| 1 | #include "sopnamsp.h"
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| 2 | #include "simplex.h"
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| 3 | #include "ntuple.h"
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| 4 | #include <math.h>
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| 5 |
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| 6 | #include "timing.h"
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| 7 |
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| 8 | //---------------------------------------------------------------
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| 9 | //------------------- Classe MinZFunction -------------------
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| 10 | //---------------------------------------------------------------
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| 11 | // Interface de classe de function multivariable pour le SimplexMinmizer
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| 12 | /*!
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| 13 | \class SOPHYA::MinZFunction
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| 14 | \ingroup NTools
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| 15 | Interface definition for a function object f(x[]) for which MinZSimplex can
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| 16 | search the minimum.
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| 17 | The pure virtual method Value() should be implemented by the derived classes.
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| 18 | */
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| 19 |
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| 20 | MinZFunction::MinZFunction(unsigned int nvar)
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| 21 | : mNVar(nvar)
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| 22 | {
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| 23 | }
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| 24 |
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| 25 | MinZFunction::~MinZFunction()
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| 26 | {
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| 27 | }
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| 28 |
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| 29 | //---------------------------------------------------------------
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| 30 | //------------------- Classe MinZFuncXi2 --------------------
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| 31 | //---------------------------------------------------------------
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| 32 | /*!
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| 33 | \class SOPHYA::MinZXi2
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| 34 | \ingroup NTools
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| 35 | Implements the MinZFunction interface using a xi2 calculator
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| 36 | \sa GeneralXi2 GeneralFitData
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| 37 | */
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| 38 | MinZFuncXi2::MinZFuncXi2(GeneralXi2* gxi2, GeneralFitData* gd)
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| 39 | : mGXi2(gxi2) , mGData(gd), MinZFunction(gxi2->NPar())
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| 40 | {
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| 41 | }
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| 42 |
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| 43 | MinZFuncXi2::~MinZFuncXi2()
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| 44 | {
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| 45 | }
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| 46 |
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| 47 | double MinZFuncXi2::Value(double const xp[])
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| 48 | {
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| 49 | int ndataused;
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| 50 | return mGXi2->Value(*mGData, const_cast<double *>(xp), ndataused);
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| 51 | }
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| 52 |
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| 53 | //---------------------------------------------------------------
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| 54 | //------------------- Classe MinZTestFunc -------------------
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| 55 | //---------------------------------------------------------------
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| 56 | class MinZTestFunc : public MinZFunction {
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| 57 | public:
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| 58 | MinZTestFunc(int sel);
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| 59 | virtual double Value(double const xp[]);
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| 60 | string ToString();
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| 61 | Vector OptParms();
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| 62 | protected:
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| 63 | static int ISelToNvar(int isel);
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| 64 | int mSel;
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| 65 | };
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| 66 |
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| 67 | int MinZTestFunc::ISelToNvar(int isel)
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| 68 | {
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| 69 | if (isel == 0) return 1;
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| 70 | if (isel == 1) return 1;
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| 71 | else if (isel == 2) return 1;
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| 72 | else if (isel == 3) return 2;
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| 73 | else if (isel == 4) return 3;
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| 74 | else return 1;
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| 75 | }
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| 76 |
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| 77 | MinZTestFunc::MinZTestFunc(int sel)
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| 78 | : MinZFunction(ISelToNvar(sel))
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| 79 | {
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| 80 | if ((sel < 0) || (sel > 4)) sel = 0;
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| 81 | mSel = sel;
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| 82 | }
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| 83 |
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| 84 | string MinZTestFunc::ToString()
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| 85 | {
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| 86 | string rs;
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| 87 | if (mSel == 0) {
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| 88 | rs = "-x+(x-2)^2";
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| 89 | }
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| 90 | else if (mSel == 1) {
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| 91 | rs = "0.1*x^2-3exp(-(x-2)^2)-5*exp(-0.5*(x+3)^2)";
