1.1.6
by Kees Cook
Import upstream version 0.46~pre1 |
1 |
/** root finding for sbasis functions.
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2 |
* Copyright 2006 N Hurst
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3 |
* Copyright 2007 JF Barraud
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4 |
*
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5 |
* It is more efficient to find roots of f(t) = c_0, c_1, ... all at once, rather than iterating.
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6 |
*
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7 |
* Todo/think about:
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8 |
* multi-roots using bernstein method, one approach would be:
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9 |
sort c
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10 |
take median and find roots of that
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1.1.8
by Kees Cook
Import upstream version 0.47~pre0 |
11 |
whenever a segment lies entirely on one side of the median,
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1.1.6
by Kees Cook
Import upstream version 0.46~pre1 |
12 |
find the median of the half and recurse.
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13 |
||
14 |
in essence we are implementing quicksort on a continuous function
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15 |
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16 |
* the gsl poly roots finder is faster than bernstein too, but we don't use it for 3 reasons:
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17 |
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18 |
a) it requires convertion to poly, which is numerically unstable
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19 |
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20 |
b) it requires gsl (which is currently not a dependency, and would bring in a whole slew of unrelated stuff)
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21 |
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22 |
c) it finds all roots, even complex ones. We don't want to accidently treat a nearly real root as a real root
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23 |
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24 |
From memory gsl poly roots was about 10 times faster than bernstein in the case where all the roots
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25 |
are in [0,1] for polys of order 5. I spent some time working out whether eigenvalue root finding
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26 |
could be done directly in sbasis space, but the maths was too hard for me. -- njh
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27 |
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28 |
jfbarraud: eigenvalue root finding could be done directly in sbasis space ?
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29 |
||
30 |
njh: I don't know, I think it should. You would make a matrix whose characteristic polynomial was
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31 |
correct, but do it by putting the sbasis terms in the right spots in the matrix. normal eigenvalue
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32 |
root finding makes a matrix that is a diagonal + a row along the top. This matrix has the property
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33 |
that its characteristic poly is just the poly whose coefficients are along the top row.
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34 |
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35 |
Now an sbasis function is a linear combination of the poly coeffs. So it seems to me that you
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36 |
should be able to put the sbasis coeffs directly into a matrix in the right spots so that the
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37 |
characteristic poly is the sbasis. You'll still have problems b) and c).
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38 |
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39 |
We might be able to lift an eigenvalue solver and include that directly into 2geom. Eigenvalues
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40 |
also allow you to find intersections of multiple curves but require solving n*m x n*m matrices.
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41 |
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42 |
**/
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43 |
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44 |
#include <cmath> |
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45 |
#include <map> |
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46 |
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1.1.8
by Kees Cook
Import upstream version 0.47~pre0 |
47 |
#include <2geom/sbasis.h> |
48 |
#include <2geom/sbasis-to-bezier.h> |
|
49 |
#include <2geom/solver.h> |
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1.1.6
by Kees Cook
Import upstream version 0.46~pre1 |
50 |
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51 |
using namespace std; |
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52 |
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53 |
namespace Geom{ |
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54 |
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1.1.8
by Kees Cook
Import upstream version 0.47~pre0 |
55 |
/** Find the smallest interval that bounds a
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56 |
\param a sbasis function
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57 |
\returns inteval
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58 |
||
59 |
*/
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60 |
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61 |
#ifdef USE_SBASIS_OF
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62 |
OptInterval bounds_exact(SBasisOf<double> const &a) { |
|
63 |
Interval result = Interval(a.at0(), a.at1()); |
|
64 |
SBasisOf<double> df = derivative(a); |
|
65 |
vector<double>extrema = roots(df); |
|
66 |
for (unsigned i=0; i<extrema.size(); i++){ |
|
67 |
result.extendTo(a(extrema[i])); |
|
68 |
}
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69 |
return result; |
|
70 |
}
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71 |
#else
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72 |
OptInterval bounds_exact(SBasis const &a) { |
|
1.1.6
by Kees Cook
Import upstream version 0.46~pre1 |
73 |
Interval result = Interval(a.at0(), a.at1()); |
74 |
SBasis df = derivative(a); |
|
75 |
vector<double>extrema = roots(df); |
|
76 |
for (unsigned i=0; i<extrema.size(); i++){ |
|
77 |
result.extendTo(a(extrema[i])); |
|
78 |
}
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79 |
return result; |
|
80 |
}
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1.1.8
by Kees Cook
Import upstream version 0.47~pre0 |
81 |
#endif
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82 |
||
83 |
/** Find a small interval that bounds a
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84 |
\param a sbasis function
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85 |
\returns inteval
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86 |
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87 |
*/
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88 |
// I have no idea how this works, some clever bounding argument by jfb.
