[833] | 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 | // |
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| 27 | // $Id: G4Integrator.icc,v 1.13 2006/06/29 18:59:47 gunter Exp $ |
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[1337] | 28 | // GEANT4 tag $Name: geant4-09-04-beta-01 $ |
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[833] | 29 | // |
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| 30 | // Implementation of G4Integrator methods. |
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| 31 | // |
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| 32 | // |
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| 33 | |
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| 34 | ///////////////////////////////////////////////////////////////////// |
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| 35 | // |
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| 36 | // Sympson integration method |
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| 37 | // |
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| 38 | ///////////////////////////////////////////////////////////////////// |
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| 39 | // |
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| 40 | // Integration of class member functions T::f by Simpson method. |
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| 41 | |
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| 42 | template <class T, class F> |
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| 43 | G4double G4Integrator<T,F>::Simpson( T& typeT, |
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| 44 | F f, |
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| 45 | G4double xInitial, |
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| 46 | G4double xFinal, |
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| 47 | G4int iterationNumber ) |
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| 48 | { |
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| 49 | G4int i ; |
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| 50 | G4double step = (xFinal - xInitial)/iterationNumber ; |
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| 51 | G4double x = xInitial ; |
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| 52 | G4double xPlus = xInitial + 0.5*step ; |
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| 53 | G4double mean = ( (typeT.*f)(xInitial) + (typeT.*f)(xFinal) )*0.5 ; |
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| 54 | G4double sum = (typeT.*f)(xPlus) ; |
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| 55 | |
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| 56 | for(i=1;i<iterationNumber;i++) |
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| 57 | { |
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| 58 | x += step ; |
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| 59 | xPlus += step ; |
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| 60 | mean += (typeT.*f)(x) ; |
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| 61 | sum += (typeT.*f)(xPlus) ; |
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| 62 | } |
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| 63 | mean += 2.0*sum ; |
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| 64 | |
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| 65 | return mean*step/3.0 ; |
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| 66 | } |
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| 67 | |
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| 68 | ///////////////////////////////////////////////////////////////////// |
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| 69 | // |
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| 70 | // Integration of class member functions T::f by Simpson method. |
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| 71 | // Convenient to use with 'this' pointer |
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| 72 | |
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| 73 | template <class T, class F> |
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| 74 | G4double G4Integrator<T,F>::Simpson( T* ptrT, |
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| 75 | F f, |
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| 76 | G4double xInitial, |
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| 77 | G4double xFinal, |
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| 78 | G4int iterationNumber ) |
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| 79 | { |
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| 80 | G4int i ; |
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| 81 | G4double step = (xFinal - xInitial)/iterationNumber ; |
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| 82 | G4double x = xInitial ; |
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| 83 | G4double xPlus = xInitial + 0.5*step ; |
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| 84 | G4double mean = ( (ptrT->*f)(xInitial) + (ptrT->*f)(xFinal) )*0.5 ; |
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| 85 | G4double sum = (ptrT->*f)(xPlus) ; |
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| 86 | |
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| 87 | for(i=1;i<iterationNumber;i++) |
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| 88 | { |
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| 89 | x += step ; |
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| 90 | xPlus += step ; |
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| 91 | mean += (ptrT->*f)(x) ; |
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| 92 | sum += (ptrT->*f)(xPlus) ; |
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| 93 | } |
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| 94 | mean += 2.0*sum ; |
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| 95 | |
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| 96 | return mean*step/3.0 ; |
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| 97 | } |
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| 98 | |
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| 99 | ///////////////////////////////////////////////////////////////////// |
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| 100 | // |
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| 101 | // Integration of class member functions T::f by Simpson method. |
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| 102 | // Convenient to use, when function f is defined in global scope, i.e. in main() |
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| 103 | // program |
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| 104 | |
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| 105 | template <class T, class F> |
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| 106 | G4double G4Integrator<T,F>::Simpson( G4double (*f)(G4double), |
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| 107 | G4double xInitial, |
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| 108 | G4double xFinal, |
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| 109 | G4int iterationNumber ) |
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| 110 | { |
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| 111 | G4int i ; |
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| 112 | G4double step = (xFinal - xInitial)/iterationNumber ; |
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| 113 | G4double x = xInitial ; |
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| 114 | G4double xPlus = xInitial + 0.5*step ; |
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| 115 | G4double mean = ( (*f)(xInitial) + (*f)(xFinal) )*0.5 ; |
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| 116 | G4double sum = (*f)(xPlus) ; |
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| 117 | |
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| 118 | for(i=1;i<iterationNumber;i++) |
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| 119 | { |
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| 120 | x += step ; |
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| 121 | xPlus += step ; |
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| 122 | mean += (*f)(x) ; |
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| 123 | sum += (*f)(xPlus) ; |
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| 124 | } |
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| 125 | mean += 2.0*sum ; |
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| 126 | |
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| 127 | return mean*step/3.0 ; |
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| 128 | } |
