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24 | // ******************************************************************** |
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25 | // |
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26 | // |
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27 | // $Id: G4PolynomialSolver.icc,v 1.8 2006/06/29 18:59:54 gunter Exp $ |
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28 | // GEANT4 tag $Name: geant4-09-04-beta-01 $ |
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29 | // |
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30 | // class G4PolynomialSolver |
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31 | // |
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32 | // 19.12.00 E.Medernach, First implementation |
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33 | // |
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34 | |
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35 | #define POLEPSILON 1e-12 |
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36 | #define POLINFINITY 9.0E99 |
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37 | #define ITERATION 12 // 20 But 8 is really enough for Newton with a good guess |
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38 | |
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39 | template <class T, class F> |
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40 | G4PolynomialSolver<T,F>::G4PolynomialSolver (T* typeF, F func, F deriv, |
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41 | G4double precision) |
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42 | { |
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43 | Precision = precision ; |
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44 | FunctionClass = typeF ; |
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45 | Function = func ; |
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46 | Derivative = deriv ; |
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47 | } |
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48 | |
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49 | template <class T, class F> |
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50 | G4PolynomialSolver<T,F>::~G4PolynomialSolver () |
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51 | { |
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52 | } |
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53 | |
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54 | template <class T, class F> |
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55 | G4double G4PolynomialSolver<T,F>::solve(G4double IntervalMin, |
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56 | G4double IntervalMax) |
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57 | { |
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58 | return Newton(IntervalMin,IntervalMax); |
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59 | } |
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60 | |
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61 | |
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62 | /* If we want to be general this could work for any |
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63 | polynomial of order more that 4 if we find the (ORDER + 1) |
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64 | control points |
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65 | */ |
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66 | #define NBBEZIER 5 |
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67 | |
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68 | template <class T, class F> |
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69 | G4int |
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70 | G4PolynomialSolver<T,F>::BezierClipping(/*T* typeF,F func,F deriv,*/ |
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71 | G4double *IntervalMin, |
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72 | G4double *IntervalMax) |
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73 | { |
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74 | /** BezierClipping is a clipping interval Newton method **/ |
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75 | /** It works by clipping the area where the polynomial is **/ |
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76 | |
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77 | G4double P[NBBEZIER][2],D[2]; |
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78 | G4double NewMin,NewMax; |
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79 | |
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80 | G4int IntervalIsVoid = 1; |
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81 | |
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82 | /*** Calculating Control Points ***/ |
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83 | /* We see the polynomial as a Bezier curve for some control points to find */ |
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84 | |
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85 | /* |
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86 | For 5 control points (polynomial of degree 4) this is: |
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87 | |
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88 | 0 p0 = F((*IntervalMin)) |
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89 | 1/4 p1 = F((*IntervalMin)) + ((*IntervalMax) - (*IntervalMin))/4 |
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90 | * F'((*IntervalMin)) |
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91 | 2/4 p2 = 1/6 * (16*F(((*IntervalMax) + (*IntervalMin))/2) |
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92 | - (p0 + 4*p1 + 4*p3 + p4)) |
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93 | 3/4 p3 = F((*IntervalMax)) - ((*IntervalMax) - (*IntervalMin))/4 |
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94 | * F'((*IntervalMax)) |
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95 | 1 p4 = F((*IntervalMax)) |
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96 | */ |
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97 | |
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98 | /* x,y,z,dx,dy,dz are constant during searching */ |
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99 | |
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100 | D[0] = (FunctionClass->*Derivative)(*IntervalMin); |
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101 | |
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102 | P[0][0] = (*IntervalMin); |
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103 | P[0][1] = (FunctionClass->*Function)(*IntervalMin); |
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104 | |
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105 | |
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106 | if (std::fabs(P[0][1]) < Precision) { |
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107 | return 1; |
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108 | } |
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109 | |
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110 | if (((*IntervalMax) - (*IntervalMin)) < POLEPSILON) { |
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111 | return 1; |
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112 | } |
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113 | |
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114 | P[1][0] = (*IntervalMin) + ((*IntervalMax) - (*IntervalMin))/4; |
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115 | P[1][1] = P[0][1] + (((*IntervalMax) - (*IntervalMin))/4.0) * D[0]; |
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116 | |
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117 | D[1] = (FunctionClass->*Derivative)(*IntervalMax); |
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118 | |
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119 | P[4][0] = (*IntervalMax); |