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| 92 | }
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| 93 | else if (mSel == 2) {
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| 94 | rs = "0.1*x^2-3exp(-(x-2)^2)+5*exp(-0.5*(x+3)^2)";
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| 95 | }
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| 96 | else if (mSel == 3) {
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| 97 | rs = "1.3*(x-50.35)^2+25*(y+3.14)^2";
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| 98 | }
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| 99 | else if (mSel == 4) {
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| 100 | rs = "(x-2.2)^2+2.*(y+3.6)^2+3.*(z-1.1)^2";
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| 101 | }
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| 102 | else rs = "????";
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| 103 | return rs;
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| 104 | }
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| 105 |
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| 106 | Vector MinZTestFunc::OptParms()
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| 107 | {
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| 108 | Vector xx;
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| 109 | if (mSel == 0) {
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| 110 | Vector rv(1);
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| 111 | rv = 2.5;
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| 112 | return rv;
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| 113 | }
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| 114 | else if (mSel == 1) {
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| 115 | Vector rv(1);
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| 116 | rv = -2.883;
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| 117 | return rv;
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| 118 | }
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| 119 | else if (mSel == 2) {
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| 120 | Vector rv(1);
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| 121 | rv = 1.812;
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| 122 | return rv;
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| 123 | }
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| 124 | else if (mSel == 3) {
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| 125 | Vector rv(2);
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| 126 | rv(0) = 50.35;
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| 127 | rv(1) = -3.14;
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| 128 | return rv;
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| 129 | }
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| 130 | else if (mSel == 4) {
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| 131 | Vector rv(3);
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| 132 | rv(0) = 2.2;
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| 133 | rv(1) = -3.6;
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| 134 | rv(2) = 1.1;
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| 135 | return rv;
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| 136 | }
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| 137 | else xx = 0.;
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| 138 | return xx ;
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| 139 | }
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| 140 |
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| 141 |
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| 142 | double MinZTestFunc::Value(double const xp[])
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| 143 | {
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| 144 | double retval = 0;
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| 145 | if (mSel == 0) {
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| 146 | double x = xp[0];
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| 147 | retval = -x+(x-2.)*(x-2.);
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| 148 | }
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| 149 | else if ((mSel == 1) || (mSel == 2)) {
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| 150 | double x = xp[0];
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| 151 | retval = 0.1*x*x;
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| 152 | x = xp[0]-2.;
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| 153 | x = x*x;
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| 154 | retval -= 3*exp(-x);
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| 155 | x = xp[0]+3.;
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| 156 | x = 0.5*x*x;
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| 157 | if (mSel == 1) retval -= 5*exp(-x);
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| 158 | else retval += 5*exp(-x);
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| 159 | }
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| 160 | else if (mSel == 3) {
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| 161 | double x = xp[0]-50.35;
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| 162 | double y = xp[1]+3.14;
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| 163 | retval = 1.3*x*x+25.*y*y;
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| 164 | }
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| 165 | else if (mSel == 4) {
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| 166 | double x = xp[0]-2.2;
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| 167 | double y = xp[1]+3.6;
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| 168 | double z = xp[2]-1.1;
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| 169 | retval = x*x+2.*y*y+3.*z*z;
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| 170 | }
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| 171 | else retval = 0.;
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| 172 | return retval;
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| 173 | }
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| 174 |
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| 175 | //---------------------------------------------------------------
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| 176 | //------------------- Classe MinZSimplex --------------------
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| 177 | //---------------------------------------------------------------
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| 178 | string __Vec2Str4MinZ_AutoTest(Vector& xx)
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| 179 | {
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| 180 | string rs;
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| 181 | char buff[32];
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| 182 | for(int i=0; i<xx.Size(); i++) {
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| 183 | sprintf(buff," %g " , xx(i));
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| 184 | rs += buff;
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| 185 | }
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| 186 | return rs;
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| 187 | }
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| 188 |
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| 189 | /*!