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| 129 | |
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| 130 | ////////////////////////////////////////////////////////////////////////// |
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| 131 | // |
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| 132 | // Adaptive Gauss method |
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| 133 | // |
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| 134 | ////////////////////////////////////////////////////////////////////////// |
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| 135 | // |
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| 136 | // |
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| 137 | |
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| 138 | template <class T, class F> |
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| 139 | G4double G4Integrator<T,F>::Gauss( T& typeT, F f, |
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| 140 | G4double xInitial, G4double xFinal ) |
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| 141 | { |
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| 142 | static G4double root = 1.0/std::sqrt(3.0) ; |
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| 143 | |
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| 144 | G4double xMean = (xInitial + xFinal)/2.0 ; |
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| 145 | G4double Step = (xFinal - xInitial)/2.0 ; |
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| 146 | G4double delta = Step*root ; |
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| 147 | G4double sum = ((typeT.*f)(xMean + delta) + |
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| 148 | (typeT.*f)(xMean - delta)) ; |
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| 149 | |
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| 150 | return sum*Step ; |
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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 | // |
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| 156 | |
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| 157 | template <class T, class F> G4double |
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| 158 | G4Integrator<T,F>::Gauss( T* ptrT, F f, G4double a, G4double b ) |
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| 159 | { |
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| 160 | return Gauss(*ptrT,f,a,b) ; |
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| 161 | } |
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| 162 | |
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| 163 | /////////////////////////////////////////////////////////////////////// |
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| 164 | // |
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| 165 | // |
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| 166 | |
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| 167 | template <class T, class F> |
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| 168 | G4double G4Integrator<T,F>::Gauss( G4double (*f)(G4double), |
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| 169 | G4double xInitial, G4double xFinal) |
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| 170 | { |
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| 171 | static G4double root = 1.0/std::sqrt(3.0) ; |
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| 172 | |
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| 173 | G4double xMean = (xInitial + xFinal)/2.0 ; |
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| 174 | G4double Step = (xFinal - xInitial)/2.0 ; |
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| 175 | G4double delta = Step*root ; |
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| 176 | G4double sum = ( (*f)(xMean + delta) + (*f)(xMean - delta) ) ; |
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| 177 | |
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| 178 | return sum*Step ; |
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| 179 | } |
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| 180 | |
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| 181 | /////////////////////////////////////////////////////////////////////////// |
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| 182 | // |
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| 183 | // |
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| 184 | |
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| 185 | template <class T, class F> |
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| 186 | void G4Integrator<T,F>::AdaptGauss( T& typeT, F f, G4double xInitial, |
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| 187 | G4double xFinal, G4double fTolerance, |
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| 188 | G4double& sum, |
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| 189 | G4int& depth ) |
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| 190 | { |
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| 191 | if(depth > 100) |
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| 192 | { |
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| 193 | G4cout<<"G4Integrator<T,F>::AdaptGauss: WARNING !!!"<<G4endl ; |
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| 194 | G4cout<<"Function varies too rapidly to get stated accuracy in 100 steps " |
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| 195 | <<G4endl ; |
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| 196 | |
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| 197 | return ; |
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| 198 | } |
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| 199 | G4double xMean = (xInitial + xFinal)/2.0 ; |
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| 200 | G4double leftHalf = Gauss(typeT,f,xInitial,xMean) ; |
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| 201 | G4double rightHalf = Gauss(typeT,f,xMean,xFinal) ; |
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| 202 | G4double full = Gauss(typeT,f,xInitial,xFinal) ; |
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| 203 | if(std::fabs(leftHalf+rightHalf-full) < fTolerance) |
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| 204 | { |
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| 205 | sum += full ; |
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| 206 | } |
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| 207 | else |
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| 208 | { |
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| 209 | depth++ ; |
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| 210 | AdaptGauss(typeT,f,xInitial,xMean,fTolerance,sum,depth) ; |
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| 211 | AdaptGauss(typeT,f,xMean,xFinal,fTolerance,sum,depth) ; |
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| 212 | } |
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| 213 | } |
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| 214 | |
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| 215 | template <class T, class F> |
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| 216 | void G4Integrator<T,F>::AdaptGauss( T* ptrT, F f, G4double xInitial, |
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| 217 | G4double xFinal, G4double fTolerance, |
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| 218 | G4double& sum, |
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| 219 | G4int& depth ) |
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| 220 | { |
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| 221 | AdaptGauss(*ptrT,f,xInitial,xFinal,fTolerance,sum,depth) ; |
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| 222 | } |
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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 | template <class T, class F> |
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| 228 | void G4Integrator<T,F>::AdaptGauss( G4double (*f)(G4double), |
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| 229 | G4double xInitial, G4double xFinal, |
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| 230 | G4double fTolerance, G4double& sum, |
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| 231 | G4int& depth ) |
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| 232 | { |
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| 233 | if(depth > 100) |
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| 234 | { |
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| 235 | G4cout<<"G4SimpleIntegration::AdaptGauss: WARNING !!!"<<G4endl ; |
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| 236 | G4cout<<"Function varies too rapidly to get stated accuracy in 100 steps " |
