfftw)g Fv(directories,)h(so)g(y)m(ou)h(will)c
 
8144
(need)j(to)g(set)h(y)m(our)150 1100 y(compiler)41 b Fp(#include)e
 
8145
Fv(paths)j(appropriately)-8 b(.)74 b Fp(fftw_test)40
 
8146
b Fv(is)h(compiled)f(from)i Fp(fftw_test.c)d Fv(and)150
 
8147
1210 y Fp(test_main.c)p Fv(,)25 b(while)h Fp(rfftw_test)f
 
8148
Fv(is)i(compiled)f(from)i Fp(rfftw_test.c)c Fv(and)j
 
8149
Fp(test_main.c)p Fv(.)37 b(When)150 1319 y(y)m(ou)i(run)f(these)h
8064
8150
(programs,)j(y)m(ou)d(will)d(b)s(e)j(prompted)f(in)m(teractiv)m(ely)h
8065
8151
(for)g(v)-5 b(arious)38 b(p)s(ossible)e(tests)k(to)150
8066
 
956 y(p)s(erform;)29 b(see)i(also)f Fo(tests/README)e
8067
 
Fu(for)i(more)g(information.)150 1214 y Ft(6.3)68 b(Installing)47
 
8152
1429 y(p)s(erform;)29 b(see)i(also)f Fp(tests/README)e
 
8153
Fv(for)i(more)g(information.)150 1710 y Fu(6.3)68 b(Installing)47
8068
8154
b(FFTW)c(in)i(b)t(oth)f(single)i(and)f(double)g(precision)275
8069
 
1407 y Fu(It)f(is)f(often)i(useful)e(to)i(install)d(b)s(oth)h(single-)h
 
8155
1912 y Fv(It)f(is)f(often)i(useful)e(to)i(install)d(b)s(oth)h(single-)h
8070
8156
(and)f(double-precision)f(v)m(ersions)h(of)i(the)f(FFTW)150
8071
 
1517 y(libraries)31 b(on)j(the)g(same)g(mac)m(hine,)h(and)e(w)m(e)h
 
8157
2021 y(libraries)31 b(on)j(the)g(same)g(mac)m(hine,)h(and)e(w)m(e)h
8072
8158
(pro)m(vide)f(a)i(con)m(v)m(enien)m(t)g(mec)m(hanism)e(for)h(ac)m
8073
 
(hieving)f(this)150 1626 y(on)d(Unix)f(systems.)275 1761
8074
 
y(When)h(the)i Fo(--enable-type-prefix)25 b Fu(option)31
 
8159
(hieving)f(this)150 2131 y(on)d(Unix)f(systems.)275 2275
 
8160
y(When)h(the)i Fp(--enable-type-prefix)25 b Fv(option)31
8075
8161
b(of)g(con\014gure)g(is)f(used,)h(the)g(FFTW)h(libraries)d(and)150
8076
 
1871 y(header)j(\014les)g(are)h(installed)d(with)h(a)i(pre\014x)e(of)i
8077
 
(`)p Fo(d)p Fu(')g(or)f(`)p Fo(s)p Fu(',)i(dep)s(ending)c(up)s(on)h
8078
 
(whether)h(y)m(ou)g(compiled)150 1980 y(in)c(double)g(or)h(single)f
 
8162
2384 y(header)j(\014les)g(are)h(installed)d(with)h(a)i(pre\014x)e(of)i
 
8163
(`)p Fp(d)p Fv(')g(or)f(`)p Fp(s)p Fv(',)i(dep)s(ending)c(up)s(on)h
 
8164
(whether)h(y)m(ou)g(compiled)150 2494 y(in)c(double)g(or)h(single)f
8079
8165
(precision.)38 b(Then,)29 b(instead)f(of)i(linking)c(y)m(our)j(program)
8080
 
g(with)f Fo(-lrfftw)g(-lfftw)p Fu(,)150 2090 y(for)39
8081
 
b(example,)j(y)m(ou)e(w)m(ould)f(link)e(with)h Fo(-ldrfftw)29
8082
 
b(-ldfftw)37 b Fu(to)k(use)e(the)h(double-precision)c(v)m(ersion)150
8083
 
2199 y(or)e(with)g Fo(-lsrfftw)28 b(-lsfftw)k Fu(to)k(use)e(the)g
 
8166
g(with)f Fp(-lrfftw)g(-lfftw)p Fv(,)150 2603 y(for)39
 
8167
b(example,)j(y)m(ou)e(w)m(ould)f(link)e(with)h Fp(-ldrfftw)29
 
8168
b(-ldfftw)37 b Fv(to)k(use)e(the)h(double-precision)c(v)m(ersion)150
 
8169
2713 y(or)e(with)g Fp(-lsrfftw)28 b(-lsfftw)k Fv(to)k(use)e(the)g
8084
8170
(single-precision)e(v)m(ersion.)53 b(Also,)35 b(y)m(ou)g(w)m(ould)e
8085
 
Fo(#include)150 2309 y(<drfftw.h>)28 b Fu(or)i Fo(<srfftw.h>)e
8086
 
Fu(instead)h(of)i Fo(<rfftw.h>)p Fu(,)d(and)i(so)g(on.)275
8087
 
2444 y Fq(The)j(names)g(of)g(FFTW)h(functions,)f(data)h(typ)-5
 
8171
Fp(#include)150 2823 y(<drfftw.h>)28 b Fv(or)i Fp(<srfftw.h>)e
 
8172
Fv(instead)h(of)i Fp(<rfftw.h>)p Fv(,)d(and)i(so)g(on.)275
 
8173
2966 y Fr(The)j(names)g(of)g(FFTW)h(functions,)f(data)h(typ)-5
8088
8174
b(es,)34 b(and)g(c)-5 b(onstants)34 b(r)-5 b(emain)34
8089
 
b(unchange)-5 b(d!)54 b Fu(Y)-8 b(ou)31 b(still)150 2553
8090
 
y(call,)24 b(for)f(instance,)i Fo(fftw_one)c Fu(and)i(not)h
8091
 
Fo(dfftw_one)p Fu(.)35 b(Only)22 b(the)i(names)f(of)h(header)f(\014les)
8092
 
f(and)h(libraries)150 2663 y(are)34 b(mo)s(di\014ed.)47
8093
 
b(One)32 b(consequence)i(of)g(this)e(is)g(that)i Fq(you)h
8094
 
Fm(cannot)h Fq(use)f(b)-5 b(oth)36 b(the)g(single-)f(and)h(double-)150
8095
 
2772 y(pr)-5 b(e)g(cision)27 b(FFTW)f(libr)-5 b(aries)27
 
8175
b(unchange)-5 b(d!)54 b Fv(Y)-8 b(ou)31 b(still)150 3076
 
8176
y(call,)24 b(for)f(instance,)i Fp(fftw_one)c Fv(and)i(not)h
 
8177
Fp(dfftw_one)p Fv(.)35 b(Only)22 b(the)i(names)f(of)h(header)f(\014les)
 
8178
f(and)h(libraries)150 3186 y(are)34 b(mo)s(di\014ed.)47
 
8179
b(One)32 b(consequence)i(of)g(this)e(is)g(that)i Fr(you)h
 
8180
Fn(cannot)h Fr(use)f(b)-5 b(oth)36 b(the)g(single-)f(and)h(double-)150
 
8181
3295 y(pr)-5 b(e)g(cision)27 b(FFTW)f(libr)-5 b(aries)27
8096
8182
b(in)e(the)h(same)g(pr)-5 b(o)g(gr)g(am,)30 b(simultane)-5
8097
 
b(ously,)26 b Fu(as)d(the)g(function)f(names)g(w)m(ould)150
8098
 
2882 y(con\015ict.)275 3017 y(So,)35 b(to)g(install)d(b)s(oth)i(the)g
 
8183
b(ously,)26 b Fv(as)d(the)g(function)f(names)g(w)m(ould)150
 
8184
3405 y(con\015ict.)275 3549 y(So,)35 b(to)g(install)d(b)s(oth)i(the)g
8099
8185
(single-)f(and)h(double-precision)e(libraries)f(on)j(the)h(same)f(mac)m
8100
 
(hine,)i(y)m(ou)150 3126 y(w)m(ould)29 b(do:)390 3255
8101
 
y Fo(./configure)45 b(--enable-type-prefix)d Fq([)32
8102
 
b(other)i(options)g(])390 3359 y Fo(make)390 3463 y(make)47
8103
 
b(install)390 3567 y(make)g(clean)390 3670 y(./configure)e
8104
 
(--enable-float)f(--enable-type-prefix)e Fq([)32 b(other)i(options)g(])
8105
 
390 3774 y Fo(make)390 3878 y(make)47 b(install)150 4136
8106
 
y Ft(6.4)68 b Fe(gcc)43 b Ft(and)i(P)l(en)l(tium)h(hac)l(ks)275
8107
 
4329 y Fu(The)22 b Fo(configure)e Fu(option)i Fo(--enable-i386-hacks)c
8108
 
Fu(enables)k(sp)s(eci\014c)f(optimizations)h(for)g(the)h(P)m(en-)150
8109
 
4438 y(tium)g(and)i(later)f(x86)i(CPUs)e(under)f(gcc,)k(whic)m(h)d(can)
 
8186
(hine,)i(y)m(ou)150 3658 y(w)m(ould)29 b(do:)390 3796
 
8187
y Fp(./configure)45 b(--enable-type-prefix)d Fr([)32
 
8188
b(other)i(options)g(])390 3900 y Fp(make)390 4004 y(make)47
 
8189
b(install)390 4108 y(make)g(clean)390 4211 y(./configure)e
 
8190
(--enable-float)f(--enable-type-prefix)e Fr([)32 b(other)i(options)g(])
 
8191
390 4315 y Fp(make)390 4419 y(make)47 b(install)150 4700
 
8192
y Fu(6.4)68 b Fb(gcc)43 b Fu(and)i(P)l(en)l(tium)h(hac)l(ks)275
 
8193
4902 y Fv(The)22 b Fp(configure)e Fv(option)i Fp(--enable-i386-hacks)c
 
8194
Fv(enables)k(sp)s(eci\014c)f(optimizations)h(for)g(the)h(P)m(en-)150
 
8195
5011 y(tium)g(and)i(later)f(x86)i(CPUs)e(under)f(gcc,)k(whic)m(h)d(can)
8110
8196
h(signi\014can)m(tly)e(impro)m(v)m(e)h(p)s(erformance)g(of)h(double-)
8111
 
150 4548 y(precision)f(transforms.)39 b(Sp)s(eci\014cally)-8
 
8197
150 5121 y(precision)f(transforms.)39 b(Sp)s(eci\014cally)-8
8112
8198
b(,)24 b(w)m(e)j(ha)m(v)m(e)g(tested)g(these)f(hac)m(ks)h(on)f(Lin)m
8113
 
(ux)e(with)h Fo(gcc)g Fu(2.[789])k(and)150 4657 y(v)m(ersions)j(of)i
8114
 
Fo(egcs)d Fu(since)i(1.0.3.)50 b(These)33 b(optimizations)f(a\013ect)i
8115
 
(only)f(the)g(p)s(erformance)f(and)g(not)i(the)150 4767
 
8199
(ux)e(with)h Fp(gcc)g Fv(2.[789])k(and)150 5230 y(v)m(ersions)j(of)i
 
8200
Fp(egcs)d Fv(since)i(1.0.3.)50 b(These)33 b(optimizations)f(a\013ect)i
 
8201
(only)f(the)g(p)s(erformance)f(and)g(not)i(the)150 5340
8116
8202
y(correctness)d(of)g(FFTW)g(\(i.e.)41 b(it)30 b(is)f(alw)m(a)m(ys)i
8117
 
(safe)g(to)g(try)f(them)g(out\).)275 4902 y(These)h(hac)m(ks)i(pro)m
8118
 
(vide)e(a)h(w)m(ork)-5 b(around)31 b(to)i(the)f(incorrect)g(alignmen)m
8119
 
(t)g(of)g(lo)s(cal)f Fo(double)f Fu(v)-5 b(ariables)150
8120
 
5011 y(in)31 b Fo(gcc)p Fu(.)47 b(The)32 b(compiler)f(aligns)h(these)h
8121
 
(v)-5 b(ariables)31 b(to)i(m)m(ultiples)e(of)h(4)h(b)m(ytes,)h(but)e
8122
 
(execution)h(is)f(m)m(uc)m(h)150 5121 y(faster)k(\(on)g(P)m(en)m(tium)g
8123
 
(and)f(P)m(en)m(tiumPro\))h(if)f Fo(double)p Fu(s)f(are)i(aligned)f(to)
8124
 
h(a)h(m)m(ultiple)c(of)j(8)h(b)m(ytes.)58 b(By)150 5230
8125
 
y(carefully)25 b(coun)m(ting)h(the)g(n)m(um)m(b)s(er)f(of)h(v)-5
8126
 
b(ariables)25 b(allo)s(cated)h(b)m(y)g(the)h(compiler)d(in)h(p)s
8127
 
(erformance-critical)150 5340 y(regions)i(of)g(the)h(co)s(de,)g(w)m(e)g
8128
 
(ha)m(v)m(e)h(b)s(een)d(able)h(to)h(in)m(tro)s(duce)e(dumm)m(y)h(allo)s
8129
 
(cations)f(\(using)g Fo(alloca)p Fu(\))g(that)p eop
 
8203
(safe)g(to)g(try)f(them)g(out\).)p eop end
8130
8204
%%Page: 58 60
8131
 
58 59 bop 150 -116 a Fu(58)3232 b(FFTW)150 299 y(align)34
8132
 
b(the)g(stac)m(k)j(prop)s(erly)-8 b(.)51 b(The)35 b(hac)m(k)g(dep)s
8133
 
(ends)e(crucially)g(on)h(the)h(compiler)e(\015ags)i(that)g(are)g(used.)
8134
 
150 408 y(F)-8 b(or)31 b(example,)f(it)g(w)m(on't)h(w)m(ork)g(without)e
8135
 
Fo(-fomit-frame-pointer)p Fu(.)275 584 y(In)d(principle,)f(these)j(hac)
8136
 
m(ks)h(are)f(no)f(longer)g(required)f(under)g Fo(gcc)h
8137
 
Fu(v)m(ersions)g(2.95)i(and)e(later,)h(whic)m(h)150 693
8138
 
y(automatically)d(align)f(the)h(stac)m(k)i(correctly)e(\(see)h
8139
 
Fo(-mpreferred-stack-bounda)o(ry)19 b Fu(in)24 b(the)h
8140
 
Fo(gcc)f Fu(man-)150 803 y(ual\).)37 b(Ho)m(w)m(ev)m(er,)24
8141
 
b(w)m(e)d(ha)m(v)m(e)h(encoun)m(tered)e(a)h(bug)f(\()p
8142
 
Fo(http://egcs.cygnus.com/ml)o(/gc)o(c-bu)o(gs/1)o(999)o(-11/)o(msg0)o
8143
 
(025)o(9.ht)o(ml)p Fu(\))p 4555 826 42 91 v 150 912 a(in)25
8144
 
b(the)h(stac)m(k)h(alignmen)m(t)e(of)h(v)m(ersions)f(2.95.[012])30
8145
 
b(that)c(causes)g(FFTW's)h(stac)m(k)g(to)g(b)s(e)e(misaligned)f(un-)150
8146
 
1022 y(der)k(some)g(circumstances.)40 b(The)27 b Fo(configure)f
8147
 
Fu(script)h(automatically)h(detects)h(this)e(bug)g(and)h(disables)150
8148
 
1132 y Fo(gcc)p Fu('s)i(stac)m(k)h(alignmen)m(t)f(in)f(fa)m(v)m(or)j
8149
 
(of)e(our)g(o)m(wn)h(hac)m(ks)g(when)e Fo(--enable-i386-hacks)c
8150
 
Fu(is)k(used.)275 1307 y(The)34 b Fo(fftw_test)e Fu(program)i(outputs)g
 
8205
TeXDict begin 58 59 bop 150 -116 a Fv(58)3232 b(FFTW)275
 
8206
299 y(These)31 b(hac)m(ks)i(pro)m(vide)e(a)h(w)m(ork)-5
 
8207
b(around)31 b(to)i(the)f(incorrect)g(alignmen)m(t)g(of)g(lo)s(cal)f
 
8208
Fp(double)f Fv(v)-5 b(ariables)150 408 y(in)31 b Fp(gcc)p
 
8209
Fv(.)47 b(The)32 b(compiler)f(aligns)h(these)h(v)-5 b(ariables)31
 
8210
b(to)i(m)m(ultiples)e(of)h(4)h(b)m(ytes,)h(but)e(execution)h(is)f(m)m
 
8211
(uc)m(h)150 518 y(faster)k(\(on)g(P)m(en)m(tium)g(and)f(P)m(en)m
 
8212
(tiumPro\))h(if)f Fp(double)p Fv(s)f(are)i(aligned)f(to)h(a)h(m)m
 
8213
(ultiple)c(of)j(8)h(b)m(ytes.)58 b(By)150 628 y(carefully)25
 
8214
b(coun)m(ting)h(the)g(n)m(um)m(b)s(er)f(of)h(v)-5 b(ariables)25
 
8215
b(allo)s(cated)h(b)m(y)g(the)h(compiler)d(in)h(p)s(erformance-critical)
 
8216
150 737 y(regions)i(of)g(the)h(co)s(de,)g(w)m(e)g(ha)m(v)m(e)h(b)s(een)
 
8217
d(able)h(to)h(in)m(tro)s(duce)e(dumm)m(y)h(allo)s(cations)f(\(using)g
 
8218
Fp(alloca)p Fv(\))g(that)150 847 y(align)34 b(the)g(stac)m(k)j(prop)s
 
8219
(erly)-8 b(.)51 b(The)35 b(hac)m(k)g(dep)s(ends)e(crucially)g(on)h(the)
 
8220
h(compiler)e(\015ags)i(that)g(are)g(used.)150 956 y(F)-8
 
8221
b(or)31 b(example,)f(it)g(w)m(on't)h(w)m(ork)g(without)e
 
8222
Fp(-fomit-frame-pointer)p Fv(.)275 1086 y(In)86 b(principle,)98
 
8223
b(these)87 b(hac)m(ks)h(are)f(no)g(longer)f(required)g(under)f
 
8224
Fp(gcc)h Fv(v)m(ersions)g(2.95)150 1195 y(and)96 b(later,)112
 
8225
b(whic)m(h)95 b(automatically)h(align)g(the)g(stac)m(k)i(correctly)e
 
8226
(\(see)h Fp(-mpreferred-)150 1305 y(stack-boundary)76
 
8227
b Fv(in)j(the)h Fp(gcc)f Fv(man)m(ual\).)190 b(Ho)m(w)m(ev)m(er,)95
 
8228
b(w)m(e)80 b(ha)m(v)m(e)i(encoun)m(tered)e(a)h(bug)150
 
8229
1415 y(\()p Fp(http://egcs.cygnus.com/ml/)o(gcc)o(-bug)o(s/19)o(99-)o
 
8230
(11/m)o(sg00)o(259)o(.htm)o(l)p Fv(\))145 b(in)150 b(the)g(stac)m(k)150
 
8231
1524 y(alignmen)m(t)39 b(of)g(v)m(ersions)g(2.95.[012])k(that)d(causes)
 
8232
g(FFTW's)g(stac)m(k)h(to)f(b)s(e)f(misaligned)e(under)h(some)150
 
8233
1634 y(circumstances.)72 b(The)40 b Fp(configure)f Fv(script)h
 
8234
(automatically)g(detects)i(this)e(bug)h(and)f(disables)f
 
8235
Fp(gcc)p Fv('s)150 1743 y(stac)m(k)32 b(alignmen)m(t)e(in)f(fa)m(v)m
 
8236
(or)i(of)g(our)f(o)m(wn)g(hac)m(ks)h(when)e Fp(--enable-i386-hacks)c
 
8237
Fv(is)30 b(used.)275 1873 y(The)k Fp(fftw_test)e Fv(program)i(outputs)g
8151
8238
(sp)s(eed)g(measuremen)m(ts)g(that)i(y)m(ou)e(can)h(use)g(to)g(see)g
8152
 
(if)e(these)150 1416 y(hac)m(ks)e(are)g(b)s(ene\014cial.)275
8153
 
1592 y(The)36 b Fo(configure)e Fu(option)i Fo(--enable-pentium-timer)30
8154
 
b Fu(enables)36 b(the)h(use)f(of)h(the)f(P)m(en)m(tium)h(and)150
8155
 
1701 y(P)m(en)m(tiumPro)g(cycle)g(coun)m(ter)h(for)f(timing)e(purp)s
 
8239
(if)e(these)150 1982 y(hac)m(ks)e(are)g(b)s(ene\014cial.)275
 
8240
2112 y(The)36 b Fp(configure)e Fv(option)i Fp(--enable-pentium-timer)30
 
8241
b Fv(enables)36 b(the)h(use)f(of)h(the)f(P)m(en)m(tium)h(and)150
 
8242
2221 y(P)m(en)m(tiumPro)g(cycle)g(coun)m(ter)h(for)f(timing)e(purp)s
8156
8243
(oses.)59 b(In)37 b(order)f(to)i(get)h(correct)f(results,)g(y)m(ou)f(m)
8157
 
m(ust)150 1811 y(de\014ne)f Fo(FFTW_CYCLES_PER_SEC)31
8158
 
b Fu(in)k Fo(fftw/config.h)e Fu(to)38 b(b)s(e)d(the)i(clo)s(c)m(k)g(sp)
8159
 
s(eed)f(of)h(y)m(our)f(pro)s(cessor;)150 1920 y(the)31
 
8244
m(ust)150 2331 y(de\014ne)f Fp(FFTW_CYCLES_PER_SEC)31
 
8245
b Fv(in)k Fp(fftw/config.h)e Fv(to)38 b(b)s(e)d(the)i(clo)s(c)m(k)g(sp)
 