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| 190 | \class SOPHYA::MinZSimplex
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| 191 | \ingroup NTools
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| 192 | This class implements non linear minimization (optimization)
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| 193 | in a multidimensional space following the \b Simplex method.
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| 194 | A \b Simplex is a geometrical figure made of N+1 points in a
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| 195 | N-dimensional space. (triangle in a plane, tetrahedron in 3-d space).
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| 196 | The minimization method implemented in this class is based on the
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| 197 | algorithm described in "Numerical Recipes, Chapter X".
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| 198 |
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| 199 | The algorithm has been slightly enhanced :
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| 200 | - More complex convergence / stop test
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| 201 | - A new transformation of the simplex has been included (ExpandHigh)
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| 202 |
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| 203 | For each step, on of the following geometrical transform is performed
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| 204 | on the Simplex figure:
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| 205 | - Reflection : reflection away from the high point (expansion by factor Alpha)
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| 206 | - ReflecExpand : reflection way from the high point and expansion by factor Beta2
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| 207 | - ContractHigh : Contraction along the high point (factor Beta)
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| 208 | - ContractLow : Contraction toward the low point (factor Beta2)
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| 209 | - ExpandHigh : Expansion along the high point
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| 210 |
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| 211 | \sa GeneralFit
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| 212 |
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| 213 | The following sample code shows a usage example:
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| 214 | \code
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| 215 | include "simplex.h"
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| 216 | ...
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| 217 | // Define our function to be minimized:
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| 218 | class MySFunc : public MinZFunction {
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| 219 | public:
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| 220 | MySFunc() : MinZFunction(2) {}
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| 221 | virtual double Value(double const xp[])
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| 222 | { return (xp[0]*xp[0]+2*xp[1]*xp[1]); }
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| 223 | };
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| 224 |
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| 225 | ...
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| 226 |
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| 227 | MySFunc mysf;
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| 228 | MinZSimplex simplex(&mysf);
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| 229 | // Guess the center and step for constructing the initial simplex
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| 230 | Vector x0(2); x0 = 1.;
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| 231 | Vector step(2); step = 2.;
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| 232 | simplex.SetInitialPoint(x0);
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| 233 | simplex.SetInitialStep(step);
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| 234 | Vector oparm(2);
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| 235 | int rc = simplex.Minimize(oparm);
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| 236 | if (rc != 0) {
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| 237 | string srt;
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| 238 | int sr = simplex.StopReason(srt);
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| 239 | cout << " Convergence Pb, StopReason= " << sr << " : " << srt << endl;
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| 240 | }
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| 241 | else {
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| 242 | cout << " Converged: NStep= " << simplex.NbIter()
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| 243 | << " OParm= " << oparm << endl;
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| 244 | }
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| 245 | \endcode
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| 246 | */
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| 247 |
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| 248 | /*!
