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24 | // ******************************************************************** |
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25 | // |
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26 | // |
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27 | // $Id: G4GaussLegendreQ.cc,v 1.8 2007/11/13 17:35:06 gcosmo Exp $ |
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28 | // GEANT4 tag $Name: geant4-09-03 $ |
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29 | // |
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30 | #include "G4GaussLegendreQ.hh" |
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31 | |
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32 | G4GaussLegendreQ::G4GaussLegendreQ( function pFunction ) |
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33 | : G4VGaussianQuadrature(pFunction) |
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34 | { |
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35 | } |
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36 | |
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37 | // -------------------------------------------------------------------------- |
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38 | // |
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39 | // Constructor for GaussLegendre quadrature method. The value nLegendre sets |
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40 | // the accuracy required, i.e the number of points where the function pFunction |
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41 | // will be evaluated during integration. The constructor creates the arrays for |
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42 | // abscissas and weights that are used in Gauss-Legendre quadrature method. |
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43 | // The values a and b are the limits of integration of the pFunction. |
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44 | // nLegendre MUST BE EVEN !!! |
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45 | |
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46 | G4GaussLegendreQ::G4GaussLegendreQ( function pFunction, |
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47 | G4int nLegendre ) |
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48 | : G4VGaussianQuadrature(pFunction) |
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49 | { |
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50 | const G4double tolerance = 1.6e-10 ; |
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51 | G4int k = nLegendre ; |
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52 | fNumber = (nLegendre + 1)/2 ; |
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53 | if(2*fNumber != k) |
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54 | { |
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55 | G4Exception("G4GaussLegendreQ::G4GaussLegendreQ()", "InvalidCall", |
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56 | FatalException, "Invalid nLegendre argument !") ; |
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57 | } |
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58 | G4double newton0=0.0, newton1=0.0, |
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59 | temp1=0.0, temp2=0.0, temp3=0.0, temp=0.0 ; |
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60 | |
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61 | fAbscissa = new G4double[fNumber] ; |
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62 | fWeight = new G4double[fNumber] ; |
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63 | |
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64 | for(G4int i=1;i<=fNumber;i++) // Loop over the desired roots |
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65 | { |
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66 | newton0 = std::cos(pi*(i - 0.25)/(k + 0.5)) ; // Initial root |
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67 | do // approximation |
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68 | { // loop of Newton's method |
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69 | temp1 = 1.0 ; |
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70 | temp2 = 0.0 ; |
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71 | for(G4int j=1;j<=k;j++) |
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72 | { |
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73 | temp3 = temp2 ; |
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74 | temp2 = temp1 ; |
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75 | temp1 = ((2.0*j - 1.0)*newton0*temp2 - (j - 1.0)*temp3)/j ; |
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76 | } |
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77 | temp = k*(newton0*temp1 - temp2)/(newton0*newton0 - 1.0) ; |
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78 | newton1 = newton0 ; |
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79 | newton0 = newton1 - temp1/temp ; // Newton's method |
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80 | } |
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81 | while(std::fabs(newton0 - newton1) > tolerance) ; |
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82 | |
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83 | fAbscissa[fNumber-i] = newton0 ; |
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84 | fWeight[fNumber-i] = 2.0/((1.0 - newton0*newton0)*temp*temp) ; |
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85 | } |
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86 | } |
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87 | |
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88 | // -------------------------------------------------------------------------- |
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89 | // |
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90 | // Returns the integral of the function to be pointed by fFunction between a |
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91 | // and b, by 2*fNumber point Gauss-Legendre integration: the function is |
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92 | // evaluated exactly 2*fNumber times at interior points in the range of |
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93 | // integration. Since the weights and abscissas are, in this case, symmetric |
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94 | // around the midpoint of the range of integration, there are actually only |
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95 | // fNumber distinct values of each. |
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96 | |
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97 | G4double |
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98 | G4GaussLegendreQ::Integral(G4double a, G4double b) const |
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99 | { |
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100 | G4double xMean = 0.5*(a + b), |
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101 | xDiff = 0.5*(b - a), |
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102 | integral = 0.0, dx = 0.0 ; |
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103 | for(G4int i=0;i<fNumber;i++) |
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104 | { |
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105 | dx = xDiff*fAbscissa[i] ; |
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106 | integral += fWeight[i]*(fFunction(xMean + dx) + fFunction(xMean - dx)) ; |
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107 | } |
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108 | return integral *= xDiff ; |
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109 | } |
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110 | |
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111 | // -------------------------------------------------------------------------- |
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112 | // |
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113 | // Returns the integral of the function to be pointed by fFunction between a |
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114 | // and b, by ten point Gauss-Legendre integration: the function is evaluated |
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115 | // exactly ten times at interior points in the range of integration. Since the |
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116 | // weights and abscissas are, in this case, symmetric around the midpoint of |
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117 | // the range of integration, there are actually only five distinct values of |
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118 | // each. |
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119 | |
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120 | G4double |
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121 | G4GaussLegendreQ::QuickIntegral(G4double a, G4double b) const |
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122 | { |
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123 | // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 916 |
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124 | |
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125 | static G4double abscissa[] = { 0.148874338981631, 0.433395394129247, |
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126 | 0.679409568299024, 0.865063366688985, |
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127 | 0.973906528517172 } ; |
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128 | |
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129 | static G4double weight[] = { 0.295524224714753, 0.269266719309996, |
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130 | 0.219086362515982, 0.149451349150581, |
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131 | 0.066671344308688 } ; |