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89 |
#ifdef USE_SBASIS_OF
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90 |
OptInterval bounds_fast(const SBasisOf<double> &sb, int order) { |
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91 |
#else
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92 |
OptInterval bounds_fast(const SBasis &sb, int order) { |
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93 |
#endif
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94 |
Interval res(0,0); // an empty sbasis is 0. |
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95 |
||
1.1.6
by Kees Cook
Import upstream version 0.46~pre1 |
96 |
for(int j = sb.size()-1; j>=order; j--) { |
97 |
double a=sb[j][0]; |
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98 |
double b=sb[j][1]; |
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99 |
||
100 |
double v, t = 0; |
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101 |
v = res[0]; |
|
102 |
if (v<0) t = ((b-a)/v+1)*0.5; |
|
103 |
if (v>=0 || t<0 || t>1) { |
|
104 |
res[0] = std::min(a,b); |
|
105 |
}else{ |
|
106 |
res[0]=lerp(t, a+v*t, b); |
|
107 |
}
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108 |
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109 |
v = res[1]; |
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110 |
if (v>0) t = ((b-a)/v+1)*0.5; |
|
111 |
if (v<=0 || t<0 || t>1) { |
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112 |
res[1] = std::max(a,b); |
|
113 |
}else{ |
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114 |
res[1]=lerp(t, a+v*t, b); |
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115 |
}
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116 |
}
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117 |
if (order>0) res*=pow(.25,order); |
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118 |
return res; |
|
119 |
}
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120 |
||
1.1.8
by Kees Cook
Import upstream version 0.47~pre0 |
121 |
/** Find a small interval that bounds a(t) for t in i to order order
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122 |
\param sb sbasis function
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123 |
\param i domain interval
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124 |
\param order number of terms
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125 |
\return interval
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126 |
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127 |
*/
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128 |
#ifdef USE_SBASIS_OF
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129 |
OptInterval bounds_local(const SBasisOf<double> &sb, const OptInterval &i, int order) { |
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130 |
#else
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131 |
OptInterval bounds_local(const SBasis &sb, const OptInterval &i, int order) { |
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132 |
#endif
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133 |
double t0=i->min(), t1=i->max(), lo=0., hi=0.; |
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1.1.6
by Kees Cook
Import upstream version 0.46~pre1 |
134 |
for(int j = sb.size()-1; j>=order; j--) { |
135 |
double a=sb[j][0]; |
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136 |
double b=sb[j][1]; |
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137 |
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138 |
double t = 0; |
|
139 |
if (lo<0) t = ((b-a)/lo+1)*0.5; |
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140 |
if (lo>=0 || t<t0 || t>t1) { |
|
141 |
lo = std::min(a*(1-t0)+b*t0+lo*t0*(1-t0),a*(1-t1)+b*t1+lo*t1*(1-t1)); |
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142 |
}else{ |
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143 |
lo = lerp(t, a+lo*t, b); |
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144 |
}
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145 |
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146 |
if (hi>0) t = ((b-a)/hi+1)*0.5; |
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147 |
if (hi<=0 || t<t0 || t>t1) { |
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148 |
hi = std::max(a*(1-t0)+b*t0+hi*t0*(1-t0),a*(1-t1)+b*t1+hi*t1*(1-t1)); |
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149 |
}else{ |
|
150 |
hi = lerp(t, a+hi*t, b); |
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151 |
}
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152 |
}
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153 |
Interval res = Interval(lo,hi); |
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154 |
if (order>0) res*=pow(.25,order); |
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155 |
return res; |
|
156 |
}
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157 |
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158 |
//-- multi_roots ------------------------------------
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159 |
// goal: solve f(t)=c for several c at once.
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160 |
/* algo: -compute f at both ends of the given segment [a,b].
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161 |
-compute bounds m<df(t)<M for df on the segment.
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162 |
let c and C be the levels below and above f(a):
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163 |
going from f(a) down to c with slope m takes at least time (f(a)-c)/m
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164 |
going from f(a) up to C with slope M takes at least time (C-f(a))/M
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165 |
From this we conclude there are no roots before a'=a+min((f(a)-c)/m,(C-f(a))/M).
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166 |
Do the same for b: compute some b' such that there are no roots in (b',b].