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| 249 | \brief Auto test function
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| 250 | \param tsel : select autotest (0,1,2,3,4) , tsel<0 -> all
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| 251 | \param prtlev : printlevel
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| 252 | */
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| 253 | int MinZSimplex::AutoTest(int tsel, int prtlev)
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| 254 | {
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| 255 | int rc = 0;
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| 256 | cout << " --- MinZSimplex::AutoTest() --- TSel= " << tsel << " PrtLev=" << prtlev << endl;
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| 257 | for(int i=0; i<5; i++) {
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| 258 | if ((tsel >= 0) && (tsel != i)) continue;
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| 259 | cout << " ======= Test avec ISel= " << i;
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| 260 | Vector xx;
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| 261 | MinZTestFunc mzf(i);
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| 262 | cout << " - Func= " << mzf.ToString() << endl;
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| 263 | Vector rv = mzf.OptParms();
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| 264 | xx = rv;
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| 265 | for(int j=0; j<2; j++) {
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| 266 | double vi = 50.*(j-0.5);
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| 267 | for(int k=0; k<2; k++) {
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| 268 | double vs = (k == 0) ? 1. : 10. ;
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| 269 | cout << "--[" << j << "," << k
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| 270 | << "] Initialisation avec IniPoint= " << vi << " IniStep= " << vs << endl;
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| 271 | MinZSimplex simplex(&mzf);
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| 272 | xx = vi;
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| 273 | simplex.SetInitialPoint(xx);
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| 274 | xx = vs;
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| 275 | simplex.SetInitialStep(xx);
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| 276 | simplex.SetPrtLevel(prtlev);
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| 277 | int rcs = simplex.Minimize(xx);
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| 278 | Vector diff = rv-xx;
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| 279 | double d2 = diff.Norm2();
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| 280 | cout << " Rc(simplex.Minimize() = " << rc << " NIter= "
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| 281 | << simplex.NbIter() << " ===> Distance^2= " << d2
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| 282 | << "\nConverged to " << __Vec2Str4MinZ_AutoTest(xx)
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| 283 | << " Best Value= " << __Vec2Str4MinZ_AutoTest(rv)
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| 284 | << " Diff = " << __Vec2Str4MinZ_AutoTest(diff) << endl;
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| 285 | if ((rcs > 5) || (d2 > 0.5)) rc ++;
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| 286 | }
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| 287 | }
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| 288 | }
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| 289 | cout << " --- MinZSimplex::AutoTest() --- Rc=" << rc << " -- END ----- " << endl;
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| 290 | return rc;
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| 291 | }
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| 292 |
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| 293 | //! Constructor from pointer to MinZFunction object
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| 294 | MinZSimplex::MinZSimplex(MinZFunction *mzf)
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| 295 | : mZF(mzf) , mPoint0(mZF->NVar()) , mStep0(mZF->NVar())
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| 296 | {
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| 297 | SetMaxIter();
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| 298 | SetControls();
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| 299 | Vector xx(NDim());
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| 300 | xx = 0.;
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| 301 | SetInitialPoint(xx);
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| 302 | xx = 1.0;
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| 303 | SetInitialStep(xx);
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| 304 | SetStopTolerance();
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| 305 | mIter = -1;
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| 306 | mStop = -1;
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| 307 | SetPrtLevel();
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| 308 | }
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| 309 |
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| 310 | MinZSimplex::~MinZSimplex()
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| 311 | {