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132 | G4double xMean = 0.5*(a + b), |
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133 | xDiff = 0.5*(b - a), |
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134 | integral = 0.0, dx = 0.0 ; |
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135 | for(G4int i=0;i<5;i++) |
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136 | { |
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137 | dx = xDiff*abscissa[i] ; |
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138 | integral += weight[i]*(fFunction(xMean + dx) + fFunction(xMean - dx)) ; |
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139 | } |
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140 | return integral *= xDiff ; |
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141 | } |
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142 | |
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143 | // ------------------------------------------------------------------------- |
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144 | // |
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145 | // Returns the integral of the function to be pointed by fFunction between a |
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146 | // and b, by 96 point Gauss-Legendre integration: the function is evaluated |
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147 | // exactly ten times at interior points in the range of integration. Since the |
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148 | // weights and abscissas are, in this case, symmetric around the midpoint of |
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149 | // the range of integration, there are actually only five distinct values of |
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150 | // each. |
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151 | |
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152 | G4double |
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153 | G4GaussLegendreQ::AccurateIntegral(G4double a, G4double b) const |
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154 | { |
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155 | // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 919 |
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156 | |
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157 | static |
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158 | G4double abscissa[] = { |
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159 | 0.016276744849602969579, 0.048812985136049731112, |
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160 | 0.081297495464425558994, 0.113695850110665920911, |
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161 | 0.145973714654896941989, 0.178096882367618602759, // 6 |
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162 | |
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163 | 0.210031310460567203603, 0.241743156163840012328, |
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164 | 0.273198812591049141487, 0.304364944354496353024, |
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165 | 0.335208522892625422616, 0.365696861472313635031, // 12 |
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166 | |
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167 | 0.395797649828908603285, 0.425478988407300545365, |
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168 | 0.454709422167743008636, 0.483457973920596359768, |
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169 | 0.511694177154667673586, 0.539388108324357436227, // 18 |
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170 | |
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171 | 0.566510418561397168404, 0.593032364777572080684, |
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172 | 0.618925840125468570386, 0.644163403784967106798, |
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173 | 0.668718310043916153953, 0.692564536642171561344, // 24 |
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174 | |
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175 | 0.715676812348967626225, 0.738030643744400132851, |
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176 | 0.759602341176647498703, 0.780369043867433217604, |
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177 | 0.800308744139140817229, 0.819400310737931675539, // 30 |
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178 | |
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179 | 0.837623511228187121494, 0.854959033434601455463, |
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180 | 0.871388505909296502874, 0.886894517402420416057, |
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181 | 0.901460635315852341319, 0.915071423120898074206, // 36 |
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182 | |
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183 | 0.927712456722308690965, 0.939370339752755216932, |
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184 | 0.950032717784437635756, 0.959688291448742539300, |
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185 | 0.968326828463264212174, 0.975939174585136466453, // 42 |
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186 | |
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187 | 0.982517263563014677447, 0.988054126329623799481, |
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188 | 0.992543900323762624572, 0.995981842987209290650, |
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189 | 0.998364375863181677724, 0.999689503883230766828 // 48 |
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190 | } ; |
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191 | |
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192 | static |
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193 | G4double weight[] = { |
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194 | 0.032550614492363166242, 0.032516118713868835987, |
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195 | 0.032447163714064269364, 0.032343822568575928429, |
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196 | 0.032206204794030250669, 0.032034456231992663218, // 6 |
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197 | |
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198 | 0.031828758894411006535, 0.031589330770727168558, |
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199 | 0.031316425596862355813, 0.031010332586313837423, |
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200 | 0.030671376123669149014, 0.030299915420827593794, // 12 |
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201 | |
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202 | 0.029896344136328385984, 0.029461089958167905970, |
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203 | 0.028994614150555236543, 0.028497411065085385646, |
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204 | 0.027970007616848334440, 0.027412962726029242823, // 18 |
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205 | |
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206 | 0.026826866725591762198, 0.026212340735672413913, |
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207 | 0.025570036005349361499, 0.024900633222483610288, |
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208 | 0.024204841792364691282, 0.023483399085926219842, // 24 |
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209 | |
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210 | 0.022737069658329374001, 0.021966644438744349195, |
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211 | 0.021172939892191298988, 0.020356797154333324595, |
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212 | 0.019519081140145022410, 0.018660679627411467385, // 30 |
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213 | |
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214 | 0.017782502316045260838, 0.016885479864245172450, |
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215 | 0.015970562902562291381, 0.015038721026994938006, |
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216 | 0.014090941772314860916, 0.013128229566961572637, // 36 |
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217 | |
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218 | 0.012151604671088319635, 0.011162102099838498591, |
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219 | 0.010160770535008415758, 0.009148671230783386633, |
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220 | 0.008126876925698759217, 0.007096470791153865269, // 42 |
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221 | |
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222 | 0.006058545504235961683, 0.005014202742927517693, |
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223 | 0.003964554338444686674, 0.002910731817934946408, |
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224 | 0.001853960788946921732, 0.000796792065552012429 // 48 |
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225 | } ; |
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226 | G4double xMean = 0.5*(a + b), |
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227 | xDiff = 0.5*(b - a), |
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228 | integral = 0.0, dx = 0.0 ; |
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229 | for(G4int i=0;i<48;i++) |
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230 | { |
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231 | dx = xDiff*abscissa[i] ; |
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232 | integral += weight[i]*(fFunction(xMean + dx) + fFunction(xMean - dx)) ; |
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233 | } |
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234 | return integral *= xDiff ; |
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235 | } |
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