1 | // |
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2 | // ******************************************************************** |
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3 | // * License and Disclaimer * |
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4 | // * * |
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5 | // * The Geant4 software is copyright of the Copyright Holders of * |
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6 | // * the Geant4 Collaboration. It is provided under the terms and * |
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7 | // * conditions of the Geant4 Software License, included in the file * |
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8 | // * LICENSE and available at http://cern.ch/geant4/license . These * |
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9 | // * include a list of copyright holders. * |
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10 | // * * |
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11 | // * Neither the authors of this software system, nor their employing * |
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12 | // * institutes,nor the agencies providing financial support for this * |
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13 | // * work make any representation or warranty, express or implied, * |
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14 | // * regarding this software system or assume any liability for its * |
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15 | // * use. Please see the license in the file LICENSE and URL above * |
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16 | // * for the full disclaimer and the limitation of liability. * |
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17 | // * * |
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18 | // * This code implementation is the result of the scientific and * |
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19 | // * technical work of the GEANT4 collaboration. * |
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20 | // * By using, copying, modifying or distributing the software (or * |
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21 | // * any work based on the software) you agree to acknowledge its * |
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22 | // * use in resulting scientific publications, and indicate your * |
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23 | // * acceptance of all terms of the Geant4 Software license. * |
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24 | // ******************************************************************** |
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25 | // |
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26 | // $Id: Gamma.cc,v 1.6 2006/06/29 19:14:28 gunter Exp $ |
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27 | // GEANT4 tag $Name: $ |
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28 | // |
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29 | // |
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30 | // ------------------------------------------------------------ |
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31 | // GEANT 4 class implementation |
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32 | // ------------------------------------------------------------ |
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33 | |
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34 | #include <cmath> |
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35 | #include <string.h> |
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36 | #include "Gamma.hh" |
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37 | |
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38 | MyGamma::MyGamma(){} |
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39 | |
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40 | MyGamma::~MyGamma(){} |
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41 | |
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42 | //____________________________________________________________________________ |
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43 | double MyGamma::Gamma(double z) |
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44 | { |
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45 | // Computation of gamma(z) for all z>0. |
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46 | // |
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47 | // The algorithm is based on the article by C.Lanczos [1] as denoted in |
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48 | // Numerical Recipes 2nd ed. on p. 207 (W.H.Press et al.). |
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49 | // |
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50 | // [1] C.Lanczos, SIAM Journal of Numerical Analysis B1 (1964), 86. |
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51 | // |
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52 | //--- Nve 14-nov-1998 UU-SAP Utrecht |
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53 | |
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54 | if (z<=0) return 0; |
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55 | |
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56 | double v = LnGamma(z); |
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57 | return std::exp(v); |
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58 | } |
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59 | |
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60 | //____________________________________________________________________________ |
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61 | double MyGamma::Gamma(double a,double x) |
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62 | { |
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63 | // Computation of the incomplete gamma function P(a,x) |
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64 | // |
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65 | // The algorithm is based on the formulas and code as denoted in |
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66 | // Numerical Recipes 2nd ed. on p. 210-212 (W.H.Press et al.). |
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67 | // |
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68 | //--- Nve 14-nov-1998 UU-SAP Utrecht |
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69 | |
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70 | if (a <= 0 || x <= 0) return 0; |
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71 | |
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72 | if (x < (a+1)) return GamSer(a,x); |
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73 | else return GamCf(a,x); |
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74 | } |
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75 | |
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76 | //____________________________________________________________________________ |
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77 | double MyGamma::GamCf(double a,double x) |
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78 | { |
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79 | // Computation of the incomplete gamma function P(a,x) |
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80 | // via its continued fraction representation. |
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81 | // |
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82 | // The algorithm is based on the formulas and code as denoted in |
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83 | // Numerical Recipes 2nd ed. on p. 210-212 (W.H.Press et al.). |
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84 | // |
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85 | //--- Nve 14-nov-1998 UU-SAP Utrecht |