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| 312 | }
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| 313 |
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| 314 | //! Perform the minimization
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| 315 | /*!
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| 316 | Return 0 if success
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| 317 | \param fpoint : vector containing the optimal point
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| 318 |
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| 319 | Convergence test :
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| 320 | \verbatim
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| 321 | On minimise f(x) f=mZF->Value() ,
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| 322 | f_max = max(f) sur simplex , f_min = min(f) sur simplex
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| 323 | fm = (abs(f_max)+abs(f_min))
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| 324 | [Delta f] = abs(f_max-f_min)
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| 325 | [Delta f/f]simplex = 2.*Delta f / fm
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| 326 | fm2 = (abs(f_max)+abs(f_max(iter-1)))
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| 327 | [Delta f_max/f_max]iter = [f_max(iter-1)-f_max]/fm2
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| 328 | Test d'arret :
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| 329 | fm < mTol0 OU
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| 330 | [Delta f/f]simplex < mTol1 mRep1 fois de suite OU
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| 331 | [Delta f_max/f_max]iter < mTol2 mRep2 fois de suite
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| 332 | */
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| 333 | int MinZSimplex::Minimize(Vector& fpoint)
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| 334 | {
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| 335 | // vector< TVector<r_8> > splx;
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| 336 | Vector splx[100];
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| 337 | Vector Y(NDim()+1);
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| 338 | // On calcule le simplex initial
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| 339 | // N = NDim, N+1 points (pp) ds l'espace a N dimensions
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| 340 | // Point0, Point0 + Step0(i) e_i
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| 341 | Vector pp,ppc;
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| 342 | pp = mPoint0;
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| 343 | //ppc = pp;
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| 344 | //splx.push_back(ppc);
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| 345 | splx[0] = pp;
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| 346 | int i,j,k;
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| 347 | for(i=0; i<NDim(); i++) {
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| 348 | Vector pps;
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| 349 | pps = mPoint0;
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| 350 | pps(i) += mStep0(i);
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| 351 | //splx.push_back(pps);
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| 352 | splx[i+1] = pps;
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| 353 | }
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| 354 | int mpts = NDim()+1;
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| 355 | // calcul des valeurs de la fonction sur les sommets
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| 356 | for(i=0; i<mpts; i++)
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| 357 | Y(i) = Value(splx[i]);
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| 358 |
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| 359 | int iter = 0;
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| 360 | mIter = iter;
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| 361 | mStop = 0;
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| 362 |
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| 363 | int nbugrtol2 = 0;
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| 364 | bool stop = false, stop0=false;
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| 365 | int rc = 0;
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| 366 | int ilo, ihi, inhi;
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| 367 | int move = 0;
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| 368 | const char* smov[6] = { "None", "Reflection", "ReflecExpand", "ContractHigh", "ContractLow", "ExpandHigh" };
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| 369 | int movcnt[6] = {0,0,0,0,0,0};
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| 370 |
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| 371 | int nrep1=0, nrep2=0;
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| 372 | FindMinMax12(Y, ilo, ihi, inhi);
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| 373 | double yhilast = Y(ihi);
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| 374 | yhilast += fabs(yhilast);
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| 375 |
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| 376 | while (!stop) { //