8246
s(eed)f(of)h(y)m(our)f(pro)s(cessor;)150 2440 y(the)31
8160
8247
b(resulting)e(FFTW)j(library)d(will)f(b)s(e)i(nonp)s(ortable.)41
8161
8248
b(The)31 b(use)f(of)h(this)f(option)h(is)f(deprecated.)43
8162
 
b(On)150 2030 y(serious)25 b(op)s(erating)h(systems)g(\(suc)m(h)g(as)g
8163
 
(Lin)m(ux\),)g(FFTW)h(uses)f Fo(gettimeofday\(\))p Fu(,)d(whic)m(h)i
8164
 
(has)h(enough)150 2140 y(resolution)34 b(and)g(is)g(p)s(ortable.)54
 
8249
b(On)150 2550 y(serious)25 b(op)s(erating)h(systems)g(\(suc)m(h)g(as)g
 
8250
(Lin)m(ux\),)g(FFTW)h(uses)f Fp(gettimeofday\(\))p Fv(,)d(whic)m(h)i
 
8251
(has)h(enough)150 2660 y(resolution)34 b(and)g(is)g(p)s(ortable.)54
8165
8252
b(\(Note)37 b(that)e(Win32)g(has)g(its)g(o)m(wn)g(high-resolution)d
8166
 
(timing)i(routines)150 2249 y(as)d(w)m(ell.)39 b(FFTW)31
 
8253
(timing)i(routines)150 2769 y(as)d(w)m(ell.)39 b(FFTW)31
8167
8254
b(con)m(tains)g(unsupp)s(orted)c(co)s(de)k(to)g(use)f(these)h
8168
 
(routines.\))150 2608 y Ft(6.5)68 b(Customizing)46 b(the)f(timer)275
8169
 
2842 y Fu(FFTW)32 b(needs)g(a)h(reasonably-precise)e(clo)s(c)m(k)i(in)e
 
8255
(routines.\))150 3011 y Fu(6.5)68 b(Customizing)46 b(the)f(timer)275
 
8256
3199 y Fv(FFTW)32 b(needs)g(a)h(reasonably-precise)e(clo)s(c)m(k)i(in)e
8170
8257
(order)h(to)h(\014nd)d(the)j(optimal)e(w)m(a)m(y)i(to)g(compute)g(a)150
8171
 
2951 y(transform.)39 b(On)27 b(Unix)f(systems,)i Fo(configure)d
8172
 
Fu(lo)s(oks)i(for)h Fo(gettimeofday)c Fu(and)j(other)h(system-sp)s
8173
 
(eci\014c)150 3061 y(timers.)81 b(If)43 b(it)h(do)s(es)g(not)g(\014nd)e
 
8258
3308 y(transform.)39 b(On)27 b(Unix)f(systems,)i Fp(configure)d
 
8259
Fv(lo)s(oks)i(for)h Fp(gettimeofday)c Fv(and)j(other)h(system-sp)s
 
8260
(eci\014c)150 3418 y(timers.)81 b(If)43 b(it)h(do)s(es)g(not)g(\014nd)e
8174
8261
(an)m(y)j(high)d(resolution)h(clo)s(c)m(k,)48 b(it)43
8175
 
b(defaults)g(to)i(using)e(the)h Fo(clock\(\))150 3170
8176
 
y Fu(function,)35 b(whic)m(h)e(is)h(v)m(ery)h(p)s(ortable,)g(but)f
 
8262
b(defaults)g(to)i(using)e(the)h Fp(clock\(\))150 3527
 
8263
y Fv(function,)35 b(whic)m(h)e(is)h(v)m(ery)h(p)s(ortable,)g(but)f
8177
8264
(forces)i(FFTW)f(to)h(run)d(for)i(a)g(long)f(time)h(in)e(order)h(to)i
8178
 
(get)150 3280 y(reliable)28 b(measuremen)m(ts.)275 3455
 
8265
(get)150 3637 y(reliable)28 b(measuremen)m(ts.)275 3766
8179
8266
y(If)23 b(y)m(our)i(mac)m(hine)f(supp)s(orts)e(a)j(high-resolution)d
8180
8267
(clo)s(c)m(k)j(not)f(recognized)h(b)m(y)f(FFTW,)h(it)f(is)f(therefore)
8181
 
150 3565 y(advisable)35 b(to)i(use)e(it.)58 b(Y)-8 b(ou)36
8182
 
b(m)m(ust)g(edit)g Fo(fftw/fftw-int.h)p Fu(.)54 b(There)35
8183
 
b(are)i(a)f(few)g(macros)h(y)m(ou)f(m)m(ust)150 3674
 
8268
150 3876 y(advisable)35 b(to)i(use)e(it.)58 b(Y)-8 b(ou)36
 
8269
b(m)m(ust)g(edit)g Fp(fftw/fftw-int.h)p Fv(.)54 b(There)35
 
8270
b(are)i(a)f(few)g(macros)h(y)m(ou)f(m)m(ust)150 3985
8184
8271
y(rede\014ne.)46 b(The)32 b(co)s(de)h(is)e(do)s(cumen)m(ted)h(and)g
8185
8272
(should)e(b)s(e)i(self-explanatory)-8 b(.)47 b(\(By)33
8186
 
b(the)g(w)m(a)m(y)-8 b(,)34 b Fo(fftw-int)150 3784 y
8187
 
Fu(stands)28 b(for)g Fo(fftw-internal)p Fu(,)d(but)j(for)g(some)g
 
8273
b(the)g(w)m(a)m(y)-8 b(,)34 b Fp(fftw-int)150 4095 y
 
8274
Fv(stands)28 b(for)g Fp(fftw-internal)p Fv(,)d(but)j(for)g(some)g
8188
8275
(inexplicable)e(reason)i(p)s(eople)g(are)g(still)e(using)h(primitiv)m
8189
 
(e)150 3894 y(systems)j(with)f(8.3)j(\014lenames.\))275
8190
 
4069 y(Ev)m(en)e(if)g(y)m(ou)h(don't)g(install)e(high-resolution)f
 
8276
(e)150 4205 y(systems)j(with)f(8.3)j(\014lenames.\))275
 
8277
4334 y(Ev)m(en)e(if)g(y)m(ou)h(don't)g(install)e(high-resolution)f
8191
8278
(timing)h(co)s(de,)j(w)m(e)f(still)d(recommend)j(that)g(y)m(ou)g(lo)s
8192
 
(ok)150 4178 y(at)38 b(the)f Fo(FFTW_TIME_MIN)d Fu(constan)m(t)39
8193
 
b(in)c Fo(fftw/fftw-int.h)p Fu(.)58 b(This)35 b(constan)m(t)k(holds)c
8194
 
(the)j(minim)m(um)150 4288 y(time)h(in)m(terv)-5 b(al)39
 
8279
(ok)150 4444 y(at)38 b(the)f Fp(FFTW_TIME_MIN)d Fv(constan)m(t)39
 
8280
b(in)c Fp(fftw/fftw-int.h)p Fv(.)58 b(This)35 b(constan)m(t)k(holds)c
 
8281
(the)j(minim)m(um)150 4553 y(time)h(in)m(terv)-5 b(al)39
8195
8282
b(\(in)g(seconds\))h(required)e(to)i(get)h(accurate)g(timing)d
8196
 
(measuremen)m(ts,)43 b(and)c(should)f(b)s(e)150 4398
 
8283
(measuremen)m(ts,)43 b(and)c(should)f(b)s(e)150 4663
8197
8284
y(\(at)j(least\))f(sev)m(eral)g(h)m(undred)d(times)j(the)f(resolution)g
8198
8285
(of)g(y)m(our)h(clo)s(c)m(k.)69 b(The)39 b(default)g(constan)m(ts)i
8199
 
(are)150 4507 y(on)c(the)g(conserv)-5 b(ativ)m(e)39 b(side,)f(and)e(ma)
 
8286
(are)150 4772 y(on)c(the)g(conserv)-5 b(ativ)m(e)39 b(side,)f(and)e(ma)
8200
8287
m(y)i(cause)g(FFTW)g(to)f(tak)m(e)i(longer)e(than)g(necessary)g(when)g
8201
 
(y)m(ou)150 4617 y(create)f(a)e(plan.)51 b(Set)35 b Fo(FFTW_TIME_MIN)30
8202
 
b Fu(to)35 b(whatev)m(er)g(is)f(appropriate)f(on)h(y)m(our)g(system)h
8203
 
(\(b)s(e)f(sure)f(to)150 4726 y(set)e(the)h Fq(right)40
8204
 
b Fo(FFTW_TIME_MIN)6 b Fu(.)18 b(.)k(.)12 b(there)31
8205
 
b(are)g(sev)m(eral)g(de\014nitions)e(in)g Fo(fftw-int.h)p
8206
 
Fu(,)g(corresp)s(onding)150 4836 y(to)i(di\013eren)m(t)f(platforms)f
8207
 
(and)h(timers\).)275 5011 y(As)36 b(an)f(aid)h(in)e(c)m(hec)m(king)j
 
8288
(y)m(ou)150 4882 y(create)f(a)e(plan.)51 b(Set)35 b Fp(FFTW_TIME_MIN)30
 
8289
b Fv(to)35 b(whatev)m(er)g(is)f(appropriate)f(on)h(y)m(our)g(system)h
 
8290
(\(b)s(e)f(sure)f(to)150 4991 y(set)e(the)h Fr(right)40
 
8291
b Fp(FFTW_TIME_MIN)6 b Fv(.)18 b(.)k(.)12 b(there)31
 
8292
b(are)g(sev)m(eral)g(de\014nitions)e(in)g Fp(fftw-int.h)p
 
8293
Fv(,)g(corresp)s(onding)150 5101 y(to)i(di\013eren)m(t)f(platforms)f
 
8294
(and)h(timers\).)275 5230 y(As)36 b(an)f(aid)h(in)e(c)m(hec)m(king)j
8208
8295
(the)f(resolution)f(of)h(y)m(our)g(clo)s(c)m(k,)i(y)m(ou)f(can)f(use)g
8209
 
(the)g Fo(tests/fftw_test)150 5121 y Fu(program)41 b(with)g(the)g
8210
 
Fo(-t)g Fu(option)g(\(c.f.)75 b Fo(tests/README)p Fu(\).)d(Remem)m(b)s
8211
 
(er,)44 b(the)e(mere)f(fact)i(that)f(y)m(our)150 5230
8212
 
y(clo)s(c)m(k)32 b(rep)s(orts)f(times)g(in,)f(sa)m(y)-8
 
8296
(the)g Fp(tests/fftw_test)150 5340 y Fv(program)41 b(with)g(the)g
 
8297
Fp(-t)g Fv(option)g(\(c.f.)75 b Fp(tests/README)p Fv(\).)d(Remem)m(b)s
 
8298
(er,)44 b(the)e(mere)f(fact)i(that)f(y)m(our)p eop end
 
8299
%%Page: 59 61
 
8300
TeXDict begin 59 60 bop 150 -116 a Fv(Chapter)30 b(6:)41
 
8301
b(Installation)29 b(and)g(Customization)1808 b(59)150
 
8302
299 y(clo)s(c)m(k)32 b(rep)s(orts)f(times)g(in,)f(sa)m(y)-8
8213
8303
b(,)34 b(picoseconds,)d(do)s(es)g(not)h(mean)g(that)g(it)f(is)f
8214
 
(actually)h Fq(ac)-5 b(cur)g(ate)40 b Fu(to)32 b(that)150
8215
 
5340 y(resolution.)p eop
8216
 
%%Page: 59 61
8217
 
59 60 bop 150 -116 a Fu(Chapter)30 b(6:)41 b(Installation)29
8218
 
b(and)g(Customization)1808 b(59)150 299 y Ft(6.6)68 b(Generating)46
8219
 
b(y)l(our)f(o)l(wn)g(co)t(de)275 491 y Fu(If)22 b(y)m(ou)i(kno)m(w)g
 
8304
(actually)h Fr(ac)-5 b(cur)g(ate)40 b Fv(to)32 b(that)150
 
8305
408 y(resolution.)150 666 y Fu(6.6)68 b(Generating)46
 
8306
b(y)l(our)f(o)l(wn)g(co)t(de)275 858 y Fv(If)22 b(y)m(ou)i(kno)m(w)g
8220
8307
(that)g(y)m(ou)f(will)e(only)h(use)i(transforms)e(of)i(a)f(certain)h
8221
8308
(size)f(\(sa)m(y)-8 b(,)26 b(p)s(o)m(w)m(ers)e(of)f(2\))h(and)f(w)m(an)
8222
 
m(t)150 601 y(to)28 b(reduce)f(the)h(size)f(of)g(the)h(library)-8
 
8309
m(t)150 968 y(to)28 b(reduce)f(the)h(size)f(of)g(the)h(library)-8
8223
8310
b(,)26 b(y)m(ou)i(can)f(recon\014gure)h(FFTW)g(to)g(supp)s(ort)d(only)i
8224
 
(those)h(sizes)f(y)m(ou)150 711 y(are)34 b(in)m(terested)g(in.)50
 
8311
(those)h(sizes)f(y)m(ou)150 1078 y(are)34 b(in)m(terested)g(in.)50
8225
8312
b(Y)-8 b(ou)34 b(ma)m(y)h(ev)m(en)f(generate)i(co)s(de)e(to)g(enable)f
8226
8313
(e\016cien)m(t)i(transforms)e(of)h(a)g(size)g(not)150
8227
 
820 y(supp)s(orted)28 b(b)m(y)i(the)g(default)f(distribution.)36
 
8314
1187 y(supp)s(orted)28 b(b)m(y)i(the)g(default)f(distribution.)36
8228
8315
b(The)30 b(default)f(distribution)d(supp)s(orts)i(transforms)h(of)h(an)
8229
 
m(y)150 930 y(size,)e(but)e(not)h(all)e(sizes)i(are)g(equally)e(fast.)
 
8316
m(y)150 1297 y(size,)e(but)e(not)h(all)e(sizes)i(are)g(equally)e(fast.)
8230
8317
40 b(The)26 b(default)g(installation)f(of)i(FFTW)g(is)f(b)s(est)g(at)i
8231
 
(handling)150 1039 y(sizes)33 b(of)h(the)g(form)g(2)892
8232
 
1006 y Fk(a)932 1039 y Fu(3)977 1006 y Fk(b)1011 1039
8233
 
y Fu(5)1056 1006 y Fk(c)1090 1039 y Fu(7)1135 1006 y
8234
 
Fk(d)1174 1039 y Fu(11)1264 1006 y Fk(e)1301 1039 y Fu(13)1391
8235
 
1006 y Fk(f)1435 1039 y Fu(,)h(where)e Fl(e)p Fu(+)p
8236
 
Fl(f)43 b Fu(is)33 b(either)g(0)h(or)g(1,)h(and)e(the)h(other)g(exp)s
8237
 
(onen)m(ts)g(are)150 1149 y(arbitrary)-8 b(.)39 b(Other)27
 
8318
(handling)150 1406 y(sizes)33 b(of)h(the)g(form)g(2)892
 
8319
1373 y Fl(a)932 1406 y Fv(3)977 1373 y Fl(b)1011 1406
 
8320
y Fv(5)1056 1373 y Fl(c)1090 1406 y Fv(7)1135 1373 y
 
8321
Fl(d)1174 1406 y Fv(11)1264 1373 y Fl(e)1301 1406 y Fv(13)1391
 
8322
1373 y Fl(f)1435 1406 y Fv(,)h(where)e Fm(e)p Fv(+)p
 
8323
Fm(f)43 b Fv(is)33 b(either)g(0)h(or)g(1,)h(and)e(the)h(other)g(exp)s
 
8324
(onen)m(ts)g(are)150 1516 y(arbitrary)-8 b(.)39 b(Other)27
8238
8325
b(sizes)g(are)h(computed)f(b)m(y)g(means)h(of)f(a)h(slo)m(w,)g
8239
8326
(general-purp)s(ose)e(routine.)39 b(Ho)m(w)m(ev)m(er,)150
8240
 
1259 y(if)34 b(y)m(ou)h(ha)m(v)m(e)h(an)f(application)e(that)j
 
8327
1626 y(if)34 b(y)m(ou)h(ha)m(v)m(e)h(an)f(application)e(that)j
8241
8328
(requires)d(fast)j(transforms)e(of)h(size,)h(sa)m(y)-8
8242
 
b(,)37 b Fo(17)p Fu(,)f(there)f(is)f(a)h(w)m(a)m(y)h(to)150
8243
 
1368 y(generate)c(sp)s(ecialized)c(co)s(de)j(to)g(handle)e(that.)275
8244
 
1503 y(The)i(directory)g Fo(gensrc)g Fu(con)m(tains)h(all)f(the)h
 
8329
b(,)37 b Fp(17)p Fv(,)f(there)f(is)f(a)h(w)m(a)m(y)h(to)150
 
8330
1735 y(generate)c(sp)s(ecialized)c(co)s(de)j(to)g(handle)e(that.)275
 
8331
1870 y(The)i(directory)g Fp(gensrc)g Fv(con)m(tains)h(all)f(the)h
8245
8332
(programs)f(and)h(scripts)f(that)h(w)m(ere)h(used)e(to)h(generate)150
8246
 
1612 y(FFTW.)g(In)e(particular,)f(the)i(program)f Fo(gensrc/genfft.ml)c
8247
 
Fu(w)m(as)31 b(used)f(to)h(generate)i(the)d(co)s(de)h(that)150
8248
 
1722 y(FFTW)j(uses)f(to)h(compute)g(the)g(transforms.)49
 
8333
1979 y(FFTW.)g(In)e(particular,)f(the)i(program)f Fp(gensrc/genfft.ml)c
 
8334
Fv(w)m(as)31 b(used)f(to)h(generate)i(the)d(co)s(de)h(that)150
 
8335
2089 y(FFTW)j(uses)f(to)h(compute)g(the)g(transforms.)49
8249
8336
b(W)-8 b(e)35 b(do)e(not)h(exp)s(ect)g(casual)f(users)g(to)h(use)f(it.)
8250
 
50 b Fo(genfft)150 1831 y Fu(is)27 b(a)i(rather)f(sophisticated)g
 
8337
50 b Fp(genfft)150 2198 y Fv(is)27 b(a)i(rather)f(sophisticated)g
8251
8338
(program)g(that)h(generates)h(directed)e(acyclic)g(graphs)g(of)g(FFT)h
8252
 
(algorithms)150 1941 y(and)38 b(p)s(erforms)e(algebraic)i
8253
 
(simpli\014cations)d(on)j(them.)64 b Fo(genfft)36 b Fu(is)i(written)f
8254
 
(in)g(Ob)5 b(jectiv)m(e)38 b(Caml,)i(a)150 2051 y(dialect)g(of)h(ML.)g
 
8339
(algorithms)150 2308 y(and)38 b(p)s(erforms)e(algebraic)i
 
8340
(simpli\014cations)d(on)j(them.)64 b Fp(genfft)36 b Fv(is)i(written)f
 
8341
(in)g(Ob)5 b(jectiv)m(e)38 b(Caml,)i(a)150 2418 y(dialect)g(of)h(ML.)g
8255
8342
(Ob)5 b(jectiv)m(e)41 b(Caml)f(is)f(describ)s(ed)g(at)i
8256
 
Fo(http://pauillac.inria.fr/o)o(cam)o(l/)35 b Fu(and)150
8257
 
2160 y(can)c(b)s(e)e(do)m(wnloaded)h(from)g(from)g Fo
8258
 
(ftp://ftp.inria.fr/lang)o(/cam)o(l-l)o(ight)o Fu(.)275
8259
 
2295 y(If)37 b(y)m(ou)g(ha)m(v)m(e)i(Ob)5 b(jectiv)m(e)38
8260
 
b(Caml)e(installed,)i(y)m(ou)f(can)h(t)m(yp)s(e)g Fo(sh)30
8261
 
b(bootstrap.sh)k Fu(in)i(the)i(top-lev)m(el)150 2404
 
8343
Fp(http://pauillac.inria.fr/o)o(cam)o(l/)35 b Fv(and)150
 
8344
2527 y(can)c(b)s(e)e(do)m(wnloaded)h(from)g(from)g Fp
 
8345
(ftp://ftp.inria.fr/lang)o(/cam)o(l-l)o(ight)o Fv(.)275
 
8346
2662 y(If)37 b(y)m(ou)g(ha)m(v)m(e)i(Ob)5 b(jectiv)m(e)38
 
8347
b(Caml)e(installed,)i(y)m(ou)f(can)h(t)m(yp)s(e)g Fp(sh)30
 
8348
b(bootstrap.sh)k Fv(in)i(the)i(top-lev)m(el)150 2771
8262
8349
y(directory)32 b(to)i(re-generate)h(the)e(\014les.)47
8263
 
b(If)33 b(y)m(ou)g(c)m(hange)h(the)f Fo(gensrc/config)c
8264
 
Fu(\014le,)k(y)m(ou)g(can)h(optimize)150 2514 y(FFTW)d(for)f(sizes)g
 
8350
b(If)33 b(y)m(ou)g(c)m(hange)h(the)f Fp(gensrc/config)c
 
8351
Fv(\014le,)k(y)m(ou)g(can)h(optimize)150 2881 y(FFTW)d(for)f(sizes)g
8265
8352
(that)h(are)g(not)g(curren)m(tly)e(supp)s(orted)g(e\016cien)m(tly)h
8266
 
(\(sa)m(y)-8 b(,)32 b(17)f(or)f(19\).)275 2648 y(W)-8
 
8353
(\(sa)m(y)-8 b(,)32 b(17)f(or)f(19\).)275 3015 y(W)-8
8267
8354
b(e)44 b(do)e(not)h(pro)m(vide)f(more)h(details)f(ab)s(out)g(the)h(co)s
8268
8355
(de-generation)h(pro)s(cess,)i(since)c(w)m(e)h(do)g(not)150
8269
 