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| 237 | <<G4endl ; |
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| 238 | |
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| 239 | return ; |
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| 240 | } |
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| 241 | G4double xMean = (xInitial + xFinal)/2.0 ; |
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| 242 | G4double leftHalf = Gauss(f,xInitial,xMean) ; |
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| 243 | G4double rightHalf = Gauss(f,xMean,xFinal) ; |
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| 244 | G4double full = Gauss(f,xInitial,xFinal) ; |
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| 245 | if(std::fabs(leftHalf+rightHalf-full) < fTolerance) |
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| 246 | { |
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| 247 | sum += full ; |
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| 248 | } |
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| 249 | else |
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| 250 | { |
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| 251 | depth++ ; |
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| 252 | AdaptGauss(f,xInitial,xMean,fTolerance,sum,depth) ; |
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| 253 | AdaptGauss(f,xMean,xFinal,fTolerance,sum,depth) ; |
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| 254 | } |
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| 255 | } |
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| 256 | |
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| 257 | //////////////////////////////////////////////////////////////////////// |
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| 258 | // |
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| 259 | // Adaptive Gauss integration with accuracy 'e' |
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| 260 | // Convenient for using with class object typeT |
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| 261 | |
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| 262 | template<class T, class F> |
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| 263 | G4double G4Integrator<T,F>::AdaptiveGauss( T& typeT, F f, G4double xInitial, |
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| 264 | G4double xFinal, G4double e ) |
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| 265 | { |
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| 266 | G4int depth = 0 ; |
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| 267 | G4double sum = 0.0 ; |
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| 268 | AdaptGauss(typeT,f,xInitial,xFinal,e,sum,depth) ; |
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| 269 | return sum ; |
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| 270 | } |
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| 271 | |
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| 272 | //////////////////////////////////////////////////////////////////////// |
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| 273 | // |
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| 274 | // Adaptive Gauss integration with accuracy 'e' |
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| 275 | // Convenient for using with 'this' pointer |
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| 276 | |
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| 277 | template<class T, class F> |
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| 278 | G4double G4Integrator<T,F>::AdaptiveGauss( T* ptrT, F f, G4double xInitial, |
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| 279 | G4double xFinal, G4double e ) |
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| 280 | { |
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| 281 | return AdaptiveGauss(*ptrT,f,xInitial,xFinal,e) ; |
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| 282 | } |
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| 283 | |
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| 284 | //////////////////////////////////////////////////////////////////////// |
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| 285 | // |
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| 286 | // Adaptive Gauss integration with accuracy 'e' |
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| 287 | // Convenient for using with global scope function f |
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| 288 | |
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| 289 | template <class T, class F> |
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| 290 | G4double G4Integrator<T,F>::AdaptiveGauss( G4double (*f)(G4double), |
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| 291 | G4double xInitial, G4double xFinal, G4double e ) |
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| 292 | { |
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| 293 | G4int depth = 0 ; |
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| 294 | G4double sum = 0.0 ; |
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| 295 | AdaptGauss(f,xInitial,xFinal,e,sum,depth) ; |
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| 296 | return sum ; |
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| 297 | } |
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| 298 | |
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| 299 | //////////////////////////////////////////////////////////////////////////// |
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| 300 | // Gauss integration methods involving ortogonal polynomials |
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| 301 | //////////////////////////////////////////////////////////////////////////// |
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| 302 | // |
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| 303 | // Methods involving Legendre polynomials |
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| 304 | // |
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| 305 | ///////////////////////////////////////////////////////////////////////// |
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| 306 | // |
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| 307 | // The value nLegendre set the accuracy required, i.e the number of points |
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| 308 | // where the function pFunction will be evaluated during integration. |
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| 309 | // The function creates the arrays for abscissas and weights that used |
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| 310 | // in Gauss-Legendre quadrature method. |
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| 311 | // The values a and b are the limits of integration of the function f . |
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| 312 | // nLegendre MUST BE EVEN !!! |
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| 313 | // Returns the integral of the function f between a and b, by 2*fNumber point |
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| 314 | // Gauss-Legendre integration: the function is evaluated exactly |
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| 315 | // 2*fNumber times at interior points in the range of integration. |
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| 316 | // Since the weights and abscissas are, in this case, symmetric around |
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| 317 | // the midpoint of the range of integration, there are actually only |
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| 318 | // fNumber distinct values of each. |
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| 319 | // Convenient for using with some class object dataT |
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| 320 | |
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| 321 | template <class T, class F> |
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| 322 | G4double G4Integrator<T,F>::Legendre( T& typeT, F f, G4double a, G4double b, |
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| 323 | G4int nLegendre ) |
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| 324 | { |
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| 325 | G4double newton, newton1, temp1, temp2, temp3, temp ; |
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| 326 | G4double xDiff, xMean, dx, integral ; |
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| 327 | |
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| 328 | const G4double tolerance = 1.6e-10 ; |