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86 | |
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87 | int itmax = 100; // Maximum number of iterations |
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88 | double eps = 3.e-7; // Relative accuracy |
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89 | double fpmin = 1.e-30; // Smallest double value allowed here |
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90 | |
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91 | if (a <= 0 || x <= 0) return 0; |
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92 | |
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93 | double gln = LnGamma(a); |
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94 | double b = x+1-a; |
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95 | double c = 1/fpmin; |
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96 | double d = 1/b; |
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97 | double h = d; |
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98 | double an,del; |
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99 | for (int i=1; i<=itmax; i++) { |
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100 | an = double(-i)*(double(i)-a); |
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101 | b += 2; |
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102 | d = an*d+b; |
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103 | if (Abs(d) < fpmin) d = fpmin; |
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104 | c = b+an/c; |
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105 | if (Abs(c) < fpmin) c = fpmin; |
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106 | d = 1/d; |
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107 | del = d*c; |
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108 | h = h*del; |
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109 | if (Abs(del-1) < eps) break; |
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110 | //if (i==itmax) cout << "*GamCf(a,x)* a too large or itmax too small" << endl; |
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111 | } |
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112 | double v = Exp(-x+a*Log(x)-gln)*h; |
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113 | return (1-v); |
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114 | } |
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115 | |
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116 | //____________________________________________________________________________ |
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117 | double MyGamma::GamSer(double a,double x) |
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118 | { |
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119 | // Computation of the incomplete gamma function P(a,x) |
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120 | // via its series representation. |
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121 | // |
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122 | // The algorithm is based on the formulas and code as denoted in |
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123 | // Numerical Recipes 2nd ed. on p. 210-212 (W.H.Press et al.). |
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124 | // |
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125 | //--- Nve 14-nov-1998 UU-SAP Utrecht |
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126 | |
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127 | int itmax = 100; // Maximum number of iterations |
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128 | double eps = 3.e-7; // Relative accuracy |
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129 | |
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130 | if (a <= 0 || x <= 0) return 0; |
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131 | |
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132 | double gln = LnGamma(a); |
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133 | double ap = a; |
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134 | double sum = 1/a; |
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135 | double del = sum; |
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136 | for (int n=1; n<=itmax; n++) { |
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137 | ap += 1; |
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138 | del = del*x/ap; |
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139 | sum += del; |
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140 | if (MyGamma::Abs(del) < Abs(sum*eps)) break; |
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141 | //if (n==itmax) cout << "*GamSer(a,x)* a too large or itmax too small" << endl; |
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142 | } |
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143 | double v = sum*Exp(-x+a*Log(x)-gln); |
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144 | return v; |
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145 | } |
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146 | |
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147 | |
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148 | double MyGamma::LnGamma(double z) |
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149 | { |
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150 | // Computation of ln[gamma(z)] for all z>0. |
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151 | // |
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152 | // The algorithm is based on the article by C.Lanczos [1] as denoted in |
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153 | // Numerical Recipes 2nd ed. on p. 207 (W.H.Press et al.). |
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154 | // |
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155 | // [1] C.Lanczos, SIAM Journal of Numerical Analysis B1 (1964), 86. |
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156 | // |
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157 | // The accuracy of the result is better than 2e-10. |
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158 | // |
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159 | //--- Nve 14-nov-1998 UU-SAP Utrecht |
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160 | |
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161 | if (z<=0) return 0; |
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162 | |
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163 | // Coefficients for the series expansion |
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164 | double c[7] = { 2.5066282746310005, 76.18009172947146, -86.50532032941677 |
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165 | ,24.01409824083091, -1.231739572450155, 0.1208650973866179e-2 |
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166 | ,-0.5395239384953e-5}; |
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167 | |
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168 | double x = z; |
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169 | double y = x; |
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170 | double tmp = x+5.5; |
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171 | tmp = (x+0.5)*Log(tmp)-tmp; |
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172 | double ser = 1.000000000190015; |
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173 | for (int i=1; i<7; i++) { |
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174 | y += 1; |
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175 | ser += c[i]/y; |
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176 | } |
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177 | double v = tmp+Log(c[0]*ser/x); |
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178 | return v; |
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179 | } |
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