2758 y(exp)s(ect)32 b(that)g(users)e(will)f(need)i(to)h(generate)h
 
8356
3125 y(exp)s(ect)32 b(that)g(users)e(will)f(need)i(to)h(generate)h
8270
8357
(their)d(o)m(wn)h(co)s(de.)44 b(Ho)m(w)m(ev)m(er,)34
8271
 
b(feel)d(free)g(to)h(con)m(tact)i(us)c(at)150 2868 y
8272
 
Fo(fftw@fftw.org)d Fu(if)i(y)m(ou)i(are)g(in)m(terested)f(in)f(the)i
8273
 
(sub)5 b(ject.)275 3002 y(Y)-8 b(ou)21 b(migh)m(t)g(\014nd)f(it)g(in)m
 
8358
b(feel)d(free)g(to)h(con)m(tact)i(us)c(at)150 3235 y
 
8359
Fp(fftw@fftw.org)d Fv(if)i(y)m(ou)i(are)g(in)m(terested)f(in)f(the)i
 
8360
(sub)5 b(ject.)275 3369 y(Y)-8 b(ou)21 b(migh)m(t)g(\014nd)f(it)g(in)m
8274
8361
(teresting)h(to)h(learn)e(Caml)g(and/or)h(some)h(mo)s(dern)d
8275
 
(programming)h(tec)m(hniques)150 3112 y(that)39 b(w)m(e)f(used)f(in)g
 
8362
(programming)h(tec)m(hniques)150 3479 y(that)39 b(w)m(e)f(used)f(in)g
8276
8363
(the)h(generator)h(\(including)c(monadic)i(programming\),)i(esp)s
8277
 
(ecially)d(if)h(y)m(ou)i(heard)150 3221 y(the)d(rumor)f(that)h(Ja)m(v)
 
8364
(ecially)d(if)h(y)m(ou)i(heard)150 3588 y(the)d(rumor)f(that)h(Ja)m(v)
8278
8365
-5 b(a)37 b(and)f(ob)5 b(ject-orien)m(ted)37 b(programming)d(are)j(the)
8279
 
f(latest)g(adv)-5 b(ancemen)m(t)37 b(in)e(the)150 3331
 
8366
f(latest)g(adv)-5 b(ancemen)m(t)37 b(in)e(the)150 3698
8280
8367
y(\014eld.)60 b(The)37 b(in)m(ternal)f(op)s(eration)h(of)h(the)f(co)s
8281
8368
(delet)h(generator)g(is)f(describ)s(ed)e(in)h(the)h(pap)s(er,)i(\\A)f
8282
 
(F)-8 b(ast)150 3440 y(F)g(ourier)32 b(T)-8 b(ransform)31
 
8369
(F)-8 b(ast)150 3807 y(F)g(ourier)32 b(T)-8 b(ransform)31
8283
8370
b(Compiler,")h(b)m(y)g(M.)h(F)-8 b(rigo,)33 b(whic)m(h)e(is)h(a)m(v)-5
8284
 
b(ailable)31 b(from)h(the)h(FFTW)g(home)f(page)150 3550
8285
 
y(\()p Fo(http://www.fftw.org)p Fu(\))e(and)k(will)f(app)s(ear)h(in)f
8286
 
(the)i Fp(Pro)s(ceedings)f(of)h(the)g(1999)i(A)m(CM)e(SIGPLAN)150
8287
 
3660 y(Conference)30 b(on)h(Programming)e(Language)j(Design)e(and)g
8288
 
(Implemen)m(tation)f(\(PLDI\))p Fu(.)p eop
 
8371
b(ailable)31 b(from)h(the)h(FFTW)g(home)f(page)150 3917
 
8372
y(\()p Fp(http://www.fftw.org)p Fv(\))e(and)k(will)f(app)s(ear)h(in)f
 
8373
(the)i Fq(Pro)s(ceedings)f(of)h(the)g(1999)i(A)m(CM)e(SIGPLAN)150
 
8374
4027 y(Conference)30 b(on)h(Programming)e(Language)j(Design)e(and)g
 
8375
(Implemen)m(tation)f(\(PLDI\))p Fv(.)p eop end
8289
8376
%%Page: 60 62
8290
 
60 61 bop 150 -116 a Fu(60)3232 b(FFTW)p eop
 
8377
TeXDict begin 60 61 bop 150 -116 a Fv(60)3232 b(FFTW)p
 
8378
eop end
8291
8379
%%Page: 61 63
8292
 
61 62 bop 150 -116 a Fu(Chapter)30 b(7:)41 b(Ac)m(kno)m(wledgmen)m(ts)
8293
 
2335 b(61)150 299 y Fr(7)80 b(Ac)l(kno)l(wledgmen)l(ts)275
8294
 
533 y Fu(Matteo)28 b(F)-8 b(rigo)27 b(w)m(as)f(supp)s(orted)e(in)h
8295
 
(part)h(b)m(y)g(the)g(Defense)h(Adv)-5 b(anced)26 b(Researc)m(h)h(Pro)5
8296
 
b(jects)27 b(Agency)150 643 y(\(D)m(ARP)-8 b(A\))33 b(under)c(Gran)m
8297
 
(ts)j(N00014-94-1-0985)37 b(and)30 b(F30602-97-1-0270)q(,)37
8298
 
b(and)30 b(b)m(y)h(a)g(Digital)g(Equip-)150 752 y(men)m(t)d(Corp)s
8299
 
(oration)d(F)-8 b(ello)m(wship.)38 b(Stev)m(en)28 b(G.)g(Johnson)e(w)m
8300
 
(as)h(supp)s(orted)e(in)h(part)h(b)m(y)g(a)g(DoD)i(NDSEG)150
8301
 
862 y(F)-8 b(ello)m(wship,)33 b(an)g(MIT)h(Karl)e(T)-8
8302
 
b(a)m(ylor)34 b(Compton)f(F)-8 b(ello)m(wship,)33 b(and)g(b)m(y)h(the)g
8303
 
(Materials)f(Researc)m(h)i(Sci-)150 971 y(ence)43 b(and)f(Engineering)f
8304
 
(Cen)m(ter)i(program)f(of)h(the)f(National)h(Science)f(F)-8
8305
 
b(oundation)42 b(under)f(a)m(w)m(ard)150 1081 y(DMR-9400334.)275
8306
 
1215 y(Both)34 b(authors)g(w)m(ere)g(also)g(supp)s(orted)e(in)h(part)g
8307
 
(b)m(y)h(their)f(resp)s(ectiv)m(e)h(girlfriends,)e(b)m(y)i(the)g
8308
 
(letters)150 1325 y(\\Q")d(and)f(\\R",)h(and)f(b)m(y)g(the)h(n)m(um)m
8309
 
(b)s(er)e(12.)275 1460 y(W)-8 b(e)32 b(are)g(grateful)f(to)h(SUN)f
8310
 
(Microsystems)h(Inc.)f(for)g(its)g(donation)g(of)h(a)f(cluster)g(of)h
8311
 
(9)g(8-pro)s(cessor)150 1569 y(Ultra)h(HPC)g(5000)i(SMPs)d(\(24)j
8312
 
(G\015ops)d(p)s(eak\).)50 b(These)33 b(mac)m(hines)f(serv)m(ed)i(as)f
8313
 
(the)h(primary)d(platform)150 1679 y(for)f(the)h(dev)m(elopmen)m(t)f
8314
 
(of)h(earlier)e(v)m(ersions)h(of)h(FFTW.)275 1813 y(W)-8
8315
 
b(e)39 b(thank)e(In)m(tel)h(Corp)s(oration)e(for)i(donating)f(a)h
8316
 
(four-pro)s(cessor)f(P)m(en)m(tium)g(Pro)h(mac)m(hine.)63
8317
 
b(W)-8 b(e)150 1923 y(thank)30 b(the)h(Lin)m(ux)e(comm)m(unit)m(y)h
8318
 
(for)g(giving)f(us)h(a)g(decen)m(t)i(OS)d(to)i(run)e(on)i(that)g(mac)m
8319
 
(hine.)275 2057 y(The)22 b Fo(genfft)f Fu(program)i(w)m(as)h(written)e
8320
 
(using)f(Ob)5 b(jectiv)m(e)23 b(Caml,)h(a)g(dialect)e(of)h(ML.)h(Ob)5
8321
 
b(jectiv)m(e)23 b(Caml)150 2167 y(is)h(a)i(small)e(and)h(elegan)m(t)h
8322
 
(language)g(dev)m(elop)s(ed)f(b)m(y)g(Xa)m(vier)g(Lero)m(y)-8
8323
 
b(.)40 b(The)25 b(implemen)m(tation)f(is)h(a)m(v)-5 b(ailable)150
8324
 
2276 y(from)30 b Fo(ftp.inria.fr)e Fu(in)i(the)h(directory)f
8325
 
Fo(lang/caml-light)p Fu(.)39 b(W)-8 b(e)32 b(used)e(v)m(ersions)g(1.07)
8326
 
j(and)d(2.00)j(of)150 2386 y(the)39 b(soft)m(w)m(are.)66
8327
 
b(In)38 b(previous)f(releases)i(of)f(FFTW,)i Fo(genfft)d
8328
 
Fu(w)m(as)h(written)g(in)f(Caml)h(Ligh)m(t,)i(b)m(y)f(the)150
8329
 
2496 y(same)29 b(authors.)39 b(An)28 b(ev)m(en)h(earlier)f(implemen)m
8330
 
(tation)f(of)h Fo(genfft)f Fu(w)m(as)h(written)g(in)f(Sc)m(heme,)i(but)
8331
 
f(Caml)150 2605 y(is)h(de\014nitely)g(b)s(etter)h(for)g(this)g(kind)e
8332
 
(of)j(application.)275 2740 y(FFTW)47 b(uses)g(man)m(y)f(to)s(ols)h
8333
 
(from)g(the)g(GNU)g(pro)5 b(ject,)52 b(including)43 b
8334
 
Fo(automake)p Fu(,)49 b Fo(texinfo)p Fu(,)h(and)150 2849
8335
 
y Fo(libtool)p Fu(.)275 2984 y(Prof.)21 b(Charles)g(E.)h(Leiserson)e
8336
 
(of)i(MIT)g(pro)m(vided)e(con)m(tin)m(uous)i(supp)s(ort)e(and)h
8337
 
(encouragemen)m(t.)40 b(This)150 3093 y(program)33 b(w)m(ould)e(not)j
8338
 
(exist)e(without)g(him.)47 b(Charles)32 b(also)g(prop)s(osed)g(the)h
8339
 
(name)g(\\co)s(delets")h(for)f(the)150 3203 y(basic)d(FFT)g(blo)s(c)m
8340
 
(ks.)275 3337 y(Prof.)48 b(John)g(D.)h(Joannop)s(oulos)e(of)i(MIT)g
8341
 
(demonstrated)f(con)m(tin)m(uing)g(tolerance)h(of)g(Stev)m(en's)150
8342
 
3447 y(\\extra-curricular")32 b(computer-science)h(activities.)47
8343
 
b(Stev)m(en's)33 b(c)m(hances)h(at)f(a)g(ph)m(ysics)f(degree)h(w)m
8344
 
(ould)150 3557 y(not)e(exist)f(without)f(him.)275 3691
8345
 
y(Andrew)g(Sterian)g(con)m(tributed)h(the)g(Windo)m(ws)g(timing)f(co)s
8346
 
(de.)275 3826 y(Didier)34 b(Miras)i(rep)s(orted)g(a)h(bug)f(in)f(the)i
8347
 
(test)g(pro)s(cedure)e(used)h(in)f(FFTW)i(1.2.)60 b(W)-8
8348
 
b(e)38 b(no)m(w)e(use)h(a)150 3935 y(completely)32 b(di\013eren)m(t)g
8349
 
(test)h(algorithm)e(b)m(y)h(F)-8 b(unda)32 b(Ergun)f(that)i(do)s(es)f
8350
 
(not)g(require)f(a)h(separate)i(FFT)150 4045 y(program)c(to)h(compare)g
 
8380
TeXDict begin 61 62 bop 150 -116 a Fv(Chapter)30 b(7:)41
 
8381
b(Ac)m(kno)m(wledgmen)m(ts)2335 b(61)150 299 y Fs(7)80
 
8382
b(Ac)l(kno)l(wledgmen)l(ts)275 533 y Fv(Matteo)28 b(F)-8
 
8383
b(rigo)27 b(w)m(as)f(supp)s(orted)e(in)h(part)h(b)m(y)g(the)g(Defense)h
 
8384
(Adv)-5 b(anced)26 b(Researc)m(h)h(Pro)5 b(jects)27 b(Agency)150
 
8385
643 y(\(D)m(ARP)-8 b(A\))33 b(under)c(Gran)m(ts)j(N00014-94-1-0985)37
 
8386
b(and)30 b(F30602-97-1-0270)q(,)37 b(and)30 b(b)m(y)h(a)g(Digital)g
 
8387
(Equip-)150 752 y(men)m(t)d(Corp)s(oration)d(F)-8 b(ello)m(wship.)38
 
8388
b(Stev)m(en)28 b(G.)g(Johnson)e(w)m(as)h(supp)s(orted)e(in)h(part)h(b)m
 
8389
(y)g(a)g(DoD)i(NDSEG)150 862 y(F)-8 b(ello)m(wship,)33
 
8390
b(an)g(MIT)h(Karl)e(T)-8 b(a)m(ylor)34 b(Compton)f(F)-8
 
8391
b(ello)m(wship,)33 b(and)g(b)m(y)h(the)g(Materials)f(Researc)m(h)i
 
8392
(Sci-)150 971 y(ence)43 b(and)f(Engineering)f(Cen)m(ter)i(program)f(of)
 
8393
h(the)f(National)h(Science)f(F)-8 b(oundation)42 b(under)f(a)m(w)m(ard)
 
8394
150 1081 y(DMR-9400334.)275 1215 y(Both)34 b(authors)g(w)m(ere)g(also)g
 
8395
(supp)s(orted)e(in)h(part)g(b)m(y)h(their)f(resp)s(ectiv)m(e)h
 
8396
(girlfriends,)e(b)m(y)i(the)g(letters)150 1325 y(\\Q")d(and)f(\\R",)h
 
8397
(and)f(b)m(y)g(the)h(n)m(um)m(b)s(er)e(12.)275 1460 y(W)-8
 
8398
b(e)32 b(are)g(grateful)f(to)h(SUN)f(Microsystems)h(Inc.)f(for)g(its)g
 
8399
(donation)g(of)h(a)f(cluster)g(of)h(9)g(8-pro)s(cessor)150
 
8400
1569 y(Ultra)h(HPC)g(5000)i(SMPs)d(\(24)j(G\015ops)d(p)s(eak\).)50
 
8401
b(These)33 b(mac)m(hines)f(serv)m(ed)i(as)f(the)h(primary)d(platform)
 
8402
150 1679 y(for)f(the)h(dev)m(elopmen)m(t)f(of)h(earlier)e(v)m(ersions)h
 
8403
(of)h(FFTW.)275 1813 y(W)-8 b(e)39 b(thank)e(In)m(tel)h(Corp)s(oration)
 
8404
e(for)i(donating)f(a)h(four-pro)s(cessor)f(P)m(en)m(tium)g(Pro)h(mac)m
 
8405
(hine.)63 b(W)-8 b(e)150 1923 y(thank)30 b(the)h(Lin)m(ux)e(comm)m
 
8406
(unit)m(y)h(for)g(giving)f(us)h(a)g(decen)m(t)i(OS)d(to)i(run)e(on)i
 
8407
(that)g(mac)m(hine.)275 2057 y(The)22 b Fp(genfft)f Fv(program)i(w)m
 
8408
(as)h(written)e(using)f(Ob)5 b(jectiv)m(e)23 b(Caml,)h(a)g(dialect)e
 
8409
(of)h(ML.)h(Ob)5 b(jectiv)m(e)23 b(Caml)150 2167 y(is)h(a)i(small)e
 
8410
(and)h(elegan)m(t)h(language)g(dev)m(elop)s(ed)f(b)m(y)g(Xa)m(vier)g
 
8411
(Lero)m(y)-8 b(.)40 b(The)25 b(implemen)m(tation)f(is)h(a)m(v)-5
 
8412
b(ailable)150 2276 y(from)30 b Fp(ftp.inria.fr)e Fv(in)i(the)h
 
8413
(directory)f Fp(lang/caml-light)p Fv(.)39 b(W)-8 b(e)32
 
8414
b(used)e(v)m(ersions)g(1.07)j(and)d(2.00)j(of)150 2386
 
8415
y(the)39 b(soft)m(w)m(are.)66 b(In)38 b(previous)f(releases)i(of)f
 
8416
(FFTW,)i Fp(genfft)d Fv(w)m(as)h(written)g(in)f(Caml)h(Ligh)m(t,)i(b)m
 
8417
(y)f(the)150 2496 y(same)29 b(authors.)39 b(An)28 b(ev)m(en)h(earlier)f
 
8418
(implemen)m(tation)f(of)h Fp(genfft)f Fv(w)m(as)h(written)g(in)f(Sc)m
 
8419
(heme,)i(but)f(Caml)150 2605 y(is)h(de\014nitely)g(b)s(etter)h(for)g
 
8420
(this)g(kind)e(of)j(application.)275 2740 y(FFTW)47 b(uses)g(man)m(y)f
 
8421
(to)s(ols)h(from)g(the)g(GNU)g(pro)5 b(ject,)52 b(including)43
 
8422
b Fp(automake)p Fv(,)49 b Fp(texinfo)p Fv(,)h(and)150
 
8423
2849 y Fp(libtool)p Fv(.)275 2984 y(Prof.)21 b(Charles)g(E.)h
 
8424
(Leiserson)e(of)i(MIT)g(pro)m(vided)e(con)m(tin)m(uous)i(supp)s(ort)e
 
8425
(and)h(encouragemen)m(t.)40 b(This)150 3093 y(program)33
 
8426
b(w)m(ould)e(not)j(exist)e(without)g(him.)47 b(Charles)32
 
8427
b(also)g(prop)s(osed)g(the)h(name)g(\\co)s(delets")h(for)f(the)150
 
8428
3203 y(basic)d(FFT)g(blo)s(c)m(ks.)275 3337 y(Prof.)48
 
8429
b(John)g(D.)h(Joannop)s(oulos)e(of)i(MIT)g(demonstrated)f(con)m(tin)m
 
8430
(uing)g(tolerance)h(of)g(Stev)m(en's)150 3447 y(\\extra-curricular")32
 
8431
b(computer-science)h(activities.)47 b(Stev)m(en's)33
 
8432
b(c)m(hances)h(at)f(a)g(ph)m(ysics)f(degree)h(w)m(ould)150
 
8433
3557 y(not)e(exist)f(without)f(him.)275 3691 y(Andrew)g(Sterian)g(con)m
 
8434
(tributed)h(the)g(Windo)m(ws)g(timing)f(co)s(de.)275
 
8435
3826 y(Didier)34 b(Miras)i(rep)s(orted)g(a)h(bug)f(in)f(the)i(test)g
 
8436
(pro)s(cedure)e(used)h(in)f(FFTW)i(1.2.)60 b(W)-8 b(e)38
 
8437
b(no)m(w)e(use)h(a)150 3935 y(completely)32 b(di\013eren)m(t)g(test)h
 
8438
(algorithm)e(b)m(y)h(F)-8 b(unda)32 b(Ergun)f(that)i(do)s(es)f(not)g
 
8439
(require)f(a)h(separate)i(FFT)150 4045 y(program)c(to)h(compare)g
8351
8440
(against.)275 4179 y(W)-8 b(olfgang)26 b(Reimer)f(con)m(tributed)g(the)
8352
8441
h(P)m(en)m(tium)f(cycle)h(coun)m(ter)g(and)g(a)g(few)f(\014xes)g(that)i
8353
8442
(help)d(p)s(orta-)150 4289 y(bilit)m(y)-8 b(.)275 4423
8355
8444
(in)f(the)i(complex)g(transforms)f(of)g(FFTW)i(2.0)150
8356
8445
4533 y(and)c(supplied)d(a)k(patc)m(h)g(to)g(correct)g(it.)275
8357
8446
4667 y(The)h(FFTW)h(F)-10 b(A)m(Q)33 b(w)m(as)g(written)f(in)f
8358
 
Fo(bfnn)g Fu(\(Bizarre)i(F)-8 b(ormat)34 b(With)e(No)i(Name\))f(and)f
 
8447
Fp(bfnn)g Fv(\(Bizarre)i(F)-8 b(ormat)34 b(With)e(No)i(Name\))f(and)f
8359
8448
(formatted)150 4777 y(using)d(the)h(to)s(ols)h(dev)m(elop)s(ed)e(b)m(y)
8360
8449
i(Ian)f(Jac)m(kson)h(for)f(the)h(Lin)m(ux)d(F)-10 b(A)m(Q.)275
8361
 
4912 y Fq(We)30 b(ar)-5 b(e)32 b(esp)-5 b(e)g(cial)5
 
8450
4912 y Fr(We)30 b(ar)-5 b(e)32 b(esp)-5 b(e)g(cial)5
8362
8451
b(ly)32 b(thankful)g(to)g(al)5 b(l)32 b(of)f(our)g(users)h(for)f(their)
8363
8452
h(c)-5 b(ontinuing)31 b(supp)-5 b(ort,)34 b(fe)-5 b(e)g(db)g(ack,)32
8364
8453
b(and)150 5021 y(inter)-5 b(est)33 b(during)g(our)g(development)i(of)d
8365
 