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| 329 | G4int i, j, k = nLegendre ; |
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| 330 | G4int fNumber = (nLegendre + 1)/2 ; |
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| 331 | |
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| 332 | if(2*fNumber != k) |
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| 333 | { |
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| 334 | G4Exception("G4Integrator<T,F>::Legendre(T&,F, ...)", "InvalidCall", |
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| 335 | FatalException, "Invalid (odd) nLegendre in constructor."); |
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| 336 | } |
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| 337 | |
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| 338 | G4double* fAbscissa = new G4double[fNumber] ; |
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| 339 | G4double* fWeight = new G4double[fNumber] ; |
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| 340 | |
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| 341 | for(i=1;i<=fNumber;i++) // Loop over the desired roots |
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| 342 | { |
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| 343 | newton = std::cos(pi*(i - 0.25)/(k + 0.5)) ; // Initial root approximation |
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| 344 | |
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| 345 | do // loop of Newton's method |
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| 346 | { |
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| 347 | temp1 = 1.0 ; |
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| 348 | temp2 = 0.0 ; |
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| 349 | for(j=1;j<=k;j++) |
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| 350 | { |
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| 351 | temp3 = temp2 ; |
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| 352 | temp2 = temp1 ; |
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| 353 | temp1 = ((2.0*j - 1.0)*newton*temp2 - (j - 1.0)*temp3)/j ; |
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| 354 | } |
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| 355 | temp = k*(newton*temp1 - temp2)/(newton*newton - 1.0) ; |
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| 356 | newton1 = newton ; |
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| 357 | newton = newton1 - temp1/temp ; // Newton's method |
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| 358 | } |
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| 359 | while(std::fabs(newton - newton1) > tolerance) ; |
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| 360 | |
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| 361 | fAbscissa[fNumber-i] = newton ; |
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| 362 | fWeight[fNumber-i] = 2.0/((1.0 - newton*newton)*temp*temp) ; |
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| 363 | } |
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| 364 | |
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| 365 | // |
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| 366 | // Now we ready to get integral |
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| 367 | // |
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| 368 | |
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| 369 | xMean = 0.5*(a + b) ; |
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| 370 | xDiff = 0.5*(b - a) ; |
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| 371 | integral = 0.0 ; |
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| 372 | for(i=0;i<fNumber;i++) |
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| 373 | { |
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| 374 | dx = xDiff*fAbscissa[i] ; |
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| 375 | integral += fWeight[i]*( (typeT.*f)(xMean + dx) + |
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| 376 | (typeT.*f)(xMean - dx) ) ; |
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| 377 | } |
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| 378 | delete[] fAbscissa; |
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| 379 | delete[] fWeight; |
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| 380 | return integral *= xDiff ; |
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| 381 | } |
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| 382 | |
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| 383 | /////////////////////////////////////////////////////////////////////// |
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| 384 | // |
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| 385 | // Convenient for using with the pointer 'this' |
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| 386 | |
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| 387 | template <class T, class F> |
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| 388 | G4double G4Integrator<T,F>::Legendre( T* ptrT, F f, G4double a, |
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| 389 | G4double b, G4int nLegendre ) |
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| 390 | { |
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| 391 | return Legendre(*ptrT,f,a,b,nLegendre) ; |
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| 392 | } |
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| 393 | |
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| 394 | /////////////////////////////////////////////////////////////////////// |
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| 395 | // |
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| 396 | // Convenient for using with global scope function f |
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| 397 | |
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| 398 | template <class T, class F> |
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| 399 | G4double G4Integrator<T,F>::Legendre( G4double (*f)(G4double), |
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| 400 | G4double a, G4double b, G4int nLegendre) |
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| 401 | { |
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| 402 | G4double newton, newton1, temp1, temp2, temp3, temp ; |
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| 403 | G4double xDiff, xMean, dx, integral ; |
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| 404 | |
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| 405 | const G4double tolerance = 1.6e-10 ; |
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| 406 | G4int i, j, k = nLegendre ; |
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| 407 | G4int fNumber = (nLegendre + 1)/2 ; |
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| 408 | |
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| 409 | if(2*fNumber != k) |
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| 410 | { |
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| 411 | G4Exception("G4Integrator<T,F>::Legendre(...)", "InvalidCall", |
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| 412 | FatalException, "Invalid (odd) nLegendre in constructor."); |
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| 413 | } |
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| 414 | |
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| 415 | G4double* fAbscissa = new G4double[fNumber] ; |
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| 416 | G4double* fWeight = new G4double[fNumber] ; |
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| 417 | |
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| 418 | for(i=1;i<=fNumber;i++) // Loop over the desired roots |
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| 419 | { |
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| 420 | newton = std::cos(pi*(i - 0.25)/(k + 0.5)) ; // Initial root approximation |
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| 421 | |
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| 422 | do // loop of Newton's method |
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| 423 | { |
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| 424 | temp1 = 1.0 ; |
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| 425 | temp2 = 0.0 ; |
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| 426 | for(j=1;j<=k;j++) |
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| 427 | { |
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| 428 | temp3 = temp2 ; |
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| 429 | temp2 = temp1 ; |
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| 430 | temp1 = ((2.0*j - 1.0)*newton*temp2 - (j - 1.0)*temp3)/j ; |