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120 | P[4][1] = (FunctionClass->*Function)(*IntervalMax); |
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121 | |
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122 | P[3][0] = (*IntervalMax) - ((*IntervalMax) - (*IntervalMin))/4; |
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123 | P[3][1] = P[4][1] - ((*IntervalMax) - (*IntervalMin))/4 * D[1]; |
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124 | |
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125 | P[2][0] = ((*IntervalMax) + (*IntervalMin))/2; |
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126 | P[2][1] = (16*(FunctionClass->*Function)(((*IntervalMax)+(*IntervalMin))/2) |
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127 | - (P[0][1] + 4*P[1][1] + 4*P[3][1] + P[4][1]))/6 ; |
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128 | |
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129 | { |
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130 | G4double Intersection ; |
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131 | G4int i,j; |
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132 | |
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133 | NewMin = (*IntervalMax) ; |
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134 | NewMax = (*IntervalMin) ; |
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135 | |
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136 | for (i=0;i<5;i++) |
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137 | for (j=i+1;j<5;j++) |
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138 | { |
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139 | /* there is an intersection only if each have different signs */ |
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140 | if (((P[j][1] > -Precision) && (P[i][1] < Precision)) || |
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141 | ((P[j][1] < Precision) && (P[i][1] > -Precision))) { |
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142 | IntervalIsVoid = 0; |
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143 | Intersection = P[j][0] - P[j][1]*((P[i][0] - P[j][0])/ |
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144 | (P[i][1] - P[j][1])); |
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145 | if (Intersection < NewMin) { |
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146 | NewMin = Intersection; |
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147 | } |
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148 | if (Intersection > NewMax) { |
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149 | NewMax = Intersection; |
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150 | } |
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151 | } |
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152 | } |
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153 | |
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154 | |
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155 | if (IntervalIsVoid != 1) { |
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156 | (*IntervalMax) = NewMax; |
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157 | (*IntervalMin) = NewMin; |
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158 | } |
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159 | } |
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160 | |
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161 | if (IntervalIsVoid == 1) { |
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162 | return -1; |
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163 | } |
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164 | |
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165 | return 0; |
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166 | } |
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167 | |
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168 | template <class T, class F> |
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169 | G4double G4PolynomialSolver<T,F>::Newton (G4double IntervalMin, |
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170 | G4double IntervalMax) |
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171 | { |
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172 | /* So now we have a good guess and an interval where |
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173 | if there are an intersection the root must be */ |
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174 | |
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175 | G4double Value = 0; |
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176 | G4double Gradient = 0; |
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177 | G4double Lambda ; |
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178 | |
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179 | G4int i=0; |
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180 | G4int j=0; |
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181 | |
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182 | |
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183 | /* Reduce interval before applying Newton Method */ |
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184 | { |
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185 | G4int NewtonIsSafe ; |
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186 | |
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187 | while ((NewtonIsSafe = BezierClipping(&IntervalMin,&IntervalMax)) == 0) ; |
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188 | |
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189 | if (NewtonIsSafe == -1) { |
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190 | return POLINFINITY; |
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191 | } |
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192 | } |
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193 | |
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194 | Lambda = IntervalMin; |
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195 | Value = (FunctionClass->*Function)(Lambda); |
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196 | |
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197 | |
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198 | // while ((std::fabs(Value) > Precision)) { |
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199 | while (j != -1) { |
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200 | |
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201 | Value = (FunctionClass->*Function)(Lambda); |
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202 | |
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203 | Gradient = (FunctionClass->*Derivative)(Lambda); |
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204 | |
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205 | Lambda = Lambda - Value/Gradient ; |
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206 | |
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207 | if (std::fabs(Value) <= Precision) { |
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208 | j ++; |
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209 | if (j == 2) { |
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210 | j = -1; |
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211 | } |
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212 | } else { |
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213 | i ++; |
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214 | |
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215 | if (i > ITERATION) |
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216 | return POLINFINITY; |
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217 | } |
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218 | } |
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219 | return Lambda ; |
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220 | } |
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