(FFTW.)p eop
 
8454
(FFTW.)p eop end
8366
8455
%%Page: 62 64
8367
 
62 63 bop 150 -116 a Fu(62)3232 b(FFTW)p eop
 
8456
TeXDict begin 62 63 bop 150 -116 a Fv(62)3232 b(FFTW)p
 
8457
eop end
8368
8458
%%Page: 63 65
8369
 
63 64 bop 150 -116 a Fu(Chapter)30 b(8:)41 b(License)30
8370
 
b(and)g(Cop)m(yrigh)m(t)2150 b(63)150 299 y Fr(8)80 b(License)53
8371
 
b(and)h(Cop)l(yrigh)l(t)275 533 y Fu(FFTW)31 b(is)e(cop)m(yrigh)m(t)
8372
 
1101 530 y(c)1076 533 y Fs(\015)h Fu(1997{1999)k(Massac)m(h)m(usetts)f
8373
 
(Institute)d(of)g(T)-8 b(ec)m(hnology)g(.)275 667 y(FFTW)31
8374
 
b(is)e(free)i(soft)m(w)m(are;)h(y)m(ou)f(can)g(redistribute)d(it)i
8375
 
(and/or)g(mo)s(dify)f(it)h(under)f(the)i(terms)f(of)h(the)150
8376
 
777 y(GNU)g(General)g(Public)d(License)i(as)h(published)26
8377
 
b(b)m(y)31 b(the)g(F)-8 b(ree)31 b(Soft)m(w)m(are)h(F)-8
8378
 
b(oundation;)30 b(either)g(v)m(ersion)150 887 y(2)h(of)f(the)h
8379
 
(License,)f(or)g(\(at)i(y)m(our)e(option\))g(an)m(y)h(later)f(v)m
8380
 
(ersion.)275 1021 y(This)37 b(program)i(is)g(distributed)d(in)i(the)i
8381
 
(hop)s(e)f(that)h(it)f(will)e(b)s(e)i(useful,)h(but)e(WITHOUT)h(ANY)150
8382
 
1131 y(W)-10 b(ARRANTY;)33 b(without)f(ev)m(en)h(the)f(implied)e(w)m
8383
 
(arran)m(t)m(y)j(of)g(MER)m(CHANT)-8 b(ABILITY)33 b(or)f(FITNESS)150
8384
 
1240 y(F)m(OR)e(A)g(P)-8 b(AR)g(TICULAR)30 b(PURPOSE.)f(See)h(the)g
8385
 
(GNU)g(General)g(Public)e(License)h(for)h(more)g(details.)275
8386
 
1375 y(Y)-8 b(ou)39 b(should)d(ha)m(v)m(e)k(receiv)m(ed)f(a)g(cop)m(y)h
8387
 
(of)f(the)f(GNU)i(General)e(Public)f(License)h(along)h(with)e(this)150
 
8459
TeXDict begin 63 64 bop 150 -116 a Fv(Chapter)30 b(8:)41
 
8460
b(License)30 b(and)g(Cop)m(yrigh)m(t)2150 b(63)150 299
 
8461
y Fs(8)80 b(License)53 b(and)h(Cop)l(yrigh)l(t)275 533
 
8462
y Fv(FFTW)31 b(is)e(cop)m(yrigh)m(t)1101 530 y(c)1076
 
8463
533 y Ft(\015)h Fv(1997{1999)k(Massac)m(h)m(usetts)f(Institute)d(of)g
 
8464
(T)-8 b(ec)m(hnology)g(.)275 667 y(FFTW)31 b(is)e(free)i(soft)m(w)m
 
8465
(are;)h(y)m(ou)f(can)g(redistribute)d(it)i(and/or)g(mo)s(dify)f(it)h
 
8466
(under)f(the)i(terms)f(of)h(the)150 777 y(GNU)g(General)g(Public)d
 
8467
(License)i(as)h(published)26 b(b)m(y)31 b(the)g(F)-8
 
8468
b(ree)31 b(Soft)m(w)m(are)h(F)-8 b(oundation;)30 b(either)g(v)m(ersion)
 
8469
150 887 y(2)h(of)f(the)h(License,)f(or)g(\(at)i(y)m(our)e(option\))g
 
8470
(an)m(y)h(later)f(v)m(ersion.)275 1021 y(This)37 b(program)i(is)g
 
8471
(distributed)d(in)i(the)i(hop)s(e)f(that)h(it)f(will)e(b)s(e)i(useful,)
 
8472
h(but)e(WITHOUT)h(ANY)150 1131 y(W)-10 b(ARRANTY;)33
 
8473
b(without)f(ev)m(en)h(the)f(implied)e(w)m(arran)m(t)m(y)j(of)g(MER)m
 
8474
(CHANT)-8 b(ABILITY)33 b(or)f(FITNESS)150 1240 y(F)m(OR)e(A)g(P)-8
 
8475
b(AR)g(TICULAR)30 b(PURPOSE.)f(See)h(the)g(GNU)g(General)g(Public)e
 
8476
(License)h(for)h(more)g(details.)275 1375 y(Y)-8 b(ou)39
 
8477
b(should)d(ha)m(v)m(e)k(receiv)m(ed)f(a)g(cop)m(y)h(of)f(the)f(GNU)i
 
8478
(General)e(Public)f(License)h(along)h(with)e(this)150
8388
8479
1484 y(program;)56 b(if)46 b(not,)52 b(write)47 b(to)h(the)g(F)-8
8389
8480
b(ree)48 b(Soft)m(w)m(are)h(F)-8 b(oundation,)51 b(Inc.,)h(59)c(T)-8
8390
8481
b(emple)47 b(Place,)52 b(Suite)150 1594 y(330,)g(Boston,)f(MA)c
8391
8482
(02111-1307)j(USA.)c(Y)-8 b(ou)47 b(can)g(also)f(\014nd)e(the)j(GPL)f
8392
 
(on)g(the)g(GNU)h(w)m(eb)f(site)150 1704 y(\()p Fo
8393
 
(http://www.gnu.org/copylef)o(t/g)o(pl.h)o(tml)p Fu(\).)275
 
8483
(on)g(the)g(GNU)h(w)m(eb)f(site)150 1704 y(\()p Fp
 
8484
(http://www.gnu.org/copylef)o(t/g)o(pl.h)o(tml)p Fv(\).)275
8394
8485
1838 y(In)33 b(addition,)g(w)m(e)h(kindly)d(ask)j(y)m(ou)g(to)h(ac)m
8395
8486
(kno)m(wledge)g(FFTW)g(and)e(its)g(authors)h(in)e(an)m(y)i(program)150
8396
8487
1948 y(or)j(publication)e(in)h(whic)m(h)g(y)m(ou)h(use)g(FFTW.)i(\(Y)-8
8397
 
b(ou)38 b(are)f(not)h Fq(r)-5 b(e)g(quir)g(e)g(d)48 b
8398
 
Fu(to)38 b(do)f(so;)42 b(it)36 b(is)h(up)f(to)i(y)m(our)150
 
8488
b(ou)38 b(are)f(not)h Fr(r)-5 b(e)g(quir)g(e)g(d)48 b
 
8489
Fv(to)38 b(do)f(so;)42 b(it)36 b(is)h(up)f(to)i(y)m(our)150
8399
8490
2057 y(common)31 b(sense)f(to)h(decide)f(whether)g(y)m(ou)g(w)m(an)m(t)
8400
8491
i(to)f(comply)e(with)g(this)h(request)g(or)g(not.\))275
8401
8492
2192 y(Non-free)g(v)m(ersions)f(of)h(FFTW)g(are)h(a)m(v)-5
8407
8498
m(ou)g(m)m(ust)g(purc)m(hase)f(a)i(license)d(from)150
8408
8499
2521 y(MIT's)39 b(T)-8 b(ec)m(hnology)40 b(Licensing)d(O\016ce.)67
8409
8500
b(Users)39 b(in)m(terested)g(in)f(suc)m(h)h(a)g(license)f(should)f(con)
8410
 
m(tact)42 b(us)150 2630 y(\()p Fo(fftw@fftw.org)p Fu(\))27
8411
 
b(for)k(more)f(information.)p eop
 
8501
m(tact)42 b(us)150 2630 y(\()p Fp(fftw@fftw.org)p Fv(\))27
 
8502
b(for)k(more)f(information.)p eop end
8412
8503
%%Page: 64 66
8413
 
64 65 bop 150 -116 a Fu(64)3232 b(FFTW)p eop
 
8504
TeXDict begin 64 65 bop 150 -116 a Fv(64)3232 b(FFTW)p
 
8505
eop end
8414
8506
%%Page: 65 67
8415
 
65 66 bop 150 -116 a Fu(Chapter)30 b(9:)41 b(Concept)30
8416
 
b(Index)2463 b(65)150 299 y Fr(9)80 b(Concept)52 b(Index)150
8417
 
651 y Ft(A)150 785 y Fd(algorithm)21 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)
8418
 
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h
8419
 
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)47 b Fd(2)150
8420
 
1058 y Ft(B)150 1193 y Fb(benchfft)8 b Fc(.)14 b(.)e(.)h(.)f(.)g(.)h(.)
8421
 
f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8422
 
(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)34
8423
 
b Fd(1)150 1294 y(b)r(enc)n(hmark)12 b Fc(.)f(.)h(.)h(.)f(.)g(.)h(.)f
8424
 
(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
8425
 
39 b Fd(1,)26 b(40,)h(49,)f(58)150 1394 y(blo)r(c)n(king)e
8426
 
Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8427
 
(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
8428
 
g(.)h(.)49 b Fd(43)150 1667 y Ft(C)150 1802 y Fd(C)26
8429
 
b(m)n(ulti-dimensional)f(arra)n(ys)16 b Fc(.)d(.)f(.)g(.)h(.)f(.)g(.)h
8430
 
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)42 b Fd(12)150
8431
 
1902 y(Caml)12 b Fc(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8432
 
(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
8433
 
g(.)h(.)f(.)g(.)h(.)38 b Fd(59,)26 b(61)150 2003 y(Cilk)15
8434
 
b Fc(.)e(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)
8435
 
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8436
 
(.)g(.)h(.)f(.)41 b Fd(2,)26 b(37)150 2104 y Fb(clock)11
8437
 
b Fc(.)j(.)e(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
8438
 
h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8439
 
(.)g(.)h(.)f(.)g(.)37 b Fd(58)150 2204 y(co)r(de)26 b(generator)21
8440
 
b Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8441
 
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)45 b Fd(1,)26
8442
 
b(59)150 2305 y(column-ma)t(jor)e Fc(.)12 b(.)g(.)g(.)h(.)f(.)g(.)h(.)f
8443
 
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)50
8444
 
b Fd(11,)27 b(52,)f(54)150 2406 y(compiler)10 b Fc(.)j(.)f(.)g(.)h(.)f
8445
 
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
8446
 
h(.)36 b Fd(2,)26 b(51,)h(55,)g(56,)f(57)150 2507 y(compiler)g(\015ags)
8447
 
16 b Fc(.)d(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8448
 
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)43 b Fd(55,)26
 
8507
TeXDict begin 65 66 bop 150 -116 a Fv(Chapter)30 b(9:)41
 
8508
b(Concept)30 b(Index)2463 b(65)150 299 y Fs(9)80 b(Concept)52
 
8509
b(Index)150 642 y Fu(A)150 777 y Fj(algorithm)21 b Ff(.)12
 
8510
b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8511
(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
 
8512
47 b Fj(2)150 1069 y Fu(B)150 1204 y Fi(benchfft)8 b
 
8513
Ff(.)14 b(.)e(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
 
8514
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
 
8515
h(.)f(.)g(.)34 b Fj(1)150 1301 y(b)r(enc)n(hmark)12 b
 
8516
Ff(.)f(.)h(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f
 
8517
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)39 b Fj(1,)26 b(40,)h(49,)f(58)150
 
8518
1399 y(blo)r(c)n(king)e Ff(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
 
8519
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
 
8520
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)49 b Fj(43)150 1691 y
 
8521
Fu(C)150 1826 y Fj(C)26 b(m)n(ulti-dimensional)f(arra)n(ys)16
 
8522
b Ff(.)d(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
 
8523
f(.)g(.)42 b Fj(12)150 1923 y(Caml)12 b Ff(.)h(.)f(.)g(.)h(.)f(.)g(.)h
 
8524
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)
 
8525
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)38 b Fj(59,)26
 
8526
b(61)150 2021 y(Cilk)15 b Ff(.)e(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
 
8527
h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8528
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)41 b Fj(2,)26
 
8529
b(37)150 2119 y Fi(clock)11 b Ff(.)j(.)e(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)
 
8530
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8531
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)37 b Fj(58)150
 
8532
2217 y(co)r(de)26 b(generator)21 b Ff(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)
 
8533
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8534
(.)g(.)h(.)45 b Fj(1,)26 b(59)150 2314 y(column-ma)t(jor)e
 
8535
Ff(.)12 b(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
 
8536
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)50 b Fj(11,)27 b(52,)f(54)150
 
8537
2412 y(compiler)10 b Ff(.)j(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8538
(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)36 b Fj(2,)26
 
8539
b(51,)h(55,)g(56,)f(58)150 2510 y(compiler)g(\015ags)16
 
8540
b Ff(.)d(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
 
8541
g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)43 b Fj(55,)26
8449
8542
b(56)150 2607 y(complex)f(m)n(ulti-dimensional)f(transform)d
8450
 
Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)47 b Fd(4,)26 b(22)150
8451
 
2708 y(complex)f(n)n(um)n(b)r(er)13 b Fc(.)e(.)h(.)g(.)g(.)h(.)f(.)g(.)
 
8543
Ff(.)13 b(.)f(.)g(.)h(.)f(.)g(.)47 b Fj(4,)26 b(22)150
 
8544
2705 y(complex)f(n)n(um)n(b)r(er)13 b Ff(.)e(.)h(.)g(.)g(.)h(.)f(.)g(.)
8452
8545
h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8453
 
(.)h(.)f(.)g(.)39 b Fd(17)150 2809 y(complex)25 b(one-dimensional)g
8454
 
(transform)e Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)49
8455
 
b Fd(3)150 2909 y(complex)25 b(to)h(real)g(transform)12
8456
 
b Fc(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
8457
 
f(.)38 b Fd(6,)26 b(26)150 3010 y(complex)f(transform)14
8458
 
b Fc(.)e(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
8459
 
h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)40 b Fd(3)150
8460
 
3111 y Fb(configure)12 b Fc(.)i(.)f(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8461
 
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)38 b Fd(18,)26
8462
 
b(37,)h(41,)g(55,)f(57)150 3211 y(con)n(v)n(olution)7
8463
 
b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8546
(.)h(.)f(.)g(.)39 b Fj(17)150 2803 y(complex)25 b(one-dimensional)g
 
8547
(transform)e Ff(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)49
 
8548
b Fj(3)150 2901 y(complex)25 b(to)h(real)g(transform)12
 
8549
b Ff(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
 
8550
f(.)38 b Fj(6,)26 b(26)150 2998 y(complex)f(transform)14
 
8551
b Ff(.)e(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
 
8552
h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)40 b Fj(3)150
 
8553
3096 y Fi(configure)12 b Ff(.)i(.)f(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
 
8554
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)38 b Fj(18,)26
 
8555
b(37,)h(41,)g(55,)f(57)150 3194 y(con)n(v)n(olution)7
 
8556
b Ff(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8464
8557
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
8465
 
32 b Fd(10)150 3312 y(cyclic)26 b(con)n(v)n(olution)10
8466
 
b Fc(.)j(.)f(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
8467
 
f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)36 b Fd(10)150
8468
 
3585 y Ft(D)150 3720 y Fd(Discrete)26 b(F)-6 b(ourier)26
8469
 
b(T)-6 b(ransform)24 b Fc(.)13 b(.)f(.)g(.)h(.)50 b Fd(21,)26
8470
 
b(25,)h(28,)g(33,)f(34)150 3820 y(distributed)f(arra)n(y)h(format)12
8471
 
b Fc(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)38
8472
 
b Fd(43,)27 b(46,)f(48)150 3921 y(distributed)f(memory)17
8473
 
b Fc(.)10 b(.)i(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8474
 
(.)g(.)h(.)42 b Fd(37,)27 b(40,)f(43)150 4193 y Ft(E)150
8475
 
4328 y Fd(Ecclesiastes)f Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
 
8558
32 b Fj(10)150 3291 y(cyclic)26 b(con)n(v)n(olution)10
 
8559
b Ff(.)j(.)f(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
 
8560
f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)36 b Fj(10)150
 
8561
3583 y Fu(D)150 3718 y Fj(Discrete)26 b(F)-6 b(ourier)26
 
8562
b(T)-6 b(ransform)13 b Ff(.)g(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)39
 
8563
b Fj(21,)27 b(25,)g(28,)f(33)150 3816 y(distributed)f(arra)n(y)h
 
8564
(format)12 b Ff(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8565
(.)38 b Fj(43,)27 b(46,)f(48)150 3913 y(distributed)f(memory)17
 
8566
b Ff(.)10 b(.)i(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8567
(.)g(.)h(.)42 b Fj(37,)27 b(40,)f(43)150 4205 y Fu(E)150
 
8568
4340 y Fj(Ecclesiastes)f Ff(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8476
8569
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)
8477
 
g(.)h(.)f(.)g(.)h(.)48 b Fd(14)150 4429 y(executor)23
8478
 
b Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h
 
8570
g(.)h(.)f(.)g(.)h(.)48 b Fj(14)150 4438 y(executor)23
 
8571
b Ff(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h
8479
8572
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
8480
 
h(.)f(.)g(.)49 b Fd(1)150 4702 y Ft(F)150 4836 y Fd(FFTW)11
8481
 
b Fc(.)i(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)
 
8573
h(.)f(.)g(.)49 b Fj(1)150 4717 y Fu(F)150 4851 y Fj(FFTW)11
 
8574
b Ff(.)i(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)
8482
8575
f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8483
 
(.)f(.)g(.)h(.)f(.)37 b Fd(1)150 4937 y(FFTWND)6 b Fc(.)13
 
8576
(.)f(.)g(.)h(.)f(.)37 b Fj(1)150 4949 y(FFTWND)6 b Ff(.)13
8484
8577
b(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8485
8578
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)32
8486
 
b Fd(22)150 5038 y(\015ags)22 b Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8487
 
(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)47 b Fd(4,)26
8488
 
b(5,)h(19,)g(23,)f(27,)h(30,)g(47,)f(52)150 5139 y(\015oating-p)r(oin)n
8489
 
(t)g(precision)21 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)46
8490
 
b Fd(18,)26 b(52,)h(55,)g(56,)f(57)150 5239 y(F)-6 b(ortran-callable)27
8491
 
b(wrapp)r(ers)8 b Fc(.)14 b(.)e(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8492
 
(.)h(.)34 b Fd(12,)27 b(51,)f(56)150 5340 y(frequency)8
8493
 
b Fc(.)k(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
 
8579
b Fj(22)150 5047 y(\015ags)22 b Ff(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8580
(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)47 b Fj(4,)26
 
8581
b(5,)h(19,)g(23,)f(27,)h(30,)g(47,)f(52)150 5145 y(\015oating-p)r(oin)n
 
8582
(t)g(precision)21 b Ff(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)46
 
8583
b Fj(18,)26 b(52,)h(55,)g(56,)f(57)150 5242 y(F)-6 b(ortran-callable)27
 
8584
b(wrapp)r(ers)8 b Ff(.)14 b(.)e(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8585
(.)h(.)34 b Fj(12,)27 b(51,)f(56)150 5340 y(frequency)8
 
8586
b Ff(.)k(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
8494
8587
h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)34
8495
 
b Fd(4,)26 b(5,)g(21)2025 651 y Ft(G)2025 798 y Fb(gettimeofday)c
8496
 
Fc(.)12 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
 
8588
b Fj(4,)26 b(5,)g(21)2025 642 y Fu(G)2025 765 y Fi(gettimeofday)c
 
8589
Ff(.)12 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8497
8590
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)45
8498
 
b Fd(58)2025 905 y(girlfriends)14 b Fc(.)g(.)e(.)g(.)h(.)f(.)g(.)h(.)f
8499
 
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)
8500
 
h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)40 b Fd(61)2025 1212
8501
 
y Ft(H)2025 1360 y Fd(halfcomplex)25 b(arra)n(y)20 b
8502
 
Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8503
 
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)46 b Fd(6,)26 b(17)2025
8504
 
1466 y(hermitian)f(arra)n(y)c Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8505
 
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)46
8506
 
b Fd(17,)27 b(34)2025 1774 y Ft(I)2025 1922 y Fd(in-place)c(transform)8
8507
 
b Fc(.)13 b(.)32 b Fd(5,)24 b(19,)h(21,)g(25,)g(30,)g(32,)g(40,)g(42,)g
8508
 
(51)2025 2028 y(installation)12 b Fc(.)i(.)e(.)h(.)f(.)g(.)h(.)f(.)g(.)
8509
 
h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8510
 
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)38 b Fd(55)2025 2336 y
8511
 
Ft(L)2025 2483 y Fd(linking)26 b(on)f(Unix)13 b Fc(.)e(.)i(.)f(.)g(.)h
8512
 
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)38 b Fd(4,)27
8513
 
b(7,)f(39,)h(43,)f(47,)h(48)2025 2590 y(LISP)16 b Fc(.)c(.)g(.)h(.)f(.)
8514
 
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8515
 
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)42
8516
 
b Fd(15,)27 b(61)2025 2696 y(load-balancing)11 b Fc(.)j(.)e(.)g(.)h(.)f
8517
 
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)
8518
 
h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)37 b Fd(48)2025 3004
8519
 
y Ft(M)2025 3151 y Fb(malloc)22 b Fc(.)12 b(.)g(.)g(.)h(.)f(.)g(.)h(.)f
8520
 
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
8521
 
f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)46 b Fd(12,)27 b(36)2025
8522
 
3258 y(ML)14 b Fc(.)e(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8523
 
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)
8524
 
h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)39 b Fd(59)2025
8525
 
3364 y(monadic)25 b(programming)c Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)g(.)h
8526
 
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)48
8527
 
b Fd(59)2025 3471 y(MPI)13 b Fc(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8528
 
(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
8529
 
f(.)g(.)h(.)38 b Fd(2,)27 b(37,)f(40,)h(56)2025 3577
8530
 
y Fb(MPI_Alltoall)e Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8531
 
(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)49
8532
 
b Fd(41,)26 b(42,)h(48)2025 3684 y Fb(MPI_Barrier)c Fc(.)12
 
8591
b Fj(58)2025 857 y(girlfriends)14 b Ff(.)g(.)e(.)g(.)h(.)f(.)g(.)h(.)f
 
8592
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)
 
