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{ ******************************************************************
DiGamma and TriGamma functions.
Contributed by Philip Fletcher (FLETCHP@WESTAT.com)
****************************************************************** }
unit udigamma;
interface
uses
utypes;
function DiGamma(X : Float ) : Float;
function TriGamma(X : Float ) : Float;
implementation
function DiGamma(X : Float ) : Float;
{ ------------------------------------------------------------------
Digamma calculates the Digamma or Psi function =
d ( LOG ( GAMMA ( X ) ) ) / dX
Reference:
J Bernardo,
Psi ( Digamma ) Function,
Algorithm AS 103,
Applied Statistics,
Volume 25, Number 3, pages 315-317, 1976.
Modified:
03 January 2000
Parameters:
Input, real X, the argument of the Digamma function.
0 < X.
Output, real Digamma, the value of the Digamma function at X.
------------------------------------------------------------------ }
const
c = 20 ;
d1 = -0.57721566490153286061; { DiGamma(1) }
s = 0.00001 ;
{ Sterling coefficient S(n) = B(n) / 2n
where B(n) = Bernoulli number }
const
S2 = 0.08333333333333333333 ; { B(2)/2 }
S4 = -0.83333333333333333333E-2 ; { B(4)/4 }
S6 = 0.39682539682539682541E-2 ; { B(6)/6 }
S8 = -0.41666666666666666666E-2 ; { B(8)/8 }
S10 = 0.75757575757575757576E-2 ; { B(10)/10 }
S12 = -0.21092796092796092796E-1 ; { B(12)/12 }
S14 = 0.83333333333333333335E-1 ; { B(14)/14 }
S16 = -0.44325980392156862745 ; { B(16)/16 }
var
dg, p, r, y : Float ;
begin
if X <= 0.0 then
begin
DiGamma := DefaultVal(FSing, MaxNum);
Exit;
end;
SetErrCode(FOk);
if X = 1.0 then
begin
DiGamma := D1;
Exit;
end;
{ Use approximation if argument <= S }
if X <= s then
dg := d1 - 1.0 / x
else
{ Reduce the argument to dg(X + N) where (X + N) >= C }
begin
dg := 0.0;
y := x ;
while y < c do
begin
dg := dg - 1.0 / y;
y := y + 1.0;
end ;
{ Use Stirling's (actually de Moivre's) expansion if argument > C }
r := 1.0 / sqr ( y ) ;
p := (((((((S16 * r + S14) * r + S12) * r + S10) * r + S8) * r +
S6) * r + S4) * r + S2) * r ;
dg := dg + ln ( y ) - 0.5 / y - p ;
end ;
DiGamma := dg ;
end ;
function TriGamma(X : Float) : Float;
{ ------------------------------------------------------------------
Trigamma calculates the Trigamma or Psi Prime function =
d**2 ( LOG ( GAMMA ( X ) ) ) / dX**2
Reference:
Algorithm As121 Appl. Statist. (1978) vol 27, no. 1
******************************************************************** }
const
a = 1.0E-4 ;
b = 20 ;
zero = 0 ;
one = 1 ;
half = 0.5 ;
{ Bernoulli numbers }
const
B2 = 0.1666666666666667 ;
B4 = -3.333333333333333E-002 ;
B6 = 2.380952380952381E-002 ;
B8 = -3.333333333333333E-002 ;
B10 = 7.575757575757576E-002 ;
B12 = -0.2531135531135531 ;
var
y, z, Res : Float ;
begin
if X <= 0.0 then
begin
TriGamma := DefaultVal(FSing, MaxNum);
Exit;
end;
SetErrCode(FOk);
Res := 0 ;
z := x ;
if z <= a then { Use small value approximation }
begin
TriGamma := one / sqr ( z ) ;
Exit ;
end ;
while z < b do { Increase argument to (x+i) >= b }
begin
Res := Res + one / sqr ( z ) ;
z := z + one ;
end ;
{ Apply asymptotic formula where argument >= b }
y := one / sqr ( z ) ;
Res := Res + Half * y + (One + y * (B2 + y * (B4 + y * (B6 + y *
(B8 + y* (B10 + y * B12)))))) / z;
TriGamma := Res;
end ;
end.
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