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1.1.8
by Kees Cook
Import upstream version 0.47~pre0 |
167 |
-if [a',b'] is not empty, repeat the process with [a',(a'+b')/2] and [(a'+b')/2,b'].
|
1.1.6
by Kees Cook
Import upstream version 0.46~pre1 |
168 |
unfortunately, extra care is needed about rounding errors, and also to avoid the repetition of roots,
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169 |
making things tricky and unpleasant...
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170 |
*/
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171 |
//TODO: Make sure the code is "rounding-errors proof" and take care about repetition of roots!
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172 |
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173 |
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174 |
static int upper_level(vector<double> const &levels,double x,double tol=0.){ |
|
175 |
return(upper_bound(levels.begin(),levels.end(),x-tol)-levels.begin()); |
|
176 |
}
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177 |
||
1.1.8
by Kees Cook
Import upstream version 0.47~pre0 |
178 |
#ifdef USE_SBASIS_OF
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179 |
static void multi_roots_internal(SBasis const &f, |
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180 |
SBasis const &df, |
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181 |
#else
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182 |
static void multi_roots_internal(SBasis const &f, |
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183 |
SBasis const &df, |
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184 |
#endif
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1.1.6
by Kees Cook
Import upstream version 0.46~pre1 |
185 |
std::vector<double> const &levels, |
186 |
std::vector<std::vector<double> > &roots, |
|
187 |
double htol, |
|
188 |
double vtol, |
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189 |
double a, |
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190 |
double fa, |
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191 |
double b, |
|
192 |
double fb){ |
|
1.1.8
by Kees Cook
Import upstream version 0.47~pre0 |
193 |
|
1.1.6
by Kees Cook
Import upstream version 0.46~pre1 |
194 |
if (f.size()==0){ |
195 |
int idx; |
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196 |
idx=upper_level(levels,0,vtol); |
|
197 |
if (idx<(int)levels.size()&&fabs(levels.at(idx))<=vtol){ |
|
198 |
roots[idx].push_back(a); |
|
199 |
roots[idx].push_back(b); |
|
200 |
}
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201 |
return; |
|
202 |
}
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1.1.8
by Kees Cook
Import upstream version 0.47~pre0 |
203 |
////usefull?
|
1.1.6
by Kees Cook
Import upstream version 0.46~pre1 |
204 |
// if (f.size()==1){
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205 |
// int idxa=upper_level(levels,fa);
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206 |
// int idxb=upper_level(levels,fb);
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207 |
// if (fa==fb){
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208 |
// if (fa==levels[idxa]){
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209 |
// roots[a]=idxa;
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210 |
// roots[b]=idxa;
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211 |
// }
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212 |
// return;
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213 |
// }
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214 |
// int idx_min=std::min(idxa,idxb);
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215 |
// int idx_max=std::max(idxa,idxb);
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216 |
// if (idx_max==levels.size()) idx_max-=1;
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217 |
// for(int i=idx_min;i<=idx_max; i++){
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218 |
// double t=a+(b-a)*(levels[i]-fa)/(fb-fa);
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219 |
// if(a<t&&t<b) roots[t]=i;
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220 |
// }
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221 |
// return;
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|
222 |
// }
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223 |
if ((b-a)<htol){ |
|
224 |
//TODO: use different tol for t and f ?
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225 |
//TODO: unsigned idx ? (remove int casts when fixed)
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|
226 |
int idx=std::min(upper_level(levels,fa,vtol),upper_level(levels,fb,vtol)); |
|
227 |
if (idx==(int)levels.size()) idx-=1; |
|
228 |
double c=levels.at(idx); |
|
229 |
if((fa-c)*(fb-c)<=0||fabs(fa-c)<vtol||fabs(fb-c)<vtol){ |
|
230 |
roots[idx].push_back((a+b)/2); |
|
231 |
}
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|
232 |
return; |
|
233 |
}
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|
1.1.8
by Kees Cook
Import upstream version 0.47~pre0 |
234 |
|
1.1.6
by Kees Cook
Import upstream version 0.46~pre1 |
235 |
int idxa=upper_level(levels,fa,vtol); |
236 |
int idxb=upper_level(levels,fb,vtol); |
|
237 |
||
1.1.8
by Kees Cook
Import upstream version 0.47~pre0 |
238 |
Interval bs = *bounds_local(df,Interval(a,b)); |
1.1.6
by Kees Cook
Import upstream version 0.46~pre1 |
239 |
|
240 |
//first times when a level (higher or lower) can be reached from a or b.