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| 377 | FindMinMax12(Y, ilo, ihi, inhi);
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| 378 | double ymean = (fabs(Y(ihi))+fabs(Y(ilo)));
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| 379 | if (ymean < mTol0) { stop0 = true; ymean = mTol0; }
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| 380 | double rtol1 = 2.*fabs(Y(ihi)-Y(ilo))/ymean;
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| 381 | double ym2 = (fabs(yhilast)+fabs(Y(ihi)));
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| 382 | if (ym2 < mTol0) ym2 = mTol0;
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| 383 | double rtol2 = 2.*(yhilast-Y(ihi))/ym2;
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| 384 | yhilast = Y(ihi);
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| 385 | if (rtol2 < 0.) {
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| 386 | if (move != 40) {
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| 387 | cout << " !!!! MinZSimplex::Minimize() BUG RTol2< 0. --> Chs " << endl;
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| 388 | nbugrtol2++;
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| 389 | }
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| 390 | else nrep2 = 0;
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| 391 | rtol2 = -rtol2;
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| 392 | }
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| 393 | if (PrtLevel() > 1)
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| 394 | cout << "--MinZSimplex::Minimize() - Iter=" << iter
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| 395 | << " Move= " << move << " (" << smov[move/10] << ")" << endl;
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| 396 | if (PrtLevel() > 2)
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| 397 | cout << "..ILO=" << ilo << " IHI=" << ihi << " INHI=" << inhi
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| 398 | << " Y(ILO)=" << Y(ilo) << " Y(IHI)=" << Y(ihi) << "\n"
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| 399 | << "...YMean_Abs=" << ymean << " RTOL1=" << rtol1 << " RTOL2=" << rtol2 << endl;
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| 400 | if (PrtLevel() > 3) {
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| 401 | for(i=0; i<mpts; i++) {
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| 402 | cout << "....Simplex[" << i << "]= ";
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| 403 | for(j=0; j<NDim(); j++) cout << splx[i](j) << " , ";
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| 404 | cout << " Y=Value= " << Y(i) << endl;
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| 405 | }
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| 406 | }
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| 407 | if (rtol1 < mTol1) nrep1++;
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| 408 | else nrep1 = 0;
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| 409 | if (rtol2 < mTol2) nrep2++;
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| 410 | else nrep2 = 0;
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| 411 |
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| 412 | if (stop0) { mStop = 1; rc = 0; stop = true; break; }
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| 413 | if (nrep1 > mRep1) { mStop = 2; rc = 0; stop = true; break; }
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| 414 | if (nrep2 > mRep2) { mStop = 3; rc = 0; stop = true; break; }
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| 415 | if (iter > MaxIter() ) { mStop = 0, rc = iter; break; }
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| 416 | iter++;
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| 417 | if (iter > 0) movcnt[move/10]++;
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| 418 |
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| 419 | // Next iteration, on modifie le simplex
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| 420 | // Calcul du centre de gravite su simplex, hors le point le + haut
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| 421 | Vector pbar(NDim());
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| 422 | pbar = 0.;
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| 423 | for(i=0; i<mpts; i++) {
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| 424 | if (i == ihi) continue;
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| 425 | pbar += splx[i];
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| 426 | }
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| 427 | pbar /= (double)NDim();
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| 428 | // On calcule le sommet oppose a point IHI (le + haut)
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| 429 | Vector pr, prr;
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| 430 | double YPR, YPRR;
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| 431 | pr = (1.+Alpha())*pbar-Alpha()*splx[ihi];
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| 432 | YPR = Value(pr);
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| 433 | if (YPR < Y(ilo)) { // Amelioaration par rapport au meilleur point,
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| 434 | // on va plus loin d'un facteur gamma
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| 435 | prr = Gamma()*pr+(1.-Gamma())*pbar;