8593
h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)40 b Fj(61)2025 1116
 
8594
y Fu(H)2025 1238 y Fj(halfcomplex)25 b(arra)n(y)20 b
 
8595
Ff(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
 
8596
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)46 b Fj(6,)26 b(17)2025
 
8597
1330 y(hermitian)f(arra)n(y)13 b Ff(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8598
(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)39
 
8599
b Fj(17,)26 b(33,)h(34)2025 1590 y Fu(I)2025 1712 y Fj(in-place)c
 
8600
(transform)8 b Ff(.)13 b(.)32 b Fj(5,)24 b(19,)h(21,)g(24,)g(30,)g(32,)
 
8601
g(40,)g(42,)g(51)2025 1804 y(installation)12 b Ff(.)i(.)e(.)h(.)f(.)g
 
8602
(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
 
8603
f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)38 b Fj(55)2025
 
8604
2050 y Fu(L)2025 2173 y Fj(linking)26 b(on)f(Unix)13
 
8605
b Ff(.)e(.)i(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)38
 
8606
b Fj(4,)27 b(7,)f(39,)h(43,)f(47,)h(48)2025 2265 y(LISP)16
 
8607
b Ff(.)c(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
 
8608
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h
 
8609
(.)f(.)42 b Fj(15,)27 b(61)2025 2356 y(load-balancing)11
 
8610
b Ff(.)j(.)e(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
 
8611
f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)37
 
8612
b Fj(48)2025 2616 y Fu(M)2025 2738 y Fi(malloc)22 b Ff(.)12
 
8613
b(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8614
(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)46
 
8615
b Fj(12,)27 b(35)2025 2830 y(ML)14 b Ff(.)e(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8616
(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
 
8617
g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)39
 
8618
b Fj(59)2025 2922 y(monadic)25 b(programming)c Ff(.)13
 
8619
b(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8620
(.)g(.)h(.)f(.)g(.)48 b Fj(59)2025 3014 y(MPI)13 b Ff(.)f(.)h(.)f(.)g
 
8621
(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
 
8622
f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)38 b Fj(2,)27 b(37,)f(40,)h(56)2025
 
8623
3106 y Fi(MPI_Alltoall)e Ff(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8624
(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)49
 
8625
b Fj(41,)26 b(42,)h(48)2025 3197 y Fi(MPI_Barrier)c Ff(.)12
8533
8626
b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8534
8627
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)46
8535
 
b Fd(43)2025 3791 y Fb(MPI_COMM_WORLD)9 b Fc(.)15 b(.)d(.)h(.)f(.)g(.)h
 
8628
b Fj(43)2025 3289 y Fi(MPI_COMM_WORLD)9 b Ff(.)15 b(.)d(.)h(.)f(.)g(.)h
8536
8629
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)
8537
 
g(.)h(.)34 b Fd(42,)27 b(47)2025 3897 y Fb(MPI_Finalize)22
8538
 
b Fc(.)12 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
 
8630
g(.)h(.)34 b Fj(42,)27 b(47)2025 3381 y Fi(MPI_Finalize)22
 
8631
b Ff(.)12 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8539
8632
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)45
8540
 
b Fd(42)2025 4004 y Fb(MPI_Init)7 b Fc(.)14 b(.)e(.)g(.)h(.)f(.)g(.)h
 
8633
b Fj(42)2025 3473 y Fi(MPI_Init)7 b Ff(.)14 b(.)e(.)g(.)h(.)f(.)g(.)h
8541
8634
(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
8542
 
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)32 b Fd(42)2025
8543
 
4110 y(m)n(ulti-dimensional)24 b(transform)19 b Fc(.)12
8544
 
b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)44 b Fd(4,)26
8545
 
b(22,)h(29)2025 4418 y Ft(N)2025 4565 y Fd(n)p 2073 4565
8546
 
24 4 v 27 w(\014elds)17 b Fc(.)c(.)f(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)
 
8635
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)32 b Fj(42)2025
 
8636
3565 y(m)n(ulti-dimensional)24 b(transform)19 b Ff(.)12
 
8637
b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)44 b Fj(4,)26
 
8638
b(22,)h(29)2025 3824 y Fu(N)2025 3946 y Fj(n)p 2073 3946
 
8639
24 4 v 27 w(\014elds)17 b Ff(.)c(.)f(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)
8547
8640
f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8548
 
(.)f(.)g(.)h(.)f(.)g(.)43 b Fd(42,)27 b(47)2025 4672
8549
 
y(nerd-readable)e(text)18 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8641
(.)f(.)g(.)h(.)f(.)g(.)43 b Fj(42,)27 b(47)2025 4038
 
8642
y(nerd-readable)e(text)18 b Ff(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8550
8643
(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
8551
 
44 b Fd(15)2025 4778 y(normalization)13 b Fc(.)g(.)f(.)g(.)h(.)f(.)g(.)
8552
 
h(.)f(.)g(.)h(.)f(.)39 b Fd(4,)26 b(5,)g(7,)h(8,)f(11,)h(21,)f(29,)h
8553
 
(43)2025 4885 y(n)n(um)n(b)r(er)c(of)k(threads)19 b Fc(.)13
 
8644
44 b Fj(15)2025 4130 y(normalization)13 b Ff(.)g(.)f(.)g(.)h(.)f(.)g(.)
 
8645
h(.)f(.)g(.)h(.)f(.)39 b Fj(4,)26 b(5,)g(7,)h(8,)f(11,)h(21,)f(29,)h
 
8646
(43)2025 4222 y(n)n(um)n(b)r(er)c(of)k(threads)19 b Ff(.)13
8554
8647
b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8555
 
(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)45 b Fd(39)2025 5192
8556
 
y Ft(O)2025 5340 y Fd(out-of-place)26 b(transform)9 b
8557
 
Fc(.)j(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8558
 
(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)35 b Fd(5)p eop
 
8648
(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)45 b Fj(39)2025 4468
 
8649
y Fu(O)2025 4591 y Fj(out-of-place)26 b(transform)9 b
 
8650
Ff(.)j(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
 
8651
(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)35 b Fj(5)2025 4850 y
 
8652
Fu(P)2025 4973 y Fj(padding)16 b Ff(.)11 b(.)i(.)f(.)g(.)h(.)f(.)g(.)h
 
8653
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)
 
8654
g(.)h(.)f(.)g(.)h(.)41 b Fj(8,)26 b(32,)h(46)2025 5065
 
8655
y(parallel)g(transform)21 b Ff(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
 
8656
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)47
 
8657
b Fj(2,)26 b(37)2025 5156 y(P)n(en)n(tium)e(hac)n(k)f
 
8658
Ff(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8659
(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)49
 
8660
b Fj(57)2025 5248 y(plan)24 b Ff(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
 
8661
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)
 
8662
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)50 b Fj(1,)26 b(3,)g(42)2025
 
8663
5340 y(planner)20 b Ff(.)13 b(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h
 
8664
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
 
8665
h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)47 b Fj(1)p eop end
8559
8666
%%Page: 66 68
8560
 
66 67 bop 150 -116 a Fu(66)3232 b(FFTW)150 299 y Ft(P)150
8561
 
440 y Fd(padding)16 b Fc(.)c(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
 
8667
TeXDict begin 66 67 bop 150 -116 a Fv(66)3232 b(FFTW)150
 
8668
299 y Fj(p)r(o)n(w)n(er)26 b(sp)r(ectrum)7 b Ff(.)k(.)h(.)h(.)f(.)g(.)h
 
8669
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)
 
8670
g(.)h(.)f(.)g(.)h(.)f(.)g(.)33 b Fj(7)150 556 y Fu(R)150
 
8671
678 y Fj(rank)16 b Ff(.)c(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
 
8672
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
 
8673
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)42 b Fj(4)150
 
8674
769 y(real)27 b(m)n(ulti-dimensional)d(transform)g Ff(.)12
 
8675
b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)50 b Fj(7,)26
 
8676
b(29)150 860 y(real)h(n)n(um)n(b)r(er)16 b Ff(.)10 b(.)j(.)f(.)g(.)h(.)
 
8677
f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8678
(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)42 b Fj(17)150
 
8679
952 y(real)27 b(transform)10 b Ff(.)i(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8680
(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)
 
8681
f(.)g(.)36 b Fj(6,)26 b(26)150 1043 y(RFFTW)12 b Ff(.)g(.)g(.)h(.)f(.)g
 
8682
(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
 
8683
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)38 b Fj(6,)26
 
8684
b(26)150 1135 y(RFFTWND)14 b Ff(.)e(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8685
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
 
8686
f(.)g(.)g(.)h(.)f(.)40 b Fj(29)150 1226 y(r\013t)n(wnd)25
 
8687
b(arra)n(y)h(format)13 b Ff(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8688
(.)g(.)h(.)f(.)g(.)40 b Fj(8,)26 b(32,)h(46,)f(54)150
 
8689
1318 y(ro)n(w-ma)t(jor)12 b Ff(.)h(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8690
(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
 
8691
38 b Fj(11,)27 b(23,)f(44)150 1575 y Fu(S)150 1696 y
 
8692
Fj(sa)n(ving)g(plans)g(to)g(disk)13 b Ff(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
8562
8693
g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8563
 
(.)f(.)41 b Fd(8,)27 b(32,)f(46)150 543 y(parallel)h(transform)21
8564
 
b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f
8565
 
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)46 b Fd(2,)26 b(37)150
8566
 
646 y(P)n(en)n(tium)e(hac)n(k)f Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8567
 
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
8568
 
h(.)f(.)g(.)h(.)f(.)49 b Fd(57)150 750 y(plan)24 b Fc(.)12
8569
 
b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h
8570
 
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)50
8571
 
b Fd(1,)26 b(3,)g(42)150 853 y(planner)20 b Fc(.)13 b(.)f(.)g(.)h(.)f
8572
 
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
8573
 
f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)47
8574
 
b Fd(1)150 956 y(p)r(o)n(w)n(er)26 b(sp)r(ectrum)7 b
8575
 
Fc(.)k(.)h(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8576
 
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)33
8577
 
b Fd(7)150 1245 y Ft(R)150 1385 y Fd(rank)16 b Fc(.)c(.)g(.)h(.)f(.)g
8578
 
(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
8579
 
g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8580
 
(.)42 b Fd(4)150 1489 y(real)27 b(m)n(ulti-dimensional)d(transform)g
8581
 
Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)50 b
8582
 
Fd(7,)26 b(29)150 1592 y(real)h(n)n(um)n(b)r(er)16 b
8583
 
Fc(.)10 b(.)j(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8584
 
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)42
8585
 
b Fd(17)150 1695 y(real)27 b(transform)10 b Fc(.)i(.)g(.)g(.)h(.)f(.)g
8586
 
(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
8587
 
g(.)g(.)h(.)f(.)g(.)36 b Fd(6,)26 b(26)150 1799 y(RFFTW)12
8588
 
b Fc(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
8589
 
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)38
8590
 
b Fd(6,)26 b(26)150 1902 y(RFFTWND)14 b Fc(.)e(.)g(.)h(.)f(.)g(.)h(.)f
8591
 
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
8592
 
f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)40 b Fd(29)150 2005 y(r\013t)n(wnd)25
8593
 
b(arra)n(y)h(format)6 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8594
 
(.)h(.)f(.)32 b Fd(8,)26 b(32,)h(33,)g(46,)f(54)150 2109
8595
 
y(ro)n(w-ma)t(jor)12 b Fc(.)h(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8596
 
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)38
8597
 
b Fd(11,)27 b(23,)f(44)2025 299 y Ft(S)2025 438 y Fd(sa)n(ving)g(plans)
8598
 
g(to)g(disk)13 b Fc(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8599
 
(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)39
8600
 
b Fd(13)2025 541 y(slab)26 b(decomp)r(osition)8 b Fc(.)13
8601
 
b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8602
 
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)33 b Fd(43)2025 644
8603
 
y(sp)r(eci\014c)26 b(planner)13 b Fc(.)f(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)
8604
 
h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8605
 
(.)h(.)f(.)g(.)39 b Fd(20)2025 747 y(stride)7 b Fc(.)12
 
8694
(.)38 b Fj(13)2025 299 y(slab)26 b(decomp)r(osition)8
 
8695
b Ff(.)13 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8696
(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)33 b Fj(43)2025
 
8697
391 y(sp)r(eci\014c)26 b(planner)13 b Ff(.)f(.)h(.)f(.)g(.)g(.)h(.)f(.)
 
8698
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8699
(.)g(.)h(.)f(.)g(.)39 b Fj(20)2025 483 y(stride)7 b Ff(.)12
8606
8700
b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8607
 
(.)33 b Fd(11,)27 b(20,)f(24,)h(28,)g(31,)f(33,)h(42)2025
8608
 
1031 y Ft(T)2025 1171 y Fd(thread)e(safet)n(y)10 b Fc(.)j(.)f(.)h(.)f
8609
 
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
8610
 
h(.)f(.)g(.)h(.)f(.)g(.)h(.)35 b Fd(36,)27 b(40)2025
8611
 
1273 y(threads)18 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8612
 
(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)44
8613
 
b Fd(2,)27 b(36,)f(37,)h(56)2025 1376 y(timer,)e(customization)h(of)10
8614
 
b Fc(.)j(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
8615
 
h(.)f(.)g(.)h(.)f(.)g(.)36 b Fd(58)2025 1479 y(T)-6 b(utorial)11
8616
 
b Fc(.)i(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
 
8701
(.)33 b Fj(11,)27 b(20,)f(24,)h(27,)g(31,)f(32,)h(42)2025
 
8702
743 y Fu(T)2025 866 y Fj(thread)e(safet)n(y)10 b Ff(.)j(.)f(.)h(.)f(.)g
 
8703
(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
 
8704
f(.)g(.)h(.)f(.)g(.)h(.)35 b Fj(36,)27 b(40)2025 958
 
8705
y(threads)18 b Ff(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
 
8706
h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)44
 
8707
b Fj(2,)27 b(36,)f(37,)h(56)2025 1050 y(timer,)e(customization)h(of)10
 
8708
b Ff(.)j(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
 
8709
h(.)f(.)g(.)h(.)f(.)g(.)36 b Fj(58)2025 1142 y(T)-6 b(utorial)11
 
8710
b Ff(.)i(.)f(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
8617
8711
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h
8618
 
(.)f(.)g(.)h(.)37 b Fd(3)2025 1764 y Ft(W)2025 1903 y
8619
 
Fd(wisdom)8 b Fc(.)k(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)
8620
 
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)34 b Fd(13,)27
8621
 
b(19,)g(24,)f(30,)h(34)2025 2006 y(wisdom,)f(imp)r(ort)f(and)g(exp)r
8622
 
(ort)12 b Fc(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
8623
 
f(.)g(.)h(.)f(.)38 b Fd(14)2025 2109 y(wisdom,)26 b(problems)f(with)11
8624
 
b Fc(.)i(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
8625
 
h(.)f(.)g(.)h(.)f(.)g(.)37 b Fd(14)p eop
 
8712
(.)f(.)g(.)h(.)37 b Fj(3)2025 1389 y Fu(W)2025 1512 y
 
8713
Fj(wisdom)8 b Ff(.)k(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)
 
8714
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)34 b Fj(13,)27
 
8715
b(19,)g(23,)f(30,)h(34)2025 1604 y(wisdom,)f(imp)r(ort)f(and)g(exp)r
 
8716
(ort)12 b Ff(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
 
8717
f(.)g(.)h(.)f(.)38 b Fj(14)2025 1696 y(wisdom,)26 b(problems)f(with)11
 
8718
b Ff(.)i(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
 
8719
h(.)f(.)g(.)h(.)f(.)g(.)37 b Fj(14)p eop end
8626
8720
%%Page: 67 69
8627
 
67 68 bop 150 -116 a Fu(Chapter)30 b(10:)41 b(Library)29
8628
 
b(Index)2448 b(67)150 299 y Fr(10)80 b(Library)55 b(Index)150
8629
 
651 y Ft(F)150 786 y Fb(fftw)13 b Fc(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
8630
 
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h
8631
 
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)38
8632
 
b Fd(20)150 888 y Fb(fftw_threads)22 b Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)
 
8721
TeXDict begin 67 68 bop 150 -116 a Fv(Chapter)30 b(10:)41
 
8722
b(Library)29 b(Index)2448 b(67)150 299 y Fs(10)80 b(Library)55
 
8723
b(Index)150 642 y Fu(F)150 764 y Fi(fftw)13 b Ff(.)g(.)f(.)g(.)h(.)f(.)
 
8724
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8725
(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)38
 
8726
b Fj(20)150 856 y Fi(fftw_threads)22 b Ff(.)12 b(.)g(.)h(.)f(.)g(.)h(.)
8633
8727
f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8634
 
(.)f(.)g(.)g(.)h(.)f(.)g(.)45 b Fd(38)150 989 y Fb(fftw_threads_one)13
8635
 
b Fc(.)j(.)d(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
8636
 
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)39 b Fd(38)150
8637
 
1090 y Fb(FFTW_BACKWARD)22 b Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
 
8728
(.)f(.)g(.)g(.)h(.)f(.)g(.)45 b Fj(38)150 947 y Fi(fftw_threads_one)13
 
8729
b Ff(.)j(.)d(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
 
8730
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)39 b Fj(38)150
 
8731
1039 y Fi(FFTW_BACKWARD)22 b Ff(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8638
8732
(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
8639
 
g(.)h(.)f(.)g(.)45 b Fd(4)150 1191 y Fb(fftw_complex)13
8640
 
b Fc(.)i(.)d(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
 
8733
g(.)h(.)f(.)g(.)45 b Fj(4)150 1130 y Fi(fftw_complex)13
 
8734
b Ff(.)i(.)d(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
8641
8735
f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)39 b
8642
 
Fd(4,)26 b(17)150 1292 y Fb(FFTW_COMPLEX)c Fc(.)12 b(.)g(.)h(.)f(.)g(.)
 
8736
Fj(4,)26 b(17)150 1222 y Fi(FFTW_COMPLEX)c Ff(.)12 b(.)g(.)h(.)f(.)g(.)
8643
8737
h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8644
 
(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)45 b Fd(18)150 1394 y
8645
 
Fb(FFTW_COMPLEX_TO_REAL)12 b Fc(.)17 b(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f
8646
 
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)37 b Fd(6,)27 b(27,)f(46)150
8647
 
1495 y Fb(fftw_create_plan)7 b Fc(.)16 b(.)c(.)h(.)f(.)g(.)g(.)h(.)f(.)
 
8738
(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)45 b Fj(18)150 1313 y
 
8739
Fi(FFTW_COMPLEX_TO_REAL)12 b Ff(.)17 b(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f
 
8740
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)37 b Fj(6,)27 b(26,)f(46)150
 
8741
1405 y Fi(fftw_create_plan)7 b Ff(.)16 b(.)c(.)h(.)f(.)g(.)g(.)h(.)f(.)
8648
8742
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)33
8649
 
b Fd(3,)26 b(18)150 1596 y Fb(fftw_create_plan_specific)18
8650
 
b Fc(.)g(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8651
 
(.)44 b Fd(18)150 1697 y Fb(fftw_destroy_plan)27 b Fc(.)13
 
8743
b Fj(3,)26 b(18)150 1496 y Fi(fftw_create_plan_specific)18
 
8744
b Ff(.)g(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8745
(.)44 b Fj(18)150 1588 y Fi(fftw_destroy_plan)27 b Ff(.)13
8652
8746
b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8653
 
(.)h(.)f(.)g(.)h(.)49 b Fd(4,)26 b(21)150 1798 y Fb(fftw_direction)c
8654
 
Fc(.)12 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)45
8655
 
b Fd(3,)26 b(4,)h(6,)f(18,)h(22,)f(29)150 1900 y Fb(FFTW_ENABLE_FLOAT)
8656
 
12 b Fc(.)k(.)c(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8657
 
(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)37 b Fd(18)150
8658
 
2001 y Fb(FFTW_ESTIMATE)22 b Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
 
8747
(.)h(.)f(.)g(.)h(.)49 b Fj(4,)26 b(21)150 1679 y Fi(fftw_direction)c
 
8748
Ff(.)12 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)45
 
8749
b Fj(3,)26 b(4,)h(6,)f(18,)h(22,)f(29)150 1771 y Fi(FFTW_ENABLE_FLOAT)
 
8750
12 b Ff(.)k(.)c(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8751
(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)37 b Fj(18)150
 
8752
1862 y Fi(FFTW_ESTIMATE)22 b Ff(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8659
8753
(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
8660
 
g(.)h(.)f(.)g(.)45 b Fd(4)150 2102 y Fb(fftw_export_wisdom)25
8661
 
b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8662
 
(.)h(.)f(.)g(.)h(.)47 b Fd(13,)26 b(34)150 2203 y Fb
8663
 
(fftw_export_wisdom_to_file)10 b Fc(.)18 b(.)12 b(.)g(.)h(.)f(.)g(.)h
8664
 
(.)f(.)g(.)h(.)f(.)g(.)36 b Fd(14,)26 b(34)150 2304 y
8665
 
Fb(fftw_export_wisdom_to_string)7 b Fc(.)18 b(.)13 b(.)f(.)g(.)h(.)f(.)
8666
 
g(.)h(.)f(.)g(.)33 b Fd(15,)26 b(34)150 2406 y Fb(fftw_f77_create_plan)
8667
 
7 b Fc(.)17 b(.)c(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8668
 
(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)33 b Fd(53)150 2507 y
8669
 
Fb(fftw_f77_destroy_plan)c Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
8670
 
g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)50 b Fd(53)150
8671
 
2608 y Fb(fftw_f77_one)12 b Fc(.)j(.)d(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
 