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241 |
double ta_hi,tb_hi,ta_lo,tb_lo; |
|
242 |
ta_hi=ta_lo=b+1;//default values => no root there. |
|
243 |
tb_hi=tb_lo=a-1;//default values => no root there. |
|
244 |
||
245 |
if (idxa<(int)levels.size() && fabs(fa-levels.at(idxa))<vtol){//a can be considered a root. |
|
246 |
//ta_hi=ta_lo=a;
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|
247 |
roots[idxa].push_back(a); |
|
248 |
ta_hi=ta_lo=a+htol; |
|
249 |
}else{ |
|
250 |
if (bs.max()>0 && idxa<(int)levels.size()) |
|
251 |
ta_hi=a+(levels.at(idxa )-fa)/bs.max(); |
|
252 |
if (bs.min()<0 && idxa>0) |
|
253 |
ta_lo=a+(levels.at(idxa-1)-fa)/bs.min(); |
|
254 |
}
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|
255 |
if (idxb<(int)levels.size() && fabs(fb-levels.at(idxb))<vtol){//b can be considered a root. |
|
256 |
//tb_hi=tb_lo=b;
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|
257 |
roots[idxb].push_back(b); |
|
258 |
tb_hi=tb_lo=b-htol; |
|
259 |
}else{ |
|
260 |
if (bs.min()<0 && idxb<(int)levels.size()) |
|
261 |
tb_hi=b+(levels.at(idxb )-fb)/bs.min(); |
|
262 |
if (bs.max()>0 && idxb>0) |
|
263 |
tb_lo=b+(levels.at(idxb-1)-fb)/bs.max(); |
|
264 |
}
|
|
1.1.8
by Kees Cook
Import upstream version 0.47~pre0 |
265 |
|
1.1.6
by Kees Cook
Import upstream version 0.46~pre1 |
266 |
double t0,t1; |
1.1.8
by Kees Cook
Import upstream version 0.47~pre0 |
267 |
t0=std::min(ta_hi,ta_lo); |
1.1.6
by Kees Cook
Import upstream version 0.46~pre1 |
268 |
t1=std::max(tb_hi,tb_lo); |
269 |
//hum, rounding errors frighten me! so I add this +tol...
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|
270 |
if (t0>t1+htol) return;//no root here. |
|
271 |
||
272 |
if (fabs(t1-t0)<htol){ |
|
273 |
multi_roots_internal(f,df,levels,roots,htol,vtol,t0,f(t0),t1,f(t1)); |
|
274 |
}else{ |
|
275 |
double t,t_left,t_right,ft,ft_left,ft_right; |
|
276 |
t_left =t_right =t =(t0+t1)/2; |
|
277 |
ft_left=ft_right=ft=f(t); |
|
278 |
int idx=upper_level(levels,ft,vtol); |
|
279 |
if (idx<(int)levels.size() && fabs(ft-levels.at(idx))<vtol){//t can be considered a root. |
|
280 |
roots[idx].push_back(t); |
|
281 |
//we do not want to count it twice (from the left and from the right)
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|
282 |
t_left =t-htol/2; |
|
283 |
t_right=t+htol/2; |
|
284 |
ft_left =f(t_left); |
|
285 |
ft_right=f(t_right); |
|
286 |
}
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|
287 |
multi_roots_internal(f,df,levels,roots,htol,vtol,t0 ,f(t0) ,t_left,ft_left); |
|
288 |
multi_roots_internal(f,df,levels,roots,htol,vtol,t_right,ft_right,t1 ,f(t1) ); |
|
289 |
}
|
|
290 |
}
|
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291 |
||
1.1.8
by Kees Cook
Import upstream version 0.47~pre0 |
292 |
/** Solve f(t)=c for several c at once.
|
293 |
\param f sbasis function
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|
294 |
\param levels vector of 'y' values
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|
295 |
\param htol, vtol
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|
296 |
\param a, b left and right bounds
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|
297 |
\returns a vector of vectors, one for each y giving roots
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|
298 |
||
299 |
Effectively computes:
|
|
300 |
results = roots(f(y_i)) for all y_i
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|
301 |
||
302 |
* algo: -compute f at both ends of the given segment [a,b].
|
|
303 |
-compute bounds m<df(t)<M for df on the segment.
|
|
304 |
let c and C be the levels below and above f(a):
|
|
305 |
going from f(a) down to c with slope m takes at least time (f(a)-c)/m
|
|
306 |
going from f(a) up to C with slope M takes at least time (C-f(a))/M
|
|
307 |
From this we conclude there are no roots before a'=a+min((f(a)-c)/m,(C-f(a))/M).
|
|
308 |
Do the same for b: compute some b' such that there are no roots in (b',b].
|
|
309 |
-if [a',b'] is not empty, repeat the process with [a',(a'+b')/2] and [(a'+b')/2,b'].