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| 436 | YPRR = Value(prr);
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| 437 | if (YPRR < Y(ilo)) { // On remplace le IHI par YPRR
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| 438 | splx[ihi] = prr;
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| 439 | Y(ihi) = YPRR;
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| 440 | move = 20;
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| 441 | }
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| 442 | else { // sinon, on remplace par YPR
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| 443 | splx[ihi] = pr;
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| 444 | Y(ihi) = YPR;
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| 445 | move = 10;
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| 446 | }
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| 447 | }
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| 448 | else { // Moins bon que le meilleur point ..
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| 449 | if (YPR > Y(inhi)) { // Plus mauvais que le second plus haut (INHI)
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| 450 | if (YPR < Y(ihi)) { // Mais meilleur que le plus haut (IHI)
|
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| 451 | splx[ihi] = pr; // On remplace donc le plus haut
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| 452 | Y(ihi) = YPR;
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| 453 | move = 11;
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| 454 | }
|
|---|
| 455 | else { // Plus mauvais que le plus mauvais IHI
|
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| 456 | // on tente avec un point intermediaire
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|---|
| 457 | prr = Beta()*splx[ihi]+(1.-Beta())*pbar;
|
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| 458 | YPRR = Value(prr);
|
|---|
| 459 | if (YPRR < Y(ihi)) { // Le point intermediaire ameliore les choses
|
|---|
| 460 | splx[ihi] = prr; // On remplace donc le point le + haut
|
|---|
| 461 | Y(ihi) = YPRR;
|
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| 462 | move = 30;
|
|---|
| 463 | }
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| 464 | else {
|
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| 465 | // On tente aussi de rester du meme cote, mais aller plus loin
|
|---|
| 466 | prr = Gamma2()*splx[ihi]+(1.-Gamma2())*pbar;
|
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| 467 | YPRR = Value(prr);
|
|---|
| 468 | if (YPRR < Y(ihi)) { // Le point intermediaire ameliore les choses
|
|---|
| 469 | splx[ihi] = prr; // On remplace donc le point le + haut
|
|---|
| 470 | Y(ihi) = YPRR;
|
|---|
| 471 | move = 50;
|
|---|
| 472 | }
|
|---|
| 473 | else {
|
|---|
| 474 | // Rien n'y fait, on contracte autour du meilleur point
|
|---|
| 475 | for(i=0; i<mpts; i++) {
|
|---|
| 476 | if (i == ilo) continue;
|
|---|
| 477 | splx[i] = Beta2()*splx[i]+(1.-Beta())*splx[ilo];
|
|---|
| 478 | Y(i) = Value(splx[i]);
|
|---|
| 479 | move = 40;
|
|---|
| 480 | }
|
|---|
| 481 | }
|
|---|
| 482 | }
|
|---|
| 483 | }
|
|---|
| 484 | }
|
|---|
| 485 | else { // Meilleur que le IHI et le INHI
|
|---|
| 486 | splx[ihi] = pr; // On remplace le plus haut
|
|---|
| 487 | Y(ihi) = YPR;
|
|---|
| 488 | move = 12;
|
|---|
| 489 | }
|
|---|
| 490 | }
|
|---|
| 491 | } // Fin de la boucle while principale
|
|---|
| 492 |
|
|---|
| 493 | fpoint = splx[ilo];
|
|---|
| 494 | mIter = iter;
|
|---|
| 495 |
|
|---|
| 496 | if (PrtLevel() > 0) {
|
|---|
| 497 | string sr;
|
|---|
| 498 | StopReason(sr);
|
|---|
| 499 | cout << "-----MinZSimplex::Minimize()/Ended - NIter=" << iter
|
|---|
| 500 | << " Moves[0..5]= " << movcnt[0] << "," << movcnt[1] << ","
|
|---|
| 501 | << movcnt[2] << "," << movcnt[3] << ","
|
|---|
| 502 | << movcnt[4] << "," << movcnt[5]
|
|---|
| 503 | << "\n..MinZSimplex Stop=" << StopReason() << " -> " << sr << endl;
|
|---|
| 504 |
|
|---|
| 505 | if (nbugrtol2 > 0) cout << "MinZSimplex::Minimize()/Warning - nbugrtol2= " << nbugrtol2 << endl;
|
|---|
| 506 | }
|
|---|
| 507 | return rc;
|
|---|
| 508 | }
|
|---|
| 509 |
|
|---|
| 510 | //! Return the stop reason and fills the corresponding string description
|
|---|
| 511 | int MinZSimplex::StopReason(string& s)
|
|---|
| 512 | {
|
|---|
| 513 | const char* sr[5] = { "NoConverg, MaxIterReached", "OK, fm<Tol0", "OK, Df/f<Tol1",
|
|---|
| 514 | "OK, [Df/f max]Iter<Tol2" "Error - Wrong StopReason" };
|
|---|
| 515 | int stop = mStop;
|
|---|
| 516 | if ((stop < 0) || (stop > 3)) stop = 4;
|
|---|
| 517 | s = sr[stop];
|
|---|
| 518 | return mStop;
|
|---|
| 519 | }
|
|---|
| 520 |
|
|---|
| 521 | int MinZSimplex::FindMinMax12(Vector& fval, int& ilo, int& ihi, int& inhi)
|
|---|
| 522 | {
|
|---|
| 523 | ilo = 0;
|
|---|
| 524 | if (fval(0) > fval(1)) { ihi = 0; inhi = 1; }
|
|---|
| 525 | else { ihi = 1; inhi = 0; }
|
|---|
| 526 |
|
|---|
| 527 | for(int k=0; k<fval.Size(); k++) {
|
|---|
| 528 | if (fval(k) < fval(ilo)) ilo = k;
|
|---|
| 529 | if (fval(k) > fval(ihi)) {
|
|---|
| 530 | inhi = ihi;
|
|---|
| 531 | ihi = k;
|
|---|
| 532 | }
|
|---|
| 533 | else if (fval(k) > fval(inhi)) {
|
|---|
| 534 | if (k != ihi) inhi = k; // ce test n'est peut-etre pas necessaire ???
|
|---|
| 535 | }
|
|---|
| 536 | }
|
|---|
| 537 | return ilo;
|
|---|
| 538 | }
|
|---|
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