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| 431 | } |
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| 432 | temp = k*(newton*temp1 - temp2)/(newton*newton - 1.0) ; |
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| 433 | newton1 = newton ; |
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| 434 | newton = newton1 - temp1/temp ; // Newton's method |
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| 435 | } |
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| 436 | while(std::fabs(newton - newton1) > tolerance) ; |
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| 437 | |
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| 438 | fAbscissa[fNumber-i] = newton ; |
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| 439 | fWeight[fNumber-i] = 2.0/((1.0 - newton*newton)*temp*temp) ; |
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| 440 | } |
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| 441 | |
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| 442 | // |
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| 443 | // Now we ready to get integral |
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| 444 | // |
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| 445 | |
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| 446 | xMean = 0.5*(a + b) ; |
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| 447 | xDiff = 0.5*(b - a) ; |
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| 448 | integral = 0.0 ; |
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| 449 | for(i=0;i<fNumber;i++) |
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| 450 | { |
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| 451 | dx = xDiff*fAbscissa[i] ; |
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| 452 | integral += fWeight[i]*( (*f)(xMean + dx) + (*f)(xMean - dx) ) ; |
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| 453 | } |
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| 454 | delete[] fAbscissa; |
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| 455 | delete[] fWeight; |
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| 456 | |
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| 457 | return integral *= xDiff ; |
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| 458 | } |
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| 459 | |
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| 460 | //////////////////////////////////////////////////////////////////////////// |
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| 461 | // |
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| 462 | // Returns the integral of the function to be pointed by T::f between a and b, |
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| 463 | // by ten point Gauss-Legendre integration: the function is evaluated exactly |
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| 464 | // ten times at interior points in the range of integration. Since the weights |
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| 465 | // and abscissas are, in this case, symmetric around the midpoint of the |
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| 466 | // range of integration, there are actually only five distinct values of each |
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| 467 | // Convenient for using with class object typeT |
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| 468 | |
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| 469 | template <class T, class F> |
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| 470 | G4double G4Integrator<T,F>::Legendre10( T& typeT, F f,G4double a, G4double b) |
---|
| 471 | { |
---|
| 472 | G4int i ; |
---|
| 473 | G4double xDiff, xMean, dx, integral ; |
---|
| 474 | |
---|
| 475 | // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 916 |
---|
| 476 | |
---|
| 477 | static G4double abscissa[] = { 0.148874338981631, 0.433395394129247, |
---|
| 478 | 0.679409568299024, 0.865063366688985, |
---|
| 479 | 0.973906528517172 } ; |
---|
| 480 | |
---|
| 481 | static G4double weight[] = { 0.295524224714753, 0.269266719309996, |
---|
| 482 | 0.219086362515982, 0.149451349150581, |
---|
| 483 | 0.066671344308688 } ; |
---|
| 484 | xMean = 0.5*(a + b) ; |
---|
| 485 | xDiff = 0.5*(b - a) ; |
---|
| 486 | integral = 0.0 ; |
---|
| 487 | for(i=0;i<5;i++) |
---|
| 488 | { |
---|
| 489 | dx = xDiff*abscissa[i] ; |
---|
| 490 | integral += weight[i]*( (typeT.*f)(xMean + dx) + (typeT.*f)(xMean - dx)) ; |
---|
| 491 | } |
---|
| 492 | return integral *= xDiff ; |
---|
| 493 | } |
---|
| 494 | |
---|
| 495 | /////////////////////////////////////////////////////////////////////////// |
---|
| 496 | // |
---|
| 497 | // Convenient for using with the pointer 'this' |
---|
| 498 | |
---|
| 499 | template <class T, class F> |
---|
| 500 | G4double G4Integrator<T,F>::Legendre10( T* ptrT, F f,G4double a, G4double b) |
---|
| 501 | { |
---|
| 502 | return Legendre10(*ptrT,f,a,b) ; |
---|
| 503 | } |
---|
| 504 | |
---|
| 505 | ////////////////////////////////////////////////////////////////////////// |
---|
| 506 | // |
---|
| 507 | // Convenient for using with global scope functions |
---|
| 508 | |
---|
| 509 | template <class T, class F> |
---|
| 510 | G4double G4Integrator<T,F>::Legendre10( G4double (*f)(G4double), |
---|
| 511 | G4double a, G4double b ) |
---|
| 512 | { |
---|
| 513 | G4int i ; |
---|
| 514 | G4double xDiff, xMean, dx, integral ; |
---|
| 515 | |
---|
| 516 | // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 916 |
---|
| 517 | |
---|
| 518 | static G4double abscissa[] = { 0.148874338981631, 0.433395394129247, |
---|
| 519 | 0.679409568299024, 0.865063366688985, |
---|
| 520 | 0.973906528517172 } ; |
---|
| 521 | |
---|
| 522 | static G4double weight[] = { 0.295524224714753, 0.269266719309996, |
---|
| 523 | 0.219086362515982, 0.149451349150581, |
---|
| 524 | 0.066671344308688 } ; |
---|
| 525 | xMean = 0.5*(a + b) ; |
---|
| 526 | xDiff = 0.5*(b - a) ; |
---|
| 527 | integral = 0.0 ; |
---|
| 528 | for(i=0;i<5;i++) |
---|
| 529 | { |
---|
| 530 | dx = xDiff*abscissa[i] ; |
---|
| 531 | integral += weight[i]*( (*f)(xMean + dx) + (*f)(xMean - dx)) ; |
---|
| 532 | } |
---|
| 533 | return integral *= xDiff ; |
---|
| 534 | } |
---|
| 535 | |
---|
| 536 | /////////////////////////////////////////////////////////////////////// |
---|
| 537 | // |
---|
| 538 | // Returns the integral of the function to be pointed by T::f between a and b, |
---|
| 539 | // by 96 point Gauss-Legendre integration: the function is evaluated exactly |
---|
| 540 | // ten Times at interior points in the range of integration. Since the weights |
---|
| 541 | // and abscissas are, in this case, symmetric around the midpoint of the |
---|
| 542 | // range of integration, there are actually only five distinct values of each |
---|
| 543 | // Convenient for using with some class object typeT |
---|
| 544 | |
---|
| 545 | template <class T, class F> |
---|
| 546 | G4double G4Integrator<T,F>::Legendre96( T& typeT, F f,G4double a, G4double b) |
---|
| 547 | { |
---|
| 548 | G4int i ; |
---|
| 549 | G4double xDiff, xMean, dx, integral ; |
---|
| 550 | |
---|
| 551 | // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 919 |
---|
| 552 | |
---|
| 553 | static G4double |
---|
| 554 | abscissa[] = { |
---|
| 555 | 0.016276744849602969579, 0.048812985136049731112, |
---|
| 556 | 0.081297495464425558994, 0.113695850110665920911, |
---|
| 557 | 0.145973714654896941989, 0.178096882367618602759, // 6 |
---|
| 558 | |
---|
| 559 | 0.210031310460567203603, 0.241743156163840012328, |
---|
| 560 | 0.273198812591049141487, 0.304364944354496353024, |
---|
| 561 | 0.335208522892625422616, 0.365696861472313635031, // 12 |
---|
| 562 | |
---|
| 563 | 0.395797649828908603285, 0.425478988407300545365, |
---|
| 564 | 0.454709422167743008636, 0.483457973920596359768, |
---|
| 565 | 0.511694177154667673586, 0.539388108324357436227, // 18 |
---|
| 566 | |
---|
| 567 | 0.566510418561397168404, 0.593032364777572080684, |
---|
| 568 | 0.618925840125468570386, 0.644163403784967106798, |
---|
| 569 | 0.668718310043916153953, 0.692564536642171561344, // 24 |
---|
| 570 | |
---|
| 571 | 0.715676812348967626225, 0.738030643744400132851, |
---|
| 572 | 0.759602341176647498703, 0.780369043867433217604, |
---|
| 573 | 0.800308744139140817229, 0.819400310737931675539, // 30 |
---|
| 574 | |
---|
| 575 | 0.837623511228187121494, 0.854959033434601455463, |
---|
| 576 | 0.871388505909296502874, 0.886894517402420416057, |
---|
| 577 | 0.901460635315852341319, 0.915071423120898074206, // 36 |
---|
| 578 | |
---|
| 579 | 0.927712456722308690965, 0.939370339752755216932, |
---|
| 580 | 0.950032717784437635756, 0.959688291448742539300, |