8754
g(.)h(.)f(.)g(.)45 b Fj(4)150 1954 y Fi(fftw_export_wisdom)25
 
8755
b Ff(.)13 b(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8756
(.)h(.)f(.)g(.)h(.)47 b Fj(13,)26 b(34)150 2045 y Fi
 
8757
(fftw_export_wisdom_to_file)10 b Ff(.)18 b(.)12 b(.)g(.)h(.)f(.)g(.)h
 
8758
(.)f(.)g(.)h(.)f(.)g(.)36 b Fj(14,)26 b(34)150 2137 y
 
8759
Fi(fftw_export_wisdom_to_string)7 b Ff(.)18 b(.)13 b(.)f(.)g(.)h(.)f(.)
 
8760
g(.)h(.)f(.)g(.)33 b Fj(15,)26 b(34)150 2228 y Fi(fftw_f77_create_plan)
 
8761
7 b Ff(.)17 b(.)c(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8762
(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)33 b Fj(53)150 2320 y
 
8763
Fi(fftw_f77_destroy_plan)c Ff(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
 
8764
g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)50 b Fj(53)150
 
8765
2411 y Fi(fftw_f77_one)12 b Ff(.)j(.)d(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8672
8766
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)
8673
 
38 b Fd(51,)26 b(53)150 2709 y Fb(fftw_f77_threads_one)7
8674
 
b Fc(.)17 b(.)c(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8675
 
(.)g(.)h(.)f(.)g(.)h(.)f(.)33 b Fd(53)150 2810 y Fb(fftw_forget_wisdom)
8676
 
10 b Fc(.)17 b(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8677
 
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)36 b Fd(35)150
8678
 
2911 y Fb(FFTW_FORWARD)23 b Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
 
8767
38 b Fj(51,)26 b(53)150 2503 y Fi(fftw_f77_threads_one)7
 
8768
b Ff(.)17 b(.)c(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8769
(.)g(.)h(.)f(.)g(.)h(.)f(.)33 b Fj(53)150 2594 y Fi(fftw_forget_wisdom)
 
8770
10 b Ff(.)17 b(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8771
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)36 b Fj(35)150
 
8772
2686 y Fi(FFTW_FORWARD)23 b Ff(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8679
8773
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
8680
 
h(.)f(.)g(.)g(.)h(.)46 b Fd(4)150 3013 y Fb(fftw_free_hook)16
8681
 
b Fc(.)g(.)c(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)
 
8774
h(.)f(.)g(.)g(.)h(.)46 b Fj(4)150 2777 y Fi(fftw_free_hook)16
 
8775
b Ff(.)g(.)c(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)
8682
8776
h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)42 b
8683
 
Fd(36)150 3114 y Fb(fftw_import_wisdom)25 b Fc(.)13 b(.)f(.)g(.)h(.)f
 
8777
Fj(35)150 2869 y Fi(fftw_import_wisdom)25 b Ff(.)13 b(.)f(.)g(.)h(.)f
8684
8778
(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)47
8685
 
b Fd(13,)26 b(35)150 3215 y Fb(fftw_import_wisdom_from_file)7
8686
 
b Fc(.)18 b(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)33
8687
 
b Fd(14,)26 b(35)150 3316 y Fb(fftw_import_wisdom_from_strin)q(g)i
8688
 
Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)48 b Fd(15,)26 b(35)150
8689
 
3417 y Fb(FFTW_IN_PLACE)c Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8779
b Fj(13,)26 b(35)150 2961 y Fi(fftw_import_wisdom_from_file)7
 
8780
b Ff(.)18 b(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)33
 
8781
b Fj(14,)26 b(35)150 3052 y Fi(fftw_import_wisdom_from_strin)q(g)i
 
8782
Ff(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)48 b Fj(15,)26 b(35)150
 
8783
3144 y Fi(FFTW_IN_PLACE)c Ff(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8690
8784
(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
8691
 
h(.)f(.)g(.)45 b Fd(5)150 3519 y Fb(fftw_malloc)23 b
8692
 
Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8785
h(.)f(.)g(.)45 b Fj(5)150 3235 y Fi(fftw_malloc)23 b
 
8786
Ff(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8693
8787
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)46
8694
 
b Fd(36)150 3620 y Fb(fftw_malloc_hook)13 b Fc(.)j(.)d(.)f(.)g(.)h(.)f
 
8788
b Fj(35)150 3327 y Fi(fftw_malloc_hook)13 b Ff(.)j(.)d(.)f(.)g(.)h(.)f
8695
8789
(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
8696
 
h(.)f(.)g(.)39 b Fd(36)150 3721 y Fb(FFTW_MEASURE)23
8697
 
b Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8790
h(.)f(.)g(.)39 b Fj(35)150 3418 y Fi(FFTW_MEASURE)23
 
8791
b Ff(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8698
8792
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)46
8699
 
b Fd(4)150 3822 y Fb(fftw_mpi)7 b Fc(.)14 b(.)e(.)g(.)h(.)f(.)g(.)h(.)f
 
8793
b Fj(4)150 3510 y Fi(fftw_mpi)7 b Ff(.)14 b(.)e(.)g(.)h(.)f(.)g(.)h(.)f
8700
8794
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
8701
 
f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)32 b Fd(47)150
8702
 
3923 y Fb(fftw_mpi_create_plan)7 b Fc(.)17 b(.)c(.)f(.)g(.)h(.)f(.)g(.)
 
8795
f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)32 b Fj(47)150
 
8796
3601 y Fi(fftw_mpi_create_plan)7 b Ff(.)17 b(.)c(.)f(.)g(.)h(.)f(.)g(.)
8703
8797
h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)33
8704
 
b Fd(47)150 4025 y Fb(fftw_mpi_destroy_plan)c Fc(.)12
 
8798
b Fj(47)150 3693 y Fi(fftw_mpi_destroy_plan)c Ff(.)12
8705
8799
b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8706
 
(.)f(.)g(.)50 b Fd(47)150 4126 y Fb(fftw_mpi_local_sizes)7
8707
 
b Fc(.)17 b(.)c(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8708
 
(.)g(.)h(.)f(.)g(.)h(.)f(.)33 b Fd(48)150 4227 y Fb(fftw_mpi_plan)18
8709
 
b Fc(.)d(.)d(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
 
8800
(.)f(.)g(.)50 b Fj(47)150 3784 y Fi(fftw_mpi_local_sizes)7
 
8801
b Ff(.)17 b(.)c(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8802
(.)g(.)h(.)f(.)g(.)h(.)f(.)33 b Fj(48)150 3876 y Fi(fftw_mpi_plan)18
 
8803
b Ff(.)d(.)d(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
8710
8804
f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)43
8711
 
b Fd(47)150 4328 y Fb(fftw_mpi_test)10 b Fc(.)16 b(.)c(.)g(.)g(.)h(.)f
 
8805
b Fj(47)150 3967 y Fi(fftw_mpi_test)10 b Ff(.)16 b(.)c(.)g(.)g(.)h(.)f
8712
8806
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
8713
 
f(.)g(.)h(.)36 b Fd(41,)26 b(49)150 4429 y Fb(FFTW_NORMAL_ORDER)12
8714
 
b Fc(.)k(.)c(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
8715
 
g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)37 b Fd(42)150 4530
8716
 
y Fb(fftw_one)20 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
 
8807
f(.)g(.)h(.)36 b Fj(41,)26 b(49)150 4059 y Fi(FFTW_NORMAL_ORDER)12
 
8808
b Ff(.)k(.)c(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
 
8809
g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)37 b Fj(42)150 4150
 
8810
y Fi(fftw_one)20 b Ff(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
8717
8811
g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8718
 
(.)f(.)g(.)h(.)44 b Fd(4,)26 b(20)150 4632 y Fb(FFTW_OUT_OF_PLACE)12
8719
 
b Fc(.)k(.)c(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
8720
 
g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)37 b Fd(19)150 4733
8721
 
y Fb(fftw_plan)17 b Fc(.)d(.)f(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8812
(.)f(.)g(.)h(.)44 b Fj(4,)26 b(20)150 4242 y Fi(FFTW_OUT_OF_PLACE)12
 
8813
b Ff(.)k(.)c(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
 
8814
g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)37 b Fj(19)150 4333
 
8815
y Fi(fftw_plan)17 b Ff(.)d(.)f(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8722
8816
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)
8723
 
h(.)f(.)43 b Fd(3,)26 b(18)150 4834 y Fb(fftw_real)f
8724
 
Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
 
8817
h(.)f(.)43 b Fj(3,)26 b(18)150 4425 y Fi(fftw_real)f
 
8818
Ff(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8725
8819
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)
8726
 
49 b Fd(17)150 4935 y Fb(FFTW_REAL)25 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g
 
8820
49 b Fj(17)150 4516 y Fi(FFTW_REAL)25 b Ff(.)13 b(.)f(.)g(.)h(.)f(.)g
8727
8821
(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
8728
 
g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)49 b Fd(18)150
8729
 
5036 y Fb(FFTW_REAL_TO_COMPLEX)24 b Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f
 
8822
g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)49 b Fj(18)150
 
8823
4608 y Fi(FFTW_REAL_TO_COMPLEX)24 b Ff(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f
8730
8824
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)45
8731
 
b Fd(6,)26 b(27)150 5138 y Fb(FFTW_SCRAMBLED_INPUT)7
8732
 
b Fc(.)17 b(.)c(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8733
 
(.)g(.)h(.)f(.)g(.)h(.)f(.)33 b Fd(47)150 5239 y Fb
8734
 
(FFTW_SCRAMBLED_OUTPUT)c Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8735
 
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)50 b Fd(47)150
8736
 
5340 y Fb(fftw_test)16 b Fc(.)e(.)f(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
 
8825
b Fj(6,)26 b(26)150 4699 y Fi(FFTW_SCRAMBLED_INPUT)7
 
8826
b Ff(.)17 b(.)c(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8827
(.)g(.)h(.)f(.)g(.)h(.)f(.)33 b Fj(47)150 4791 y Fi
 
8828
(FFTW_SCRAMBLED_OUTPUT)c Ff(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8829
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)50 b Fj(47)150
 
8830
4882 y Fi(fftw_test)16 b Ff(.)e(.)f(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8737
8831
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)
8738
 
g(.)h(.)42 b Fd(57,)26 b(58)2025 651 y Fb(fftw_threads_init)12
8739
 
b Fc(.)k(.)c(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
8740
 
f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)38 b Fd(38)2025 750
8741
 
y Fb(fftw_threads_test)12 b Fc(.)k(.)c(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8742
 
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)38
8743
 
b Fd(39)2025 850 y Fb(FFTW_THREADSAFE)15 b Fc(.)g(.)e(.)f(.)g(.)g(.)h
 
8832
g(.)h(.)42 b Fj(57,)26 b(58)150 4974 y Fi(fftw_threads_init)12
 
8833
b Ff(.)k(.)c(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
 
8834
g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)37 b Fj(38)150 5065
 
8835
y Fi(fftw_threads_test)12 b Ff(.)k(.)c(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8836
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)37
 
8837
b Fj(39)150 5157 y Fi(FFTW_THREADSAFE)15 b Ff(.)g(.)e(.)f(.)g(.)h(.)f
 
8838
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
 
8839
f(.)g(.)g(.)h(.)40 b Fj(36)150 5248 y Fi(FFTW_TIME_MIN)18
 
8840
b Ff(.)d(.)d(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
 
8841
f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)43
 
8842
b Fj(58)150 5340 y Fi(FFTW_TRANSPOSED_ORDER)25 b Ff(.)12
 
8843
b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)46 b Fj(42,)27
 
8844
b(44,)g(46,)f(48)2025 642 y Fi(FFTW_USE_WISDOM)15 b Ff(.)g(.)e(.)f(.)g
 
8845
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
 
8846
f(.)g(.)h(.)f(.)g(.)h(.)40 b Fj(13)2025 734 y Fi(fftw2d_create_plan)26
 
8847
b Ff(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8848
(.)h(.)f(.)g(.)g(.)h(.)48 b Fj(5,)26 b(22)2025 825 y
 
8849
Fi(fftw2d_create_plan_specific)16 b Ff(.)i(.)12 b(.)g(.)h(.)f(.)g(.)h
 
8850
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)41 b Fj(22)2025 917 y
 
8851
Fi(fftw2d_mpi_create_plan)27 b Ff(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
 
8852
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)48 b Fj(42)2025
 
8853
1008 y Fi(fftw3d_create_plan)26 b Ff(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h
 
8854
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)48
 
8855
b Fj(5,)26 b(22)2025 1100 y Fi(fftw3d_create_plan_specific)16
 
8856
b Ff(.)i(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)41
 
8857
b Fj(22)2025 1191 y Fi(fftw3d_f77_create_plan)27 b Ff(.)13
 
8858
b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8859
(.)h(.)48 b Fj(53)2025 1283 y Fi(fftw3d_mpi_create_plan)27
 
8860
b Ff(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
 
8861
(.)f(.)g(.)h(.)48 b Fj(42)2025 1374 y Fi(fftwnd)23 b
 
8862
Ff(.)12 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8744
8863
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
8745
 
h(.)f(.)g(.)h(.)40 b Fd(36)2025 950 y Fb(FFTW_TIME_MIN)18
8746
 
b Fc(.)d(.)d(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)
8747
 
h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)43
8748
 
b Fd(58)2025 1050 y Fb(FFTW_TRANSPOSED_ORDER)25 b Fc(.)12
8749
 
b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)46 b Fd(42,)27
8750
 
b(44,)f(46,)h(48)2025 1149 y Fb(FFTW_USE_WISDOM)15 b
8751
 
Fc(.)g(.)e(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8752
 
(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)40 b Fd(13)2025
8753
 
1249 y Fb(fftw2d_create_plan)26 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8754
 
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)48
8755
 
b Fd(5,)26 b(22)2025 1349 y Fb(fftw2d_create_plan_specific)16
8756
 
b Fc(.)i(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)41
8757
 
b Fd(22)2025 1449 y Fb(fftw2d_mpi_create_plan)27 b Fc(.)13
8758
 
b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8759
 
(.)h(.)48 b Fd(42)2025 1548 y Fb(fftw3d_create_plan)26
8760
 
b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8761
 
(.)h(.)f(.)g(.)g(.)h(.)48 b Fd(5,)26 b(22)2025 1648 y
8762
 
Fb(fftw3d_create_plan_specific)16 b Fc(.)i(.)12 b(.)g(.)h(.)f(.)g(.)h
8763
 
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)41 b Fd(22)2025 1748 y
8764
 
Fb(fftw3d_f77_create_plan)27 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
8765
 
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)48 b Fd(53)2025
8766
 
1848 y Fb(fftw3d_mpi_create_plan)27 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h
8767
 
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)48
8768
 
b Fd(42)2025 1947 y Fb(fftwnd)23 b Fc(.)12 b(.)g(.)h(.)f(.)g(.)g(.)h(.)
8769
 
f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8770
 
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)47 b Fd(5,)26
8771
 
b(24)2025 2047 y Fb(fftwnd_threads)16 b Fc(.)f(.)e(.)f(.)g(.)h(.)f(.)g
8772
 
(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
8773
 
g(.)g(.)h(.)f(.)42 b Fd(38)2025 2147 y Fb(fftwnd_threads_one)10
8774
 
b Fc(.)17 b(.)12 b(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8775
 
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)36 b Fd(38)2025
8776
 
2247 y Fb(fftwnd_create_plan)26 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8777
 
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)48
8778
 
b Fd(4,)26 b(22)2025 2346 y Fb(fftwnd_create_plan_specific)16
8779
 
b Fc(.)i(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)41
8780
 
b Fd(22)2025 2446 y Fb(fftwnd_destroy_plan)25 b Fc(.)12
 
8864
h(.)47 b Fj(5,)26 b(24)2025 1466 y Fi(fftwnd_threads)16
 
8865
b Ff(.)f(.)e(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
 
8866
h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)42 b
 
8867
Fj(38)2025 1557 y Fi(fftwnd_threads_one)10 b Ff(.)17
 
8868
b(.)12 b(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
 
8869
f(.)g(.)h(.)f(.)g(.)h(.)f(.)36 b Fj(38)2025 1649 y Fi
 
8870
(fftwnd_create_plan)26 b Ff(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8871
(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)48 b Fj(4,)26
 
8872
b(22)2025 1740 y Fi(fftwnd_create_plan_specific)16 b
 
8873
Ff(.)i(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)41
 
8874
b Fj(22)2025 1832 y Fi(fftwnd_destroy_plan)25 b Ff(.)12
8781
8875
b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8782
 
(.)f(.)g(.)47 b Fd(5,)26 b(25)2025 2546 y Fb(fftwnd_f77_create_plan)h
8783
 
Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8784
 
(.)f(.)g(.)h(.)48 b Fd(53)2025 2646 y Fb(fftwnd_f77_destroy_plan)26
8785
 
b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8786
 
(.)h(.)f(.)47 b Fd(53)2025 2745 y Fb(fftwnd_f77_one)16
8787
 
b Fc(.)f(.)e(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
 
8876
(.)f(.)g(.)47 b Fj(5,)26 b(25)2025 1923 y Fi(fftwnd_f77_create_plan)h
 
8877
Ff(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
 
8878
(.)f(.)g(.)h(.)48 b Fj(53)2025 2015 y Fi(fftwnd_f77_destroy_plan)26
 
8879
b Ff(.)13 b(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8880
(.)h(.)f(.)47 b Fj(53)2025 2106 y Fi(fftwnd_f77_one)16
 
8881
b Ff(.)f(.)e(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
8788
8882
h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)42 b
8789
 
Fd(53)2025 2845 y Fb(fftwnd_mpi)24 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g
 
8883
Fj(53)2025 2198 y Fi(fftwnd_mpi)24 b Ff(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g
8790
8884
(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
8791
 
f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)48 b Fd(42)2025 2945
8792
 
y Fb(fftwnd_mpi_create_plan)27 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8793
 
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)48 b Fd(42)2025
8794
 
3045 y Fb(fftwnd_mpi_destroy_plan)26 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)
 
8885
f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)48 b Fj(42)2025 2289
 
8886
y Fi(fftwnd_mpi_create_plan)27 b Ff(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8887
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)48 b Fj(42)2025
 
8888
2381 y Fi(fftwnd_mpi_destroy_plan)26 b Ff(.)13 b(.)f(.)g(.)h(.)f(.)g(.)
8795
8889
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)47 b
8796
 
Fd(43)2025 3144 y Fb(fftwnd_mpi_local_sizes)27 b Fc(.)13
 
8890
Fj(43)2025 2472 y Fi(fftwnd_mpi_local_sizes)27 b Ff(.)13
8797
8891
b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8798
 
(.)h(.)48 b Fd(43)2025 3244 y Fb(fftwnd_mpi_plan)15 b
8799
 
Fc(.)g(.)e(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8800
 
(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)40 b Fd(42)2025
8801
 
3344 y Fb(fftwnd_one)16 b Fc(.)e(.)e(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
 
8892
(.)h(.)48 b Fj(43)2025 2564 y Fi(fftwnd_mpi_plan)15 b
 
8893
Ff(.)g(.)e(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8894
(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)40 b Fj(42)2025
 
8895
2655 y Fi(fftwnd_one)16 b Ff(.)e(.)e(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
8802
8896
g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8803
 
(.)f(.)g(.)42 b Fd(5,)26 b(24)2025 3444 y Fb(fftwnd_plan)14
8804
 
b Fc(.)h(.)d(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
 
8897
(.)f(.)g(.)42 b Fj(5,)26 b(24)2025 2747 y Fi(fftwnd_plan)14
 
8898
b Ff(.)h(.)d(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
8805
8899
f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)40
8806
 
b Fd(4,)26 b(22)2025 3710 y Ft(G)2025 3843 y Fb(genfft)c
8807
 
Fc(.)12 b(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
 
8900
b Fj(4,)26 b(22)2025 3007 y Fu(G)2025 3128 y Fi(genfft)c
 
8901
Ff(.)12 b(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8808
8902
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
8809
 
46 b Fd(59,)27 b(61)2025 4110 y Ft(R)2025 4243 y Fb(rfftw)11
8810
 
b Fc(.)i(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)
 
8903
46 b Fj(59,)27 b(61)2025 3388 y Fu(R)2025 3510 y Fi(rfftw)11
 
8904
b Ff(.)i(.)g(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)
8811
8905
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8812
 
(.)g(.)h(.)f(.)g(.)37 b Fd(27)2025 4342 y Fb(rfftw_threads)18
8813
 
b Fc(.)d(.)d(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)
 
8906
(.)g(.)h(.)f(.)g(.)37 b Fj(27)2025 3601 y Fi(rfftw_threads)18
 
8907
b Ff(.)d(.)d(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)
8814
8908
h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)43
8815
 
b Fd(38)2025 4442 y Fb(rfftw_threads_one)12 b Fc(.)k(.)c(.)g(.)h(.)f(.)
 