|
|
310 |
unfortunately, extra care is needed about rounding errors, and also to avoid the repetition of roots,
|
|
311 |
making things tricky and unpleasant...
|
|
312 |
||
313 |
TODO: Make sure the code is "rounding-errors proof" and take care about repetition of roots!
|
|
314 |
*/
|
|
1.1.6
by Kees Cook
Import upstream version 0.46~pre1 |
315 |
std::vector<std::vector<double> > multi_roots(SBasis const &f, |
316 |
std::vector<double> const &levels, |
|
317 |
double htol, |
|
318 |
double vtol, |
|
319 |
double a, |
|
320 |
double b){ |
|
321 |
||
322 |
std::vector<std::vector<double> > roots(levels.size(), std::vector<double>()); |
|
323 |
||
324 |
SBasis df=derivative(f); |
|
1.1.8
by Kees Cook
Import upstream version 0.47~pre0 |
325 |
multi_roots_internal(f,df,levels,roots,htol,vtol,a,f(a),b,f(b)); |
1.1.6
by Kees Cook
Import upstream version 0.46~pre1 |
326 |
|
327 |
return(roots); |
|
328 |
}
|
|
329 |
//-------------------------------------
|
|
330 |
||
331 |
||
332 |
void subdiv_sbasis(SBasis const & s, |
|
1.1.8
by Kees Cook
Import upstream version 0.47~pre0 |
333 |
std::vector<double> & roots, |
1.1.6
by Kees Cook
Import upstream version 0.46~pre1 |
334 |
double left, double right) { |
1.1.8
by Kees Cook
Import upstream version 0.47~pre0 |
335 |
OptInterval bs = bounds_fast(s); |
336 |
if(!bs || bs->min() > 0 || bs->max() < 0) |
|
1.1.6
by Kees Cook
Import upstream version 0.46~pre1 |
337 |
return; // no roots here |
338 |
if(s.tailError(1) < 1e-7) { |
|
339 |
double t = s[0][0] / (s[0][0] - s[0][1]); |
|
340 |
roots.push_back(left*(1-t) + t*right); |
|
341 |
return; |
|
342 |
}
|
|
343 |
double middle = (left + right)/2; |
|
344 |
subdiv_sbasis(compose(s, Linear(0, 0.5)), roots, left, middle); |
|
345 |
subdiv_sbasis(compose(s, Linear(0.5, 1.)), roots, middle, right); |
|
346 |
}
|
|
347 |
||
348 |
// It is faster to use the bernstein root finder for small degree polynomials (<100?.
|
|
349 |
||
1.1.8
by Kees Cook
Import upstream version 0.47~pre0 |
350 |
std::vector<double> roots1(SBasis const & s) { |
351 |
std::vector<double> res; |
|
352 |
double d = s[0][0] - s[0][1]; |
|
353 |
if(d != 0) { |
|
354 |
double r = s[0][0] / d; |
|
355 |
if(0 <= r && r <= 1) |
|
356 |
res.push_back(r); |
|
357 |
}
|
|
358 |
return res; |
|
359 |
}
|
|
360 |
||
361 |
/** Find all t s.t s(t) = 0
|
|
362 |
\param a sbasis function
|
|
363 |
\returns vector of zeros (roots)
|
|
364 |
||
365 |
*/
|
|
1.1.6
by Kees Cook
Import upstream version 0.46~pre1 |
366 |
std::vector<double> roots(SBasis const & s) { |
1.1.8
by Kees Cook
Import upstream version 0.47~pre0 |
367 |
switch(s.size()) { |
368 |
case 0: |
|
369 |
return std::vector<double>(); |
|
370 |
case 1: |
|
371 |
return roots1(s); |
|
372 |
default: |
|
373 |
{
|
|
374 |
Bezier bz; |
|
375 |
sbasis_to_bezier(bz, s); |
|
376 |
return bz.roots(); |
|
377 |
}
|
|
378 |
}
|
|
1.1.6
by Kees Cook
Import upstream version 0.46~pre1 |
379 |
}
|
380 |
||
381 |
};
|
|
382 |
||
383 |
/*
|
|
384 |
Local Variables:
|
|
385 |
mode:c++
|
|
386 |
c-file-style:"stroustrup"
|
|
387 |
c-file-offsets:((innamespace . 0)(inline-open . 0)(case-label . +))
|
|
388 |
indent-tabs-mode:nil
|
|
389 |
fill-column:99
|
|
390 |
End:
|
|
391 |
*/
|
|
392 |
// vim: filetype=cpp:expandtab:shiftwidth=4:tabstop=8:softtabstop=4:encoding=utf-8:textwidth=99 :
|