---|
| 581 | 0.968326828463264212174, 0.975939174585136466453, // 42 |
---|
| 582 | |
---|
| 583 | 0.982517263563014677447, 0.988054126329623799481, |
---|
| 584 | 0.992543900323762624572, 0.995981842987209290650, |
---|
| 585 | 0.998364375863181677724, 0.999689503883230766828 // 48 |
---|
| 586 | } ; |
---|
| 587 | |
---|
| 588 | static G4double |
---|
| 589 | weight[] = { |
---|
| 590 | 0.032550614492363166242, 0.032516118713868835987, |
---|
| 591 | 0.032447163714064269364, 0.032343822568575928429, |
---|
| 592 | 0.032206204794030250669, 0.032034456231992663218, // 6 |
---|
| 593 | |
---|
| 594 | 0.031828758894411006535, 0.031589330770727168558, |
---|
| 595 | 0.031316425596862355813, 0.031010332586313837423, |
---|
| 596 | 0.030671376123669149014, 0.030299915420827593794, // 12 |
---|
| 597 | |
---|
| 598 | 0.029896344136328385984, 0.029461089958167905970, |
---|
| 599 | 0.028994614150555236543, 0.028497411065085385646, |
---|
| 600 | 0.027970007616848334440, 0.027412962726029242823, // 18 |
---|
| 601 | |
---|
| 602 | 0.026826866725591762198, 0.026212340735672413913, |
---|
| 603 | 0.025570036005349361499, 0.024900633222483610288, |
---|
| 604 | 0.024204841792364691282, 0.023483399085926219842, // 24 |
---|
| 605 | |
---|
| 606 | 0.022737069658329374001, 0.021966644438744349195, |
---|
| 607 | 0.021172939892191298988, 0.020356797154333324595, |
---|
| 608 | 0.019519081140145022410, 0.018660679627411467385, // 30 |
---|
| 609 | |
---|
| 610 | 0.017782502316045260838, 0.016885479864245172450, |
---|
| 611 | 0.015970562902562291381, 0.015038721026994938006, |
---|
| 612 | 0.014090941772314860916, 0.013128229566961572637, // 36 |
---|
| 613 | |
---|
| 614 | 0.012151604671088319635, 0.011162102099838498591, |
---|
| 615 | 0.010160770535008415758, 0.009148671230783386633, |
---|
| 616 | 0.008126876925698759217, 0.007096470791153865269, // 42 |
---|
| 617 | |
---|
| 618 | 0.006058545504235961683, 0.005014202742927517693, |
---|
| 619 | 0.003964554338444686674, 0.002910731817934946408, |
---|
| 620 | 0.001853960788946921732, 0.000796792065552012429 // 48 |
---|
| 621 | } ; |
---|
| 622 | xMean = 0.5*(a + b) ; |
---|
| 623 | xDiff = 0.5*(b - a) ; |
---|
| 624 | integral = 0.0 ; |
---|
| 625 | for(i=0;i<48;i++) |
---|
| 626 | { |
---|
| 627 | dx = xDiff*abscissa[i] ; |
---|
| 628 | integral += weight[i]*((typeT.*f)(xMean + dx) + (typeT.*f)(xMean - dx)) ; |
---|
| 629 | } |
---|
| 630 | return integral *= xDiff ; |
---|
| 631 | } |
---|
| 632 | |
---|
| 633 | /////////////////////////////////////////////////////////////////////// |
---|
| 634 | // |
---|
| 635 | // Convenient for using with the pointer 'this' |
---|
| 636 | |
---|
| 637 | template <class T, class F> |
---|
| 638 | G4double G4Integrator<T,F>::Legendre96( T* ptrT, F f,G4double a, G4double b) |
---|
| 639 | { |
---|
| 640 | return Legendre96(*ptrT,f,a,b) ; |
---|
| 641 | } |
---|
| 642 | |
---|
| 643 | /////////////////////////////////////////////////////////////////////// |
---|
| 644 | // |
---|
| 645 | // Convenient for using with global scope function f |
---|
| 646 | |
---|
| 647 | template <class T, class F> |
---|
| 648 | G4double G4Integrator<T,F>::Legendre96( G4double (*f)(G4double), |
---|
| 649 | G4double a, G4double b ) |
---|
| 650 | { |
---|
| 651 | G4int i ; |
---|
| 652 | G4double xDiff, xMean, dx, integral ; |
---|
| 653 | |
---|
| 654 | // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 919 |
---|
| 655 | |
---|
| 656 | static G4double |
---|
| 657 | abscissa[] = { |
---|
| 658 | 0.016276744849602969579, 0.048812985136049731112, |
---|
| 659 | 0.081297495464425558994, 0.113695850110665920911, |
---|
| 660 | 0.145973714654896941989, 0.178096882367618602759, // 6 |
---|
| 661 | |
---|
| 662 | 0.210031310460567203603, 0.241743156163840012328, |
---|
| 663 | 0.273198812591049141487, 0.304364944354496353024, |
---|
| 664 | 0.335208522892625422616, 0.365696861472313635031, // 12 |
---|
| 665 | |
---|
| 666 | 0.395797649828908603285, 0.425478988407300545365, |
---|
| 667 | 0.454709422167743008636, 0.483457973920596359768, |
---|
| 668 | 0.511694177154667673586, 0.539388108324357436227, // 18 |
---|
| 669 | |
---|
| 670 | 0.566510418561397168404, 0.593032364777572080684, |
---|
| 671 | 0.618925840125468570386, 0.644163403784967106798, |
---|
| 672 | 0.668718310043916153953, 0.692564536642171561344, // 24 |
---|
| 673 | |
---|
| 674 | 0.715676812348967626225, 0.738030643744400132851, |
---|
| 675 | 0.759602341176647498703, 0.780369043867433217604, |
---|
| 676 | 0.800308744139140817229, 0.819400310737931675539, // 30 |
---|
| 677 | |
---|
| 678 | 0.837623511228187121494, 0.854959033434601455463, |
---|
| 679 | 0.871388505909296502874, 0.886894517402420416057, |
---|
| 680 | 0.901460635315852341319, 0.915071423120898074206, // 36 |
---|
| 681 | |
---|
| 682 | 0.927712456722308690965, 0.939370339752755216932, |
---|
| 683 | 0.950032717784437635756, 0.959688291448742539300, |
---|
| 684 | 0.968326828463264212174, 0.975939174585136466453, // 42 |
---|
| 685 | |
---|
| 686 | 0.982517263563014677447, 0.988054126329623799481, |
---|
| 687 | 0.992543900323762624572, 0.995981842987209290650, |
---|
| 688 | 0.998364375863181677724, 0.999689503883230766828 // 48 |
---|
| 689 | } ; |
---|
| 690 | |
---|
| 691 | static G4double |
---|
| 692 | weight[] = { |
---|
| 693 | 0.032550614492363166242, 0.032516118713868835987, |
---|
| 694 | 0.032447163714064269364, 0.032343822568575928429, |
---|
| 695 | 0.032206204794030250669, 0.032034456231992663218, // 6 |
---|
| 696 | |
---|
| 697 | 0.031828758894411006535, 0.031589330770727168558, |
---|
| 698 | 0.031316425596862355813, 0.031010332586313837423, |
---|
| 699 | 0.030671376123669149014, 0.030299915420827593794, // 12 |
---|
| 700 | |
---|
| 701 | 0.029896344136328385984, 0.029461089958167905970, |
---|
| 702 | 0.028994614150555236543, 0.028497411065085385646, |
---|
| 703 | 0.027970007616848334440, 0.027412962726029242823, // 18 |
---|
| 704 | |
---|
| 705 | 0.026826866725591762198, 0.026212340735672413913, |
---|
| 706 | 0.025570036005349361499, 0.024900633222483610288, |
---|
| 707 | 0.024204841792364691282, 0.023483399085926219842, // 24 |
---|
| 708 | |
---|
| 709 | 0.022737069658329374001, 0.021966644438744349195, |
---|
| 710 | 0.021172939892191298988, 0.020356797154333324595, |
---|
| 711 | 0.019519081140145022410, 0.018660679627411467385, // 30 |
---|
| 712 | |
---|
| 713 | 0.017782502316045260838, 0.016885479864245172450, |
---|
| 714 | 0.015970562902562291381, 0.015038721026994938006, |
---|
| 715 | 0.014090941772314860916, 0.013128229566961572637, // 36 |
---|
| 716 | |
---|
| 717 | 0.012151604671088319635, 0.011162102099838498591, |
---|
| 718 | 0.010160770535008415758, 0.009148671230783386633, |
---|
| 719 | 0.008126876925698759217, 0.007096470791153865269, // 42 |
---|
| 720 | |
---|
| 721 | 0.006058545504235961683, 0.005014202742927517693, |
---|
| 722 | 0.003964554338444686674, 0.002910731817934946408, |
---|
| 723 | 0.001853960788946921732, 0.000796792065552012429 // 48 |
---|
| 724 | } ; |
---|
| 725 | xMean = 0.5*(a + b) ; |
---|
| 726 | xDiff = 0.5*(b - a) ; |
---|
| 727 | integral = 0.0 ; |
---|
| 728 | for(i=0;i<48;i++) |
---|
| 729 | { |
---|
| 730 | dx = xDiff*abscissa[i] ; |
---|
| 731 | integral += weight[i]*((*f)(xMean + dx) + (*f)(xMean - dx)) ; |
---|
| 732 | } |
---|
| 733 | return integral *= xDiff ; |
---|
| 734 | } |
---|
| 735 | |
---|
| 736 | ////////////////////////////////////////////////////////////////////////////// |
---|
| 737 | // |
---|
| 738 | // Methods involving Chebyshev polynomials |
---|
| 739 | // |
---|
| 740 | /////////////////////////////////////////////////////////////////////////// |
---|
| 741 | // |
---|
| 742 | // Integrates function pointed by T::f from a to b by Gauss-Chebyshev |
---|
| 743 | // quadrature method. |
---|
| 744 | // Convenient for using with class object typeT |
---|
| 745 | |
---|
| 746 | template <class T, class F> |
---|
| 747 | G4double G4Integrator<T,F>::Chebyshev( T& typeT, F f, G4double a, |
---|
| 748 | G4double b, G4int nChebyshev ) |
---|
| 749 | { |
---|
| 750 | G4int i ; |
---|
| 751 | G4double xDiff, xMean, dx, integral = 0.0 ; |
---|
| 752 | |
---|
| 753 | G4int fNumber = nChebyshev ; // Try to reduce fNumber twice ?? |
---|
| 754 | G4double cof = pi/fNumber ; |
---|
| 755 | G4double* fAbscissa = new G4double[fNumber] ; |
---|
| 756 | G4double* fWeight = new G4double[fNumber] ; |
---|
| 757 | for(i=0;i<fNumber;i++) |
---|
| 758 | { |
---|
| 759 | fAbscissa[i] = std::cos(cof*(i + 0.5)) ; |
---|
| 760 | fWeight[i] = cof*std::sqrt(1 - fAbscissa[i]*fAbscissa[i]) ; |
---|
| 761 | } |
---|
| 762 | |
---|
| 763 | // |
---|
| 764 | // Now we ready to estimate the integral |
---|
| 765 | // |
---|
| 766 | |
---|
| 767 | xMean = 0.5*(a + b) ; |
---|
| 768 | xDiff = 0.5*(b - a) ; |