8909
b Fj(38)2025 3693 y Fi(rfftw_threads_one)12 b Ff(.)k(.)c(.)g(.)h(.)f(.)
8816
8910
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8817
 
(.)g(.)g(.)38 b Fd(38)2025 4542 y Fb(rfftw_create_plan)27
8818
 
b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8819
 
(.)h(.)f(.)g(.)h(.)f(.)g(.)50 b Fd(6,)26 b(26)2025 4642
8820
 
y Fb(rfftw_create_plan_specific)17 b Fc(.)h(.)12 b(.)h(.)f(.)g(.)g(.)h
8821
 
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)43 b Fd(26)2025 4741
8822
 
y Fb(rfftw_destroy_plan)26 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8911
(.)g(.)g(.)38 b Fj(38)2025 3784 y Fi(rfftw_create_plan)27
 
8912
b Ff(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8913
(.)h(.)f(.)g(.)h(.)f(.)g(.)50 b Fj(6,)26 b(26)2025 3876
 
8914
y Fi(rfftw_create_plan_specific)17 b Ff(.)h(.)12 b(.)h(.)f(.)g(.)g(.)h
 
8915
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)43 b Fj(26)2025 3967
 
8916
y Fi(rfftw_destroy_plan)26 b Ff(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8823
8917
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)48
8824
 
b Fd(6,)26 b(28)2025 4841 y Fb(rfftw_f77_create_plan)j
8825
 
Fc(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8826
 
(.)g(.)g(.)h(.)f(.)50 b Fd(54)2025 4941 y Fb(rfftw_f77_destroy_plan)27
8827
 
b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8828
 
(.)f(.)g(.)h(.)48 b Fd(54)2025 5041 y Fb(rfftw_f77_one)18
8829
 
b Fc(.)d(.)d(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)
 
8918
b Fj(6,)26 b(28)2025 4059 y Fi(rfftw_f77_create_plan)j
 
8919
Ff(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8920
(.)g(.)g(.)h(.)f(.)50 b Fj(54)2025 4150 y Fi(rfftw_f77_destroy_plan)27
 
8921
b Ff(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
 
8922
(.)f(.)g(.)h(.)48 b Fj(54)2025 4242 y Fi(rfftw_f77_one)18
 
8923
b Ff(.)d(.)d(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)
8830
8924
h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)43
8831
 
b Fd(54)2025 5140 y Fb(rfftw_mpi_test)9 b Fc(.)15 b(.)d(.)h(.)f(.)g(.)h
 
8925
b Fj(54)2025 4333 y Fi(rfftw_mpi_test)9 b Ff(.)15 b(.)d(.)h(.)f(.)g(.)h
8832
8926
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)
8833
 
g(.)h(.)34 b Fd(41,)27 b(49)2025 5240 y Fb(rfftw_one)17
8834
 
b Fc(.)d(.)f(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
 
8927
g(.)h(.)34 b Fj(41,)27 b(49)2025 4425 y Fi(rfftw_one)17
 
8928
b Ff(.)d(.)f(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
8835
8929
g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)43
8836
 
b Fd(6,)26 b(27)2025 5340 y Fb(rfftw_plan)16 b Fc(.)e(.)e(.)h(.)f(.)g
 
8930
b Fj(6,)26 b(27)2025 4516 y Fi(rfftw_plan)16 b Ff(.)e(.)e(.)h(.)f(.)g
8837
8931
(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)
8838
 
f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)42 b Fd(6,)26 b(26)p
8839
 
eop
 
8932
f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)42 b Fj(6,)26 b(26)2025
 
8933
4608 y Fi(rfftw_test)e Ff(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
 
8934
h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8935
(.)g(.)h(.)f(.)g(.)48 b Fj(57)2025 4699 y Fi(rfftw_threads_test)10
 
8936
b Ff(.)17 b(.)12 b(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8937
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)36 b Fj(39)2025
 
8938
4791 y Fi(rfftw2d_create_plan)13 b Ff(.)k(.)12 b(.)g(.)h(.)f(.)g(.)h(.)
 
8939
f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)39 b Fj(7,)26
 
8940
b(29,)h(47)2025 4882 y Fi(rfftw2d_f77_create_plan)f Ff(.)13
 
8941
b(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8942
(.)47 b Fj(54)2025 4974 y Fi(rfftw3d_mpi_create_plan)26
 
8943
b Ff(.)13 b(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8944
(.)h(.)f(.)47 b Fj(45)2025 5065 y Fi(rfftw3d_create_plan)25
 
8945
b Ff(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8946
(.)g(.)h(.)f(.)g(.)47 b Fj(7,)26 b(29)2025 5157 y Fi(rfftwnd_mpi)d
 
8947
Ff(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8948
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)46
 
8949
b Fj(46)2025 5248 y Fi(rfftwnd_mpi_destroy_plan)25 b
 
8950
Ff(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8951
(.)g(.)46 b Fj(46)2025 5340 y Fi(rfftwnd_mpi_local_sizes)26
 
8952
b Ff(.)13 b(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8953
(.)h(.)f(.)47 b Fj(45)p eop end
8840
8954
%%Page: 68 70
8841
 
68 69 bop 150 -116 a Fu(68)3232 b(FFTW)150 299 y Fb(rfftw_test)24
8842
 
b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8843
 
(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)47
8844
 
b Fd(57)150 399 y Fb(rfftw_threads_test)10 b Fc(.)17
8845
 
b(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)
8846
 
g(.)h(.)f(.)g(.)h(.)f(.)g(.)36 b Fd(39)150 498 y Fb
8847
 
(rfftw2d_create_plan)13 b Fc(.)k(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8848
 
(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)39 b Fd(7,)27 b(29,)f(47)150
8849
 
598 y Fb(rfftw2d_f77_create_plan)g Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)
8850
 
f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)47 b Fd(54)150
8851
 
697 y Fb(rfftw3d_mpi_create_plan)26 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h
8852
 
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)47 b Fd(45)150
8853
 
797 y Fb(rfftw3d_create_plan)25 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8854
 
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)47
8855
 
b Fd(7,)26 b(29)150 897 y Fb(rfftwnd_mpi)d Fc(.)12 b(.)h(.)f(.)g(.)h(.)
8856
 
f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g
8857
 
(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)46 b Fd(46)150 996
8858
 
y Fb(rfftwnd_mpi_destroy_plan)25 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)
8859
 
f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)45 b Fd(46)150
8860
 
1096 y Fb(rfftwnd_mpi_local_sizes)26 b Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)
8861
 
h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)47 b
8862
 
Fd(45)150 1196 y Fb(rfftwnd_threads_complex_to_re)q(al)10
8863
 
b Fc(.)18 b(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)35
8864
 
b Fd(39)150 1295 y Fb(rfftwnd_threads_one_complex_t)q(o_re)q(al)28
8865
 
b Fc(.)12 b(.)h(.)f(.)g(.)h(.)47 b Fd(39)2025 299 y Fb
8866
 
(rfftwnd_threads_one_real_to_co)q(mplex)16 b Fc(.)i(.)12
8867
 
b(.)40 b Fd(38,)27 b(39)2025 399 y Fb(rfftwnd_threads_real_to_comple)q
8868
 
(x)10 b Fc(.)18 b(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)35
8869
 
b Fd(38)2025 498 y Fb(rfftwnd_complex_to_real)26 b Fc(.)13
8870
 
b(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f
8871
 
(.)47 b Fd(31)2025 598 y Fb(rfftwnd_create_plan)13 b
8872
 
Fc(.)k(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8873
 
(.)f(.)39 b Fd(7,)26 b(29,)h(47)2025 697 y Fb(rfftwnd_destroy_plan)7
8874
 
b Fc(.)17 b(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8875
 
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)33 b Fd(33)2025 797 y
8876
 
Fb(rfftwnd_f77_destroy_plan)25 b Fc(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h
8877
 
(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)46 b Fd(54)2025
8878
 
897 y Fb(rfftwnd_f77_one_real_to_comple)q(x)10 b Fc(.)18
8879
 
b(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)35 b Fd(54)2025
8880
 
996 y Fb(rfftwnd_one_complex_to_real)9 b Fc(.)18 b(.)13
8881
 
b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)35 b Fd(8,)26
8882
 
b(31)2025 1096 y Fb(rfftwnd_one_real_to_complex)9 b Fc(.)18
8883
 
b(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)35
8884
 
b Fd(8,)26 b(31)2025 1196 y Fb(rfftwnd_plan)13 b Fc(.)i(.)d(.)g(.)h(.)f
 
8955
TeXDict begin 68 69 bop 150 -116 a Fv(68)3232 b(FFTW)150
 
8956
299 y Fi(rfftwnd_threads_complex_to_re)q(al)10 b Ff(.)18
 
8957
b(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)35 b Fj(39)150
 
8958
391 y Fi(rfftwnd_threads_one_complex_t)q(o_re)q(al)28
 
8959
b Ff(.)12 b(.)h(.)f(.)g(.)h(.)47 b Fj(39)150 483 y Fi
 
8960
(rfftwnd_threads_one_real_to_c)q(ompl)q(ex)28 b Ff(.)12
 
8961
b(.)h(.)f(.)g(.)h(.)47 b Fj(39)150 575 y Fi
 
8962
(rfftwnd_threads_real_to_compl)q(ex)10 b Ff(.)18 b(.)12
 
8963
b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)35 b Fj(38)150 667
 
8964
y Fi(rfftwnd_complex_to_real)26 b Ff(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f
 
8965
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)47 b Fj(31)150
 
8966
759 y Fi(rfftwnd_create_plan)13 b Ff(.)k(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g
 
8967
(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)39 b Fj(7,)27
 
8968
b(29,)f(47)150 851 y Fi(rfftwnd_destroy_plan)7 b Ff(.)17
 
8969
b(.)c(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h
 
8970
(.)f(.)g(.)h(.)f(.)33 b Fj(33)2025 299 y Fi(rfftwnd_f77_destroy_plan)25
 
8971
b Ff(.)12 b(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g
 
8972
(.)g(.)46 b Fj(54)2025 409 y Fi(rfftwnd_f77_one_real_to_comple)q(x)10
 
8973
b Ff(.)18 b(.)12 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)35
 
8974
b Fj(54)2025 520 y Fi(rfftwnd_one_complex_to_real)9 b
 
8975
Ff(.)18 b(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)35
 
8976
b Fj(8,)26 b(31)2025 630 y Fi(rfftwnd_one_real_to_complex)9
 
8977
b Ff(.)18 b(.)13 b(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)35
 
8978
b Fj(8,)26 b(31)2025 741 y Fi(rfftwnd_plan)13 b Ff(.)i(.)d(.)g(.)h(.)f
8885
8979
(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)
8886
 
h(.)f(.)g(.)h(.)f(.)g(.)39 b Fd(7,)26 b(29)2025 1295
8887
 
y Fb(rfftwnd_real_to_complex)g Fc(.)13 b(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)
8888
 
f(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)47 b Fd(31)p
8889
 
eop
 
8980
h(.)f(.)g(.)h(.)f(.)g(.)39 b Fj(7,)26 b(29)2025 851 y
 
8981
Fi(rfftwnd_real_to_complex)g Ff(.)13 b(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f
 
8982
(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)h(.)f(.)47 b Fj(31)p eop
 
8983
end
8890
8984
%%Page: -1 71
8891
 
-1 70 bop 3725 -116 a Fu(i)150 299 y Fr(T)-13 b(able)54
8892
 
b(of)g(Con)l(ten)l(ts)150 641 y Ft(1)135 b(In)l(tro)t(duction)15
8893
 
b Fa(.)20 b(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f
8894
 
(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)60
8895
 
b Ft(1)150 911 y(2)135 b(T)-11 b(utorial)25 b Fa(.)20
 
8985
TeXDict begin -1 70 bop 3725 -116 a Fv(i)150 299 y Fs(T)-13
 
8986
b(able)54 b(of)g(Con)l(ten)l(ts)150 641 y Fu(1)135 b(In)l(tro)t
 
8987
(duction)15 b Fa(.)20 b(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h
 
8988
(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)
 
8989
60 b Fu(1)150 911 y(2)135 b(T)-11 b(utorial)25 b Fa(.)20
8896
8990
b(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h
8897
8991
(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)69
8898
 
b Ft(3)449 1048 y Fu(2.1)92 b(Complex)29 b(One-dimensional)f(T)-8
8899
 
b(ransforms)29 b(T)-8 b(utorial)22 b Fq(.)15 b(.)g(.)g(.)h(.)f(.)g(.)g
8900
 
(.)g(.)g(.)g(.)g(.)53 b Fu(3)449 1157 y(2.2)92 b(Complex)29
 
8992
b Fu(3)449 1048 y Fv(2.1)92 b(Complex)29 b(One-dimensional)f(T)-8
 
8993
b(ransforms)29 b(T)-8 b(utorial)22 b Fr(.)15 b(.)g(.)g(.)h(.)f(.)g(.)g
 
8994
(.)g(.)g(.)g(.)g(.)53 b Fv(3)449 1157 y(2.2)92 b(Complex)29
8901
8995
b(Multi-dimensional)e(T)-8 b(ransforms)29 b(T)-8 b(utorial)16
8902
 
b Fq(.)e(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)46 b Fu(4)449
 
8996
b Fr(.)e(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)46 b Fv(4)449
8903
8997
1267 y(2.3)92 b(Real)30 b(One-dimensional)e(T)-8 b(ransforms)29
8904
 
b(T)-8 b(utorial)23 b Fq(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
8905
 
(.)g(.)g(.)g(.)g(.)54 b Fu(6)449 1377 y(2.4)92 b(Real)30
 
8998
b(T)-8 b(utorial)23 b Fr(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
 
8999
(.)g(.)g(.)g(.)g(.)54 b Fv(6)449 1377 y(2.4)92 b(Real)30
8906
9000
b(Multi-dimensional)d(T)-8 b(ransforms)29 b(T)-8 b(utorial)16
8907
 
b Fq(.)e(.)h(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)46
8908
 
b Fu(7)449 1486 y(2.5)92 b(Multi-dimensional)26 b(Arra)m(y)31
8909
 
b(F)-8 b(ormat)29 b Fq(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g
 
9001
b Fr(.)e(.)h(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)46
 
9002
b Fv(7)449 1486 y(2.5)92 b(Multi-dimensional)26 b(Arra)m(y)31
 
9003
b(F)-8 b(ormat)29 b Fr(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g
8910
9004
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)57
8911
 
b Fu(11)748 1596 y(2.5.1)93 b(Ro)m(w-ma)5 b(jor)31 b(F)-8
8912
 
b(ormat)28 b Fq(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
 
9005
b Fv(11)748 1596 y(2.5.1)93 b(Ro)m(w-ma)5 b(jor)31 b(F)-8
 
9006
b(ormat)28 b Fr(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
8913
9007
(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)56
8914
 
b Fu(11)748 1705 y(2.5.2)93 b(Column-ma)5 b(jor)28 b(F)-8
8915
 
b(ormat)23 b Fq(.)16 b(.)f(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
8916
 
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)52 b Fu(11)748
 
9008
b Fv(11)748 1705 y(2.5.2)93 b(Column-ma)5 b(jor)28 b(F)-8
 
9009
b(ormat)23 b Fr(.)16 b(.)f(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
 
9010
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)52 b Fv(11)748
8917
9011
1815 y(2.5.3)93 b(Static)30 b(Arra)m(ys)h(in)e(C)20 b
8918
 
Fq(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
 
9012
Fr(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
8919
9013
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)50
8920
 
b Fu(12)748 1924 y(2.5.4)93 b(Dynamic)30 b(Arra)m(ys)g(in)f(C)23
8921
 
b Fq(.)15 b(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
8922
 
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)53 b Fu(12)748
 
9014
b Fv(12)748 1924 y(2.5.4)93 b(Dynamic)30 b(Arra)m(ys)g(in)f(C)23
 
9015
b Fr(.)15 b(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
 
9016
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)53 b Fv(12)748
8923
9017
2034 y(2.5.5)93 b(Dynamic)30 b(Arra)m(ys)g(in)f(C|The)h(W)-8
8924
 
b(rong)31 b(W)-8 b(a)m(y)15 b Fq(.)i(.)e(.)g(.)g(.)g(.)g(.)g(.)g(.)45
8925
 
b Fu(12)449 2144 y(2.6)92 b(W)-8 b(ords)31 b(of)f(Wisdom)20
8926
 
b Fq(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
 
9018
b(rong)31 b(W)-8 b(a)m(y)15 b Fr(.)i(.)e(.)g(.)g(.)g(.)g(.)g(.)g(.)45
 
9019
b Fv(12)449 2144 y(2.6)92 b(W)-8 b(ords)31 b(of)f(Wisdom)20
 
9020
b Fr(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
8927
9021
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
8928
 
h(.)f(.)g(.)g(.)49 b Fu(13)748 2253 y(2.6.1)93 b(Ca)m(v)m(eats)32
8929
 
b(in)d(Using)h(Wisdom)22 b Fq(.)14 b(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
 
9022
h(.)f(.)g(.)g(.)49 b Fv(13)748 2253 y(2.6.1)93 b(Ca)m(v)m(eats)32
 
9023
b(in)d(Using)h(Wisdom)22 b Fr(.)14 b(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
8930
9024
g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)51 b
8931
 
Fu(14)748 2363 y(2.6.2)93 b(Imp)s(orting)28 b(and)i(Exp)s(orting)f
8932
 
(Wisdom)d Fq(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
8933
 
56 b Fu(14)150 2605 y Ft(3)135 b(FFTW)43 b(Reference)28
 
9025
Fv(14)748 2363 y(2.6.2)93 b(Imp)s(orting)28 b(and)i(Exp)s(orting)f
 
9026
(Wisdom)d Fr(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
 
9027
56 b Fv(14)150 2605 y Fu(3)135 b(FFTW)43 b(Reference)28
8934
9028
b Fa(.)21 b(.)e(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f
8935
 
(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)72 b Ft(17)449 2742
8936
 
y Fu(3.1)92 b(Data)32 b(T)m(yp)s(es)15 b Fq(.)f(.)h(.)g(.)g(.)g(.)g(.)g
 
9029
(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)72 b Fu(17)449 2742
 
9030
y Fv(3.1)92 b(Data)32 b(T)m(yp)s(es)15 b Fr(.)f(.)h(.)g(.)g(.)g(.)g(.)g
8937
9031
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)
8938
9032
g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
8939
 
(.)g(.)44 b Fu(17)449 2852 y(3.2)92 b(One-dimensional)27
8940
 
b(T)-8 b(ransforms)30 b(Reference)10 b Fq(.)15 b(.)g(.)g(.)g(.)g(.)h(.)
 
9033
(.)g(.)44 b Fv(17)449 2852 y(3.2)92 b(One-dimensional)27
 
9034
b(T)-8 b(ransforms)30 b(Reference)10 b Fr(.)15 b(.)g(.)g(.)g(.)g(.)h(.)
8941
9035
f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)39 b
8942
 
Fu(18)748 2961 y(3.2.1)93 b(Plan)29 b(Creation)h(for)g(One-dimensional)
8943
 
d(T)-8 b(ransforms)13 b Fq(.)h(.)h(.)42 b Fu(18)748 3071
 
9036
Fv(18)748 2961 y(3.2.1)93 b(Plan)29 b(Creation)h(for)g(One-dimensional)
 
9037
d(T)-8 b(ransforms)13 b Fr(.)h(.)h(.)42 b Fv(18)748 3071
8944
9038
y(3.2.2)93 b(Discussion)28 b(on)i(Sp)s(eci\014c)f(Plans)19
8945
 
b Fq(.)14 b(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g
8946
 
(.)g(.)g(.)g(.)48 b Fu(20)748 3181 y(3.2.3)93 b(Computing)28
8947
 
b(the)j(One-dimensional)c(T)-8 b(ransform)14 b Fq(.)g(.)h(.)g(.)g(.)g
8948
 
(.)43 b Fu(20)748 3290 y(3.2.4)93 b(Destro)m(ying)31
8949
 
b(a)f(One-dimensional)e(Plan)c Fq(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
8950
 
g(.)g(.)g(.)g(.)54 b Fu(21)748 3400 y(3.2.5)93 b(What)31
8951
 
b(FFTW)g(Really)e(Computes)22 b Fq(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g
8952
 
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)51 b Fu(21)449 3509
 
9039
b Fr(.)14 b(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g
 
9040
(.)g(.)g(.)g(.)48 b Fv(20)748 3181 y(3.2.3)93 b(Computing)28
 
9041
b(the)j(One-dimensional)c(T)-8 b(ransform)14 b Fr(.)g(.)h(.)g(.)g(.)g
 
9042
(.)43 b Fv(20)748 3290 y(3.2.4)93 b(Destro)m(ying)31
 
9043
b(a)f(One-dimensional)e(Plan)c Fr(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
 
9044
g(.)g(.)g(.)g(.)54 b Fv(21)748 3400 y(3.2.5)93 b(What)31
 
9045
b(FFTW)g(Really)e(Computes)22 b Fr(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g
 
9046
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)51 b Fv(21)449 3509
8953
9047
y(3.3)92 b(Multi-dimensional)26 b(T)-8 b(ransforms)30
8954
 
b(Reference)24 b Fq(.)15 b(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
8955
 
(.)g(.)g(.)g(.)g(.)53 b Fu(22)748 3619 y(3.3.1)93 b(Plan)27
 
9048
b(Reference)24 b Fr(.)15 b(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
 
9049
(.)g(.)g(.)g(.)g(.)53 b Fv(22)748 3619 y(3.3.1)93 b(Plan)27
8956
9050
b(Creation)h(for)g(Multi-dimensional)c(T)-8 b(ransforms)10
8957
 
b Fq(.)k(.)38 b Fu(22)748 3729 y(3.3.2)93 b(Computing)28
8958
 
b(the)j(Multi-dimensional)26 b(T)-8 b(ransform)27 b Fq(.)15
8959
 
b(.)g(.)g(.)57 b Fu(24)748 3838 y(3.3.3)93 b(Destro)m(ying)31
8960
 
b(a)f(Multi-dimensional)d(Plan)17 b Fq(.)d(.)h(.)g(.)g(.)g(.)g(.)g(.)g
8961
 
(.)g(.)g(.)h(.)46 b Fu(25)748 3948 y(3.3.4)93 b(What)31
8962
 
b(FFTWND)g(Really)f(Computes)18 b Fq(.)c(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)
8963
 
g(.)g(.)g(.)g(.)g(.)47 b Fu(25)449 4057 y(3.4)92 b(Real)30
 
9051
b Fr(.)k(.)38 b Fv(22)748 3729 y(3.3.2)93 b(Computing)28
 
9052
b(the)j(Multi-dimensional)26 b(T)-8 b(ransform)27 b Fr(.)15
 
9053
b(.)g(.)g(.)57 b Fv(24)748 3838 y(3.3.3)93 b(Destro)m(ying)31
 
9054
b(a)f(Multi-dimensional)d(Plan)17 b Fr(.)d(.)h(.)g(.)g(.)g(.)g(.)g(.)g
 
9055
(.)g(.)g(.)h(.)46 b Fv(25)748 3948 y(3.3.4)93 b(What)31
 
9056
b(FFTWND)g(Really)f(Computes)18 b Fr(.)c(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)
 
9057
g(.)g(.)g(.)g(.)g(.)47 b Fv(25)449 4057 y(3.4)92 b(Real)30
8964
9058
b(One-dimensional)e(T)-8 b(ransforms)29 b(Reference)13
8965
 
b Fq(.)j(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)43
8966
 
b Fu(26)748 4167 y(3.4.1)93 b(Plan)29 b(Creation)h(for)g(Real)g
8967
 
(One-dimensional)e(T)-8 b(ransforms)954 4276 y Fq(.)16
8968
 
b(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
8969
 
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)
8970
 
f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)54 b
8971
 
Fu(26)748 4386 y(3.4.2)93 b(Computing)28 b(the)j(Real)f
8972
 
(One-dimensional)e(T)-8 b(ransform)954 4496 y Fq(.)16
8973
 
b(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
8974
 
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)
8975
 
f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)54 b
8976
 
Fu(27)748 4605 y(3.4.3)93 b(Destro)m(ying)31 b(a)f(Real)h
8977
 
(One-dimensional)c(Plan)g Fq(.)15 b(.)g(.)g(.)h(.)f(.)g(.)g(.)57
8978
 
b Fu(28)748 4715 y(3.4.4)93 b(What)31 b(RFFTW)g(Really)e(Computes)10
8979
 
b Fq(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
8980
 
39 b Fu(28)449 4824 y(3.5)92 b(Real)30 b(Multi-dimensional)d(T)-8
8981
 
b(ransforms)29 b(Reference)f Fq(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
8982
 
(.)g(.)g(.)57 b Fu(29)748 4934 y(3.5.1)93 b(Plan)28 b(Creation)g(for)h
 
9059
b Fr(.)j(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)43
 
9060
b Fv(26)748 4167 y(3.4.1)93 b(Plan)29 b(Creation)h(for)g(Real)g
 
9061
(One-dimensional)e(T)-8 b(ransforms)954 4276 y Fr(.)16
 
9062
b(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
 
9063
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)
 
9064
f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)54 b
 
9065
Fv(26)748 4386 y(3.4.2)93 b(Computing)28 b(the)j(Real)f
 
9066
(One-dimensional)e(T)-8 b(ransform)954 4496 y Fr(.)16
 
9067
b(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
 
9068
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)
 
9069
f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)54 b
 
9070
Fv(27)748 4605 y(3.4.3)93 b(Destro)m(ying)31 b(a)f(Real)h
 
9071
(One-dimensional)c(Plan)g Fr(.)15 b(.)g(.)g(.)h(.)f(.)g(.)g(.)57
 
9072
b Fv(28)748 4715 y(3.4.4)93 b(What)31 b(RFFTW)g(Really)e(Computes)10
 
9073
b Fr(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
 
9074
39 b Fv(28)449 4824 y(3.5)92 b(Real)30 b(Multi-dimensional)d(T)-8
 
9075
b(ransforms)29 b(Reference)f Fr(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
 
9076
(.)g(.)g(.)57 b Fv(29)748 4934 y(3.5.1)93 b(Plan)28 b(Creation)g(for)h
8983
9077
(Real)g(Multi-dimensional)c(T)-8 b(ransforms)954 5044
8984
 
y Fq(.)16 b(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
 
9078
y Fr(.)16 b(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
8985
9079
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
8986
9080
g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)54
8987
 
b Fu(29)748 5153 y(3.5.2)93 b(Computing)28 b(the)j(Real)f
8988
 
(Multi-dimensional)d(T)-8 b(ransform)954 5263 y Fq(.)16
 
9081
b Fv(29)748 5153 y(3.5.2)93 b(Computing)28 b(the)j(Real)f
 
9082
(Multi-dimensional)d(T)-8 b(ransform)954 5263 y Fr(.)16
8989
9083
b(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
8990
9084
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)
8991
9085
f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)54 b
8992
 
Fu(31)p eop
 
9086
Fv(30)p eop end
8993
9087
%%Page: -2 72
8994
 
-2 71 bop 150 -116 a Fu(ii)3270 b(FFTW)748 83 y(3.5.3)93
8995
 
b(Arra)m(y)30 b(Dimensions)f(for)h(Real)g(Multi-dimensional)930
8996
 
193 y(T)-8 b(ransforms)17 b Fq(.)d(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
 
9088
TeXDict begin -2 71 bop 150 -116 a Fv(ii)3270 b(FFTW)748
 
9089
83 y(3.5.3)93 b(Arra)m(y)30 b(Dimensions)f(for)h(Real)g
 
9090
(Multi-dimensional)930 193 y(T)-8 b(ransforms)17 b Fr(.)d(.)h(.)g(.)g
8997
9091
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
8998
 
g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)46 b Fu(32)748
8999
 
302 y(3.5.4)93 b(Strides)28 b(in)h(In-place)h(RFFTWND)c
9000
 
Fq(.)16 b(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
9001
 
(.)54 b Fu(33)748 412 y(3.5.5)93 b(Destro)m(ying)31 b(a)f
9002
 
(Multi-dimensional)d(Plan)17 b Fq(.)d(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g
9003
 
(.)g(.)h(.)46 b Fu(33)748 521 y(3.5.6)93 b(What)31 b(RFFTWND)g(Really)f
9004
 
(Computes)c Fq(.)15 b(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)56
9005
 
b Fu(33)449 631 y(3.6)92 b(Wisdom)30 b(Reference)f Fq(.)15
 
9092
g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)46
 
9093
b Fv(32)748 302 y(3.5.4)93 b(Strides)28 b(in)h(In-place)h(RFFTWND)c
 
9094
Fr(.)16 b(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
 
9095
(.)54 b Fv(32)748 412 y(3.5.5)93 b(Destro)m(ying)31 b(a)f
 
9096
(Multi-dimensional)d(Plan)17 b Fr(.)d(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g
 
9097
(.)g(.)h(.)46 b Fv(33)748 521 y(3.5.6)93 b(What)31 b(RFFTWND)g(Really)f
 
9098
(Computes)c Fr(.)15 b(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)56
 
9099
b Fv(33)449 631 y(3.6)92 b(Wisdom)30 b(Reference)f Fr(.)15
9006
9100
b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f
9007
9101
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
9008
 
g(.)58 b Fu(34)748 741 y(3.6.1)93 b(Exp)s(orting)28 b(Wisdom)g
9009
 
Fq(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g
9010
 
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)58 b Fu(34)748
9011
 
850 y(3.6.2)93 b(Imp)s(orting)28 b(Wisdom)h Fq(.)15 b(.)g(.)g(.)g(.)g
9012
 
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
9013
 
g(.)g(.)g(.)g(.)g(.)59 b Fu(35)748 960 y(3.6.3)93 b(F)-8
9014
 
b(orgetting)31 b(Wisdom)23 b Fq(.)15 b(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g
9015
 
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
9016
 
g(.)53 b Fu(35)449 1069 y(3.7)92 b(Memory)31 b(Allo)s(cator)f
9017
 
(Reference)22 b Fq(.)15 b(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g
 
9102
g(.)58 b Fv(34)748 741 y(3.6.1)93 b(Exp)s(orting)28 b(Wisdom)g
 
9103
Fr(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g
 
9104
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)58 b Fv(34)748
 
9105
850 y(3.6.2)93 b(Imp)s(orting)28 b(Wisdom)h Fr(.)15 b(.)g(.)g(.)g(.)g
 
9106
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
 
9107
g(.)g(.)g(.)g(.)g(.)59 b Fv(35)748 960 y(3.6.3)93 b(F)-8
 
9108
b(orgetting)31 b(Wisdom)23 b Fr(.)15 b(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g
 
9109
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
 
9110
g(.)53 b Fv(35)449 1069 y(3.7)92 b(Memory)31 b(Allo)s(cator)f
 
9111
(Reference)22 b Fr(.)15 b(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g
9018
9112
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)51
9019
 
b Fu(36)449 1179 y(3.8)92 b(Thread)29 b(safet)m(y)16
9020
 
b Fq(.)h(.)e(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)
 
9113
b Fv(35)449 1179 y(3.8)92 b(Thread)29 b(safet)m(y)16
 
9114
b Fr(.)h(.)e(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)
9021
9115
g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
9022
 
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)46 b Fu(36)150 1419 y
9023
 
Ft(4)135 b(P)l(arallel)47 b(FFTW)30 b Fa(.)18 b(.)h(.)h(.)f(.)g(.)h(.)f
 
9116
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)46 b Fv(36)150 1419 y
 
9117
Fu(4)135 b(P)l(arallel)47 b(FFTW)30 b Fa(.)18 b(.)h(.)h(.)f(.)g(.)h(.)f
9024
9118
(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)
9025
 
f(.)g(.)75 b Ft(37)449 1556 y Fu(4.1)92 b(Multi-threaded)29
9026
 
b(FFTW)10 b Fq(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g
 
9119
f(.)g(.)75 b Fu(37)449 1556 y Fv(4.1)92 b(Multi-threaded)29
 
9120
b(FFTW)10 b Fr(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g
9027
9121
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
9028
 
g(.)g(.)g(.)39 b Fu(37)748 1665 y(4.1.1)93 b(Installation)28
 
9122
g(.)g(.)g(.)39 b Fv(37)748 1665 y(4.1.1)93 b(Installation)28
9029
9123
b(and)i(Supp)s(orted)e(Hardw)m(are/Soft)m(w)m(are)9 b
9030
 
Fq(.)17 b(.)e(.)38 b Fu(37)748 1775 y(4.1.2)93 b(Usage)31
9031
 
b(of)g(Multi-threaded)e(FFTW)f Fq(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
9032
 
g(.)g(.)g(.)g(.)g(.)g(.)g(.)57 b Fu(38)748 1885 y(4.1.3)93
9033
 
b(Ho)m(w)31 b(Man)m(y)g(Threads)e(to)i(Use?)22 b Fq(.)15
 
9124
Fr(.)17 b(.)e(.)38 b Fv(37)748 1775 y(4.1.2)93 b(Usage)31
 
9125
b(of)g(Multi-threaded)e(FFTW)f Fr(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
 
9126
g(.)g(.)g(.)g(.)g(.)g(.)g(.)57 b Fv(38)748 1885 y(4.1.3)93
 
9127
b(Ho)m(w)31 b(Man)m(y)g(Threads)e(to)i(Use?)22 b Fr(.)15
9034
9128
b(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
9035
 
(.)g(.)51 b Fu(39)748 1994 y(4.1.4)93 b(Using)29 b(Multi-threaded)g
 
9129
(.)g(.)51 b Fv(39)748 1994 y(4.1.4)93 b(Using)29 b(Multi-threaded)g
9036
9130
(FFTW)i(in)e(a)i(Multi-threaded)930 2104 y(Program)24
9037
 
b Fq(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g
 
9131
b Fr(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g
9038
9132
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
9039
 
g(.)g(.)g(.)g(.)g(.)g(.)54 b Fu(40)748 2213 y(4.1.5)93
9040
 
b(Tips)28 b(for)i(Optimal)f(Threading)22 b Fq(.)13 b(.)i(.)g(.)g(.)g(.)
 
9133
g(.)g(.)g(.)g(.)g(.)g(.)54 b Fv(40)748 2213 y(4.1.5)93
 
9134
b(Tips)28 b(for)i(Optimal)f(Threading)22 b Fr(.)13 b(.)i(.)g(.)g(.)g(.)
9041
9135
g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)51
9042
 
b Fu(40)748 2323 y(4.1.6)93 b(P)m(arallelization)29 b(de\014ciencies)g
9043
 
(in)g(one-dimensional)930 2433 y(transforms)e Fq(.)15
 
9136
b Fv(40)748 2323 y(4.1.6)93 b(P)m(arallelization)29 b(de\014ciencies)g
 
9137
(in)g(one-dimensional)930 2433 y(transforms)e Fr(.)15
9044
9138
b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
9045
9139
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
9046
 
g(.)g(.)58 b Fu(40)449 2542 y(4.2)92 b(MPI)30 b(FFTW)d
9047
 
Fq(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g
 
9140
g(.)g(.)58 b Fv(40)449 2542 y(4.2)92 b(MPI)30 b(FFTW)d
 
9141
Fr(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g
9048
9142
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
9049
 
g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)55 b Fu(40)748 2652
9050
 
y(4.2.1)93 b(MPI)30 b(FFTW)h(Installation)9 b Fq(.)k(.)i(.)g(.)g(.)g(.)
 
9143
g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)55 b Fv(40)748 2652
 
9144
y(4.2.1)93 b(MPI)30 b(FFTW)h(Installation)9 b Fr(.)k(.)i(.)g(.)g(.)g(.)
9051
9145
g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
9052
 
(.)38 b Fu(41)748 2761 y(4.2.2)93 b(Usage)31 b(of)f(MPI)g(FFTW)h(for)f
 
9146
(.)38 b Fv(41)748 2761 y(4.2.2)93 b(Usage)31 b(of)f(MPI)g(FFTW)h(for)f
9053
9147
(Complex)f(Multi-dimensional)930 2871 y(T)-8 b(ransforms)17
9054
 
b Fq(.)d(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
 
9148
b Fr(.)d(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
9055
9149
g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g
9056
 
(.)g(.)g(.)g(.)46 b Fu(41)748 2980 y(4.2.3)93 b(MPI)30
9057
 
b(Data)i(La)m(y)m(out)9 b Fq(.)16 b(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
 
9150
(.)g(.)g(.)g(.)46 b Fv(41)748 2980 y(4.2.3)93 b(MPI)30
 
9151
b(Data)i(La)m(y)m(out)9 b Fr(.)16 b(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
9058
9152
(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
9059
 
g(.)g(.)38 b Fu(43)748 3090 y(4.2.4)93 b(Usage)31 b(of)g(MPI)f(FFTW)h
 
9153
g(.)g(.)38 b Fv(43)748 3090 y(4.2.4)93 b(Usage)31 b(of)g(MPI)f(FFTW)h
9060
9154
(for)f(Real)h(Multi-dimensional)930 3200 y(T)-8 b(ransforms)17
9061
 
b Fq(.)d(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
 
9155
b Fr(.)d(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
9062
9156
g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g
9063
 
(.)g(.)g(.)g(.)46 b Fu(44)748 3309 y(4.2.5)93 b(Usage)31
 
9157
(.)g(.)g(.)g(.)46 b Fv(44)748 3309 y(4.2.5)93 b(Usage)31
9064
9158
b(of)g(MPI)f(FFTW)h(for)f(Complex)g(One-dimensional)930
9065
 
3419 y(T)-8 b(ransforms)17 b Fq(.)d(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
 
9159
3419 y(T)-8 b(ransforms)17 b Fr(.)d(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
9066
9160
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
9067
 
g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)46 b Fu(47)748
9068
 
3528 y(4.2.6)93 b(MPI)30 b(Tips)21 b Fq(.)13 b(.)i(.)g(.)g(.)g(.)g(.)g
 
9161
g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)46 b Fv(47)748
 
9162
3528 y(4.2.6)93 b(MPI)30 b(Tips)21 b Fr(.)13 b(.)i(.)g(.)g(.)g(.)g(.)g
9069
9163
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)
9070
9164
g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)50 b
9071
 
Fu(48)150 3768 y Ft(5)135 b(Calling)46 b(FFTW)e(from)h(F)-11
 
9165
Fv(48)150 3768 y Fu(5)135 b(Calling)46 b(FFTW)e(from)h(F)-11
9072
9166
b(ortran)16 b Fa(.)j(.)g(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)
9073
 
g(.)61 b Ft(51)449 3905 y Fu(5.1)92 b(W)-8 b(rapp)s(er)30
9074
 
b(Routines)13 b Fq(.)h(.)h(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g
 
9167
g(.)61 b Fu(51)449 3905 y Fv(5.1)92 b(W)-8 b(rapp)s(er)30
 
9168
b(Routines)13 b Fr(.)h(.)h(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g
9075
9169
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
9076
 
g(.)g(.)g(.)g(.)g(.)g(.)g(.)43 b Fu(51)449 4015 y(5.2)92
9077
 
b(FFTW)31 b(Constan)m(ts)g(in)e(F)-8 b(ortran)21 b Fq(.)16
 
9170
g(.)g(.)g(.)g(.)g(.)g(.)g(.)43 b Fv(51)449 4015 y(5.2)92
 
9171
b(FFTW)31 b(Constan)m(ts)g(in)e(F)-8 b(ortran)21 b Fr(.)16
9078
9172
b(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
9079
 
(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)50 b Fu(52)449
9080
 
4125 y(5.3)92 b(F)-8 b(ortran)31 b(Examples)20 b Fq(.)14
 
9173
(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)50 b Fv(52)449
 
9174
4125 y(5.3)92 b(F)-8 b(ortran)31 b(Examples)20 b Fr(.)14
9081
9175
b(.)h(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
9082
9176
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
9083
 
g(.)g(.)50 b Fu(52)150 4364 y Ft(6)135 b(Installation)47
 
9177
g(.)g(.)50 b Fv(52)150 4364 y Fu(6)135 b(Installation)47
9084
9178
b(and)e(Customization)35 b Fa(.)19 b(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)
9085
 
f(.)h(.)78 b Ft(55)449 4501 y Fu(6.1)92 b(Installation)29
9086
 
b(on)h(Unix)19 b Fq(.)14 b(.)h(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g
 
9179
f(.)h(.)78 b Fu(55)449 4501 y Fv(6.1)92 b(Installation)29
 
9180
b(on)h(Unix)19 b Fr(.)14 b(.)h(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g
9087
9181
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
9088
 
g(.)g(.)g(.)g(.)g(.)g(.)49 b Fu(55)449 4611 y(6.2)92
9089
 
b(Installation)29 b(on)h(non-Unix)f(Systems)13 b Fq(.)i(.)g(.)g(.)g(.)g
 
9182
g(.)g(.)g(.)g(.)g(.)g(.)49 b Fv(55)449 4611 y(6.2)92
 
9183
b(Installation)29 b(on)h(non-Unix)f(Systems)13 b Fr(.)i(.)g(.)g(.)g(.)g
9090
9184
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)
9091
 
g(.)42 b Fu(56)449 4721 y(6.3)92 b(Installing)28 b(FFTW)j(in)e(b)s(oth)
9092
 
h(single)f(and)h(double)e(precision)14 b Fq(.)g(.)h(.)g(.)g(.)g(.)43
9093
 
b Fu(57)449 4830 y(6.4)92 b Fo(gcc)29 b Fu(and)h(P)m(en)m(tium)g(hac)m
9094
 
(ks)18 b Fq(.)e(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
 
9185
g(.)42 b Fv(56)449 4721 y(6.3)92 b(Installing)28 b(FFTW)j(in)e(b)s(oth)
 
9186
h(single)f(and)h(double)e(precision)14 b Fr(.)g(.)h(.)g(.)g(.)g(.)43
 
9187
b Fv(57)449 4830 y(6.4)92 b Fp(gcc)29 b Fv(and)h(P)m(en)m(tium)g(hac)m
 
9188
(ks)18 b Fr(.)e(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g
9095
9189
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)
9096
 
47 b Fu(57)449 4940 y(6.5)92 b(Customizing)29 b(the)h(timer)16
9097
 
b Fq(.)e(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)
 
9190
47 b Fv(57)449 4940 y(6.5)92 b(Customizing)29 b(the)h(timer)16
 
9191
b Fr(.)e(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g(.)
9098
9192
g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)45
9099
 
b Fu(58)449 5049 y(6.6)92 b(Generating)31 b(y)m(our)f(o)m(wn)g(co)s(de)
9100
 
c Fq(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g
 
9193
b Fv(58)449 5049 y(6.6)92 b(Generating)31 b(y)m(our)f(o)m(wn)g(co)s(de)
 
9194
c Fr(.)15 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)g
9101
9195
(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)55
9102
 
b Fu(59)150 5289 y Ft(7)135 b(Ac)l(kno)l(wledgmen)l(ts)12
 
9196
b Fv(59)150 5289 y Fu(7)135 b(Ac)l(kno)l(wledgmen)l(ts)12
9103
9197
b Fa(.)20 b(.)g(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f
9104
 
(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)57 b Ft(61)p eop
 
9198
(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)57 b Fu(61)p eop end
9105
9199
%%Page: -3 73
9106
 
-3 72 bop 3674 -116 a Fu(iii)150 83 y Ft(8)135 b(License)45
9107
 
b(and)g(Cop)l(yrigh)l(t)35 b Fa(.)19 b(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h
9108
 
(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)79 b Ft(63)150
9109
 
353 y(9)135 b(Concept)45 b(Index)13 b Fa(.)20 b(.)f(.)g(.)h(.)f(.)h(.)f
9110
 
(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)
9111
 
f(.)g(.)h(.)58 b Ft(65)150 623 y(10)135 b(Library)45
9112
 
b(Index)35 b Fa(.)19 b(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h
9113
 
(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)80
9114
 
b Ft(67)p eop
 
9200
TeXDict begin -3 72 bop 3674 -116 a Fv(iii)150 83 y Fu(8)135
 
9201
b(License)45 b(and)g(Cop)l(yrigh)l(t)35 b Fa(.)19 b(.)h(.)f(.)h(.)f(.)h
 
9202
(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)79
 
9203
b Fu(63)150 353 y(9)135 b(Concept)45 b(Index)13 b Fa(.)20
 
9204
b(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h
 
9205
(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)58 b Fu(65)150
 
9206
623 y(10)135 b(Library)45 b(Index)35 b Fa(.)19 b(.)h(.)f(.)h(.)f(.)h(.)
 
9207
f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f
 
9208
(.)h(.)f(.)80 b Fu(67)p eop end
 
9209
%%Page: -4 74
 
9210
TeXDict begin -4 73 bop 150 -116 a Fv(iv)3248 b(FFTW)p
 
9211
eop end
9115
9212
%%Trailer
9116
9213
end
9117
9214
userdict /end-hook known{end-hook}if