---|
| 769 | for(i=0;i<fNumber;i++) |
---|
| 770 | { |
---|
| 771 | dx = xDiff*fAbscissa[i] ; |
---|
| 772 | integral += fWeight[i]*(typeT.*f)(xMean + dx) ; |
---|
| 773 | } |
---|
| 774 | delete[] fAbscissa; |
---|
| 775 | delete[] fWeight; |
---|
| 776 | return integral *= xDiff ; |
---|
| 777 | } |
---|
| 778 | |
---|
| 779 | /////////////////////////////////////////////////////////////////////// |
---|
| 780 | // |
---|
| 781 | // Convenient for using with 'this' pointer |
---|
| 782 | |
---|
| 783 | template <class T, class F> |
---|
| 784 | G4double G4Integrator<T,F>::Chebyshev( T* ptrT, F f, G4double a, |
---|
| 785 | G4double b, G4int n ) |
---|
| 786 | { |
---|
| 787 | return Chebyshev(*ptrT,f,a,b,n) ; |
---|
| 788 | } |
---|
| 789 | |
---|
| 790 | //////////////////////////////////////////////////////////////////////// |
---|
| 791 | // |
---|
| 792 | // For use with global scope functions f |
---|
| 793 | |
---|
| 794 | template <class T, class F> |
---|
| 795 | G4double G4Integrator<T,F>::Chebyshev( G4double (*f)(G4double), |
---|
| 796 | G4double a, G4double b, G4int nChebyshev ) |
---|
| 797 | { |
---|
| 798 | G4int i ; |
---|
| 799 | G4double xDiff, xMean, dx, integral = 0.0 ; |
---|
| 800 | |
---|
| 801 | G4int fNumber = nChebyshev ; // Try to reduce fNumber twice ?? |
---|
| 802 | G4double cof = pi/fNumber ; |
---|
| 803 | G4double* fAbscissa = new G4double[fNumber] ; |
---|
| 804 | G4double* fWeight = new G4double[fNumber] ; |
---|
| 805 | for(i=0;i<fNumber;i++) |
---|
| 806 | { |
---|
| 807 | fAbscissa[i] = std::cos(cof*(i + 0.5)) ; |
---|
| 808 | fWeight[i] = cof*std::sqrt(1 - fAbscissa[i]*fAbscissa[i]) ; |
---|
| 809 | } |
---|
| 810 | |
---|
| 811 | // |
---|
| 812 | // Now we ready to estimate the integral |
---|
| 813 | // |
---|
| 814 | |
---|
| 815 | xMean = 0.5*(a + b) ; |
---|
| 816 | xDiff = 0.5*(b - a) ; |
---|
| 817 | for(i=0;i<fNumber;i++) |
---|
| 818 | { |
---|
| 819 | dx = xDiff*fAbscissa[i] ; |
---|
| 820 | integral += fWeight[i]*(*f)(xMean + dx) ; |
---|
| 821 | } |
---|
| 822 | delete[] fAbscissa; |
---|
| 823 | delete[] fWeight; |
---|
| 824 | return integral *= xDiff ; |
---|
| 825 | } |
---|
| 826 | |
---|
| 827 | ////////////////////////////////////////////////////////////////////// |
---|
| 828 | // |
---|
| 829 | // Method involving Laguerre polynomials |
---|
| 830 | // |
---|
| 831 | ////////////////////////////////////////////////////////////////////// |
---|
| 832 | // |
---|
| 833 | // Integral from zero to infinity of std::pow(x,alpha)*std::exp(-x)*f(x). |
---|
| 834 | // The value of nLaguerre sets the accuracy. |
---|
| 835 | // The function creates arrays fAbscissa[0,..,nLaguerre-1] and |
---|
| 836 | // fWeight[0,..,nLaguerre-1] . |
---|
| 837 | // Convenient for using with class object 'typeT' and (typeT.*f) function |
---|
| 838 | // (T::f) |
---|
| 839 | |
---|
| 840 | template <class T, class F> |
---|
| 841 | G4double G4Integrator<T,F>::Laguerre( T& typeT, F f, G4double alpha, |
---|
| 842 | G4int nLaguerre ) |
---|
| 843 | { |
---|
| 844 | const G4double tolerance = 1.0e-10 ; |
---|
| 845 | const G4int maxNumber = 12 ; |
---|
| 846 | G4int i, j, k ; |
---|
| 847 | G4double newton=0., newton1, temp1, temp2, temp3, temp, cofi ; |
---|
| 848 | G4double integral = 0.0 ; |
---|
| 849 | |
---|
| 850 | G4int fNumber = nLaguerre ; |
---|
| 851 | G4double* fAbscissa = new G4double[fNumber] ; |
---|
| 852 | G4double* fWeight = new G4double[fNumber] ; |
---|
| 853 | |
---|
| 854 | for(i=1;i<=fNumber;i++) // Loop over the desired roots |
---|
| 855 | { |
---|
| 856 | if(i == 1) |
---|
| 857 | { |
---|
| 858 | newton = (1.0 + alpha)*(3.0 + 0.92*alpha) |
---|
| 859 | / (1.0 + 2.4*fNumber + 1.8*alpha) ; |
---|
| 860 | } |
---|
| 861 | else if(i == 2) |
---|
| 862 | { |
---|
| 863 | newton += (15.0 + 6.25*alpha)/(1.0 + 0.9*alpha + 2.5*fNumber) ; |
---|
| 864 | } |
---|
| 865 | else |
---|
| 866 | { |
---|
| 867 | cofi = i - 2 ; |
---|
| 868 | newton += ((1.0+2.55*cofi)/(1.9*cofi) |
---|
| 869 | + 1.26*cofi*alpha/(1.0+3.5*cofi)) |
---|
| 870 | * (newton - fAbscissa[i-3])/(1.0 + 0.3*alpha) ; |
---|
| 871 | } |
---|
| 872 | for(k=1;k<=maxNumber;k++) |
---|
| 873 | { |
---|
| 874 | temp1 = 1.0 ; |
---|
| 875 | temp2 = 0.0 ; |
---|
| 876 | |
---|
| 877 | for(j=1;j<=fNumber;j++) |
---|
| 878 | { |
---|
| 879 | temp3 = temp2 ; |
---|
| 880 | temp2 = temp1 ; |
---|
| 881 | temp1 = ((2*j - 1 + alpha - newton)*temp2 - (j - 1 + alpha)*temp3)/j ; |
---|
| 882 | } |
---|
| 883 | temp = (fNumber*temp1 - (fNumber +alpha)*temp2)/newton ; |
---|
| 884 | newton1 = newton ; |
---|
| 885 | newton = newton1 - temp1/temp ; |
---|
| 886 | |
---|
| 887 | if(std::fabs(newton - newton1) <= tolerance) |
---|
| 888 | { |
---|
| 889 | break ; |
---|
| 890 | } |
---|
| 891 | } |
---|
| 892 | if(k > maxNumber) |
---|
| 893 | { |
---|
| 894 | G4Exception("G4Integrator<T,F>::Laguerre(T,F, ...)", "Error", |
---|
| 895 | FatalException, "Too many (>12) iterations."); |
---|
| 896 | } |
---|
| 897 | |
---|
| 898 | fAbscissa[i-1] = newton ; |
---|
| 899 | fWeight[i-1] = -std::exp(GammaLogarithm(alpha + fNumber) - |
---|
| 900 | GammaLogarithm((G4double)fNumber))/(temp*fNumber*temp2) ; |
---|
| 901 | } |
---|
| 902 | |
---|
| 903 | // |
---|
| 904 | // Integral evaluation |
---|
| 905 | // |
---|
| 906 | |
---|
| 907 | for(i=0;i<fNumber;i++) |
---|
| 908 | { |
---|
| 909 | integral += fWeight[i]*(typeT.*f)(fAbscissa[i]) ; |
---|
| 910 | } |
---|
| 911 | delete[] fAbscissa; |
---|
| 912 | delete[] fWeight; |
---|
| 913 | return integral ; |
---|
| 914 | } |
---|
| 915 | |
---|
| 916 | |
---|
| 917 | |
---|
| 918 | ////////////////////////////////////////////////////////////////////// |
---|
| 919 | // |
---|
| 920 | // |
---|
| 921 | |
---|
| 922 | template <class T, class F> G4double |
---|
| 923 | G4Integrator<T,F>::Laguerre( T* ptrT, F f, G4double alpha, G4int nLaguerre ) |
---|
| 924 | { |
---|
| 925 | return Laguerre(*ptrT,f,alpha,nLaguerre) ; |
---|
| 926 | } |
---|
| 927 | |
---|
| 928 | //////////////////////////////////////////////////////////////////////// |
---|
| 929 | // |
---|
| 930 | // For use with global scope functions f |
---|
| 931 | |
---|
| 932 | template <class T, class F> G4double |
---|
| 933 | G4Integrator<T,F>::Laguerre( G4double (*f)(G4double), |
---|
| 934 | G4double alpha, G4int nLaguerre ) |
---|
| 935 | { |
---|
| 936 | const G4double tolerance = 1.0e-10 ; |
---|
| 937 | const G4int maxNumber = 12 ; |
---|
| 938 | G4int i, j, k ; |
---|
| 939 | G4double newton=0., newton1, temp1, temp2, temp3, temp, cofi ; |
---|
| 940 | G4double integral = 0.0 ; |
---|
| 941 | |
---|
| 942 | G4int fNumber = nLaguerre ; |
---|
| 943 | G4double* fAbscissa = new G4double[fNumber] ; |
---|
| 944 | G4double* fWeight = new G4double[fNumber] ; |
---|
| 945 | |
---|
| 946 | for(i=1;i<=fNumber;i++) // Loop over the desired roots |
---|
| 947 | { |
---|
| 948 | if(i == 1) |
---|
| 949 | { |
---|
| 950 | newton = (1.0 + alpha)*(3.0 + 0.92*alpha) |
---|
| 951 | / (1.0 + 2.4*fNumber + 1.8*alpha) ; |
---|
| 952 | } |
---|
| 953 | else if(i == 2) |
---|
| 954 | { |
---|
| 955 | newton += (15.0 + 6.25*alpha)/(1.0 + 0.9*alpha + 2.5*fNumber) ; |
---|
| 956 | } |
---|
| 957 | else |
---|
| 958 | { |
---|
| 959 | cofi = i - 2 ; |
---|
| 960 | newton += ((1.0+2.55*cofi)/(1.9*cofi) |
---|
| 961 | + 1.26*cofi*alpha/(1.0+3.5*cofi)) |
---|
| 962 | * (newton - fAbscissa[i-3])/(1.0 + 0.3*alpha) ; |
---|
| 963 | } |
---|
| 964 | for(k=1;k<=maxNumber;k++) |
---|
| 965 | { |
---|
| 966 | temp1 = 1.0 ; |
---|
| 967 | temp2 = 0.0 ; |
---|
| 968 | |
---|
| 969 | for(j=1;j<=fNumber;j++) |
---|
| 970 | { |
---|
| 971 | temp3 = temp2 ; |
---|
| 972 | temp2 = temp1 ; |
---|
| 973 | temp1 = ((2*j - 1 + alpha - newton)*temp2 - (j - 1 + alpha)*temp3)/j ; |
---|
| 974 | } |
---|
| 975 | temp = (fNumber*temp1 - (fNumber +alpha)*temp2)/newton ; |
---|
| 976 | newton1 = newton ; |
---|
| 977 | newton = newton1 - temp1/temp ; |
---|
| 978 | |
---|
| 979 | if(std::fabs(newton - newton1) <= tolerance) |
---|
| 980 | { |
---|
| 981 | break ; |
---|
| 982 | } |
---|
| 983 | } |
---|
| 984 | if(k > maxNumber) |
---|
| 985 | { |
---|
| 986 | G4Exception("G4Integrator<T,F>::Laguerre( ...)", "Error", |
---|
| 987 | FatalException, "Too many (>12) iterations."); |
---|
| 988 | } |
---|
| 989 | |
---|
| 990 | fAbscissa[i-1] = newton ; |
---|
| 991 | fWeight[i-1] = -std::exp(GammaLogarithm(alpha + fNumber) - |
---|
| 992 | GammaLogarithm((G4double)fNumber))/(temp*fNumber*temp2) ; |
---|
| 993 | } |
---|
| 994 | |
---|
| 995 | // |
---|
| 996 | // Integral evaluation |
---|
| 997 | // |
---|
| 998 | |
---|
| 999 | for(i=0;i<fNumber;i++) |
---|
| 1000 | { |
---|
| 1001 | integral += fWeight[i]*(*f)(fAbscissa[i]) ; |
---|
| 1002 | } |
---|
| 1003 | delete[] fAbscissa; |
---|
| 1004 | delete[] fWeight; |
---|
| 1005 | return integral ; |
---|
| 1006 | } |
---|
| 1007 | |
---|
| 1008 | /////////////////////////////////////////////////////////////////////// |
---|
| 1009 | // |
---|
| 1010 | // Auxiliary function which returns the value of std::log(gamma-function(x)) |
---|
| 1011 | // Returns the value ln(Gamma(xx) for xx > 0. Full accuracy is obtained for |
---|
| 1012 | // xx > 1. For 0 < xx < 1. the reflection formula (6.1.4) can be used first. |
---|
| 1013 | // (Adapted from Numerical Recipes in C) |
---|
| 1014 | // |
---|
| 1015 | |
---|
| 1016 | template <class T, class F> |
---|
| 1017 | G4double G4Integrator<T,F>::GammaLogarithm(G4double xx) |
---|
| 1018 | { |
---|
| 1019 | static G4double cof[6] = { 76.18009172947146, -86.50532032941677, |
---|
| 1020 | 24.01409824083091, -1.231739572450155, |
---|
| 1021 | 0.1208650973866179e-2, -0.5395239384953e-5 } ; |
---|
| 1022 | register G4int j; |
---|
| 1023 | G4double x = xx - 1.0 ; |
---|
| 1024 | G4double tmp = x + 5.5 ; |
---|
| 1025 | tmp -= (x + 0.5) * std::log(tmp) ; |
---|
| 1026 | G4double ser = 1.000000000190015 ; |
---|
| 1027 | |
---|
| 1028 | for ( j = 0; j <= 5; j++ ) |
---|
| 1029 | { |
---|
| 1030 | x += 1.0 ; |
---|
| 1031 | ser += cof[j]/x ; |
---|
| 1032 | } |
---|
| 1033 | return -tmp + std::log(2.5066282746310005*ser) ; |
---|
| 1034 | } |
---|
| 1035 | |
---|
| 1036 | /////////////////////////////////////////////////////////////////////// |
---|
| 1037 | // |
---|
| 1038 | // Method involving Hermite polynomials |
---|
| 1039 | // |
---|
| 1040 | /////////////////////////////////////////////////////////////////////// |
---|
| 1041 | // |
---|
| 1042 | // |
---|
| 1043 | // Gauss-Hermite method for integration of std::exp(-x*x)*f(x) |
---|
| 1044 | // from minus infinity to plus infinity . |
---|
| 1045 | // |
---|
| 1046 | |
---|
| 1047 | template <class T, class F> |
---|
| 1048 | G4double G4Integrator<T,F>::Hermite( T& typeT, F f, G4int nHermite ) |
---|
| 1049 | { |
---|
| 1050 | const G4double tolerance = 1.0e-12 ; |
---|
| 1051 | const G4int maxNumber = 12 ; |
---|
| 1052 | |
---|
| 1053 | G4int i, j, k ; |
---|
| 1054 | G4double integral = 0.0 ; |
---|
| 1055 | G4double newton=0., newton1, temp1, temp2, temp3, temp ; |
---|
| 1056 | |
---|
| 1057 | G4double piInMinusQ = std::pow(pi,-0.25) ; // 1.0/std::sqrt(std::sqrt(pi)) ?? |
---|
| 1058 | |
---|
| 1059 | G4int fNumber = (nHermite +1)/2 ; |
---|
| 1060 | G4double* fAbscissa = new G4double[fNumber] ; |
---|
| 1061 | G4double* fWeight = new G4double[fNumber] ; |
---|
| 1062 | |
---|
| 1063 | for(i=1;i<=fNumber;i++) |
---|
| 1064 | { |
---|
| 1065 | if(i == 1) |
---|
| 1066 | { |
---|
| 1067 | newton = std::sqrt((G4double)(2*nHermite + 1)) - |
---|
| 1068 | 1.85575001*std::pow((G4double)(2*nHermite + 1),-0.16666999) ; |
---|
| 1069 | } |
---|
| 1070 | else if(i == 2) |
---|
| 1071 | { |
---|
| 1072 | newton -= 1.14001*std::pow((G4double)nHermite,0.425999)/newton ; |
---|
| 1073 | } |
---|
| 1074 | else if(i == 3) |
---|
| 1075 | { |
---|
| 1076 | newton = 1.86002*newton - 0.86002*fAbscissa[0] ; |
---|
| 1077 | } |
---|
| 1078 | else if(i == 4) |
---|
| 1079 | { |
---|
| 1080 | newton = 1.91001*newton - 0.91001*fAbscissa[1] ; |
---|
| 1081 | } |
---|
| 1082 | else |
---|
| 1083 | { |
---|
| 1084 | newton = 2.0*newton - fAbscissa[i - 3] ; |
---|
| 1085 | } |
---|
| 1086 | for(k=1;k<=maxNumber;k++) |
---|
| 1087 | { |
---|
| 1088 | temp1 = piInMinusQ ; |
---|
| 1089 | temp2 = 0.0 ; |
---|
| 1090 | |
---|
| 1091 | for(j=1;j<=nHermite;j++) |
---|
| 1092 | { |
---|
| 1093 | temp3 = temp2 ; |
---|
| 1094 | temp2 = temp1 ; |
---|
| 1095 | temp1 = newton*std::sqrt(2.0/j)*temp2 - |
---|
| 1096 | std::sqrt(((G4double)(j - 1))/j)*temp3 ; |
---|
| 1097 | } |
---|
| 1098 | temp = std::sqrt((G4double)2*nHermite)*temp2 ; |
---|
| 1099 | newton1 = newton ; |
---|
| 1100 | newton = newton1 - temp1/temp ; |
---|
| 1101 | |
---|
| 1102 | if(std::fabs(newton - newton1) <= tolerance) |
---|
| 1103 | { |
---|
| 1104 | break ; |
---|
| 1105 | } |
---|
| 1106 | } |
---|
| 1107 | if(k > maxNumber) |
---|
| 1108 | { |
---|
| 1109 | G4Exception("G4Integrator<T,F>::Hermite(T,F, ...)", "Error", |
---|
| 1110 | FatalException, "Too many (>12) iterations."); |
---|
| 1111 | } |
---|
| 1112 | fAbscissa[i-1] = newton ; |
---|
| 1113 | fWeight[i-1] = 2.0/(temp*temp) ; |
---|
| 1114 | } |
---|
| 1115 | |
---|
| 1116 | // |
---|
| 1117 | // Integral calculation |
---|
| 1118 | // |
---|
| 1119 | |
---|
| 1120 | for(i=0;i<fNumber;i++) |
---|
| 1121 | { |
---|
| 1122 | integral += fWeight[i]*( (typeT.*f)(fAbscissa[i]) + |
---|
| 1123 | (typeT.*f)(-fAbscissa[i]) ) ; |
---|
| 1124 | } |
---|
| 1125 | delete[] fAbscissa; |
---|
| 1126 | delete[] fWeight; |
---|
| 1127 | return integral ; |
---|
| 1128 | } |
---|
| 1129 | |
---|
| 1130 | |
---|
| 1131 | //////////////////////////////////////////////////////////////////////// |
---|
| 1132 | // |
---|
| 1133 | // For use with 'this' pointer |
---|
| 1134 | |
---|
| 1135 | template <class T, class F> |
---|
| 1136 | G4double G4Integrator<T,F>::Hermite( T* ptrT, F f, G4int n ) |
---|
| 1137 | { |
---|
| 1138 | return Hermite(*ptrT,f,n) ; |
---|
| 1139 | } |
---|
| 1140 | |
---|
| 1141 | //////////////////////////////////////////////////////////////////////// |
---|
| 1142 | // |
---|
| 1143 | // For use with global scope f |
---|
| 1144 | |
---|
| 1145 | template <class T, class F> |
---|
| 1146 | G4double G4Integrator<T,F>::Hermite( G4double (*f)(G4double), G4int nHermite) |
---|
| 1147 | { |
---|
| 1148 | const G4double tolerance = 1.0e-12 ; |
---|
| 1149 | const G4int maxNumber = 12 ; |
---|
| 1150 | |
---|
| 1151 | G4int i, j, k ; |
---|
| 1152 | G4double integral = 0.0 ; |
---|
| 1153 | G4double newton=0., newton1, temp1, temp2, temp3, temp ; |
---|
| 1154 | |
---|
| 1155 | G4double piInMinusQ = std::pow(pi,-0.25) ; // 1.0/std::sqrt(std::sqrt(pi)) ?? |
---|
| 1156 | |
---|
| 1157 | G4int fNumber = (nHermite +1)/2 ; |
---|
| 1158 | G4double* fAbscissa = new G4double[fNumber] ; |
---|
| 1159 | G4double* fWeight = new G4double[fNumber] ; |
---|
| 1160 | |
---|
| 1161 | for(i=1;i<=fNumber;i++) |
---|
| 1162 | { |
---|
| 1163 | if(i == 1) |
---|
| 1164 | { |
---|
| 1165 | newton = std::sqrt((G4double)(2*nHermite + 1)) - |
---|
| 1166 | 1.85575001*std::pow((G4double)(2*nHermite + 1),-0.16666999) ; |
---|
| 1167 | } |
---|
| 1168 | else if(i == 2) |
---|
| 1169 | { |
---|
| 1170 | newton -= 1.14001*std::pow((G4double)nHermite,0.425999)/newton ; |
---|
| 1171 | } |
---|
| 1172 | else if(i == 3) |
---|
| 1173 | { |
---|
| 1174 | newton = 1.86002*newton - 0.86002*fAbscissa[0] ; |
---|
| 1175 | } |
---|
| 1176 | else if(i == 4) |
---|
| 1177 | { |
---|
| 1178 | newton = 1.91001*newton - 0.91001*fAbscissa[1] ; |
---|
| 1179 | } |
---|
| 1180 | else |
---|
| 1181 | { |
---|
| 1182 | newton = 2.0*newton - fAbscissa[i - 3] ; |
---|
| 1183 | } |
---|
| 1184 | for(k=1;k<=maxNumber;k++) |
---|
| 1185 | { |
---|
| 1186 | temp1 = piInMinusQ ; |
---|
| 1187 | temp2 = 0.0 ; |
---|
| 1188 | |
---|
| 1189 | for(j=1;j<=nHermite;j++) |
---|
| 1190 | { |
---|
| 1191 | temp3 = temp2 ; |
---|
| 1192 | temp2 = temp1 ; |
---|
| 1193 | temp1 = newton*std::sqrt(2.0/j)*temp2 - |
---|
| 1194 | std::sqrt(((G4double)(j - 1))/j)*temp3 ; |
---|
| 1195 | } |
---|
| 1196 | temp = std::sqrt((G4double)2*nHermite)*temp2 ; |
---|
| 1197 | newton1 = newton ; |
---|
| 1198 | newton = newton1 - temp1/temp ; |
---|
| 1199 | |
---|
| 1200 | if(std::fabs(newton - newton1) <= tolerance) |
---|
| 1201 | { |
---|
| 1202 | break ; |
---|
| 1203 | } |
---|
| 1204 | } |
---|
| 1205 | if(k > maxNumber) |
---|
| 1206 | { |
---|
| 1207 | G4Exception("G4Integrator<T,F>::Hermite(...)", "Error", |
---|
| 1208 | FatalException, "Too many (>12) iterations."); |
---|
| 1209 | } |
---|
| 1210 | fAbscissa[i-1] = newton ; |
---|
| 1211 | fWeight[i-1] = 2.0/(temp*temp) ; |
---|
| 1212 | } |
---|
| 1213 | |
---|
| 1214 | // |
---|
| 1215 | // Integral calculation |
---|
| 1216 | // |
---|
| 1217 | |
---|
| 1218 | for(i=0;i<fNumber;i++) |
---|
| 1219 | { |
---|
| 1220 | integral += fWeight[i]*( (*f)(fAbscissa[i]) + (*f)(-fAbscissa[i]) ) ; |
---|
| 1221 | } |
---|
| 1222 | delete[] fAbscissa; |
---|
| 1223 | delete[] fWeight; |
---|
| 1224 | return integral ; |
---|
| 1225 | } |
---|
| 1226 | |
---|
| 1227 | //////////////////////////////////////////////////////////////////////////// |
---|
| 1228 | // |
---|
| 1229 | // Method involving Jacobi polynomials |
---|
| 1230 | // |
---|
| 1231 | //////////////////////////////////////////////////////////////////////////// |
---|
| 1232 | // |
---|
| 1233 | // Gauss-Jacobi method for integration of ((1-x)^alpha)*((1+x)^beta)*f(x) |
---|
| 1234 | // from minus unit to plus unit . |
---|
| 1235 | // |
---|
| 1236 | |
---|
| 1237 | template <class T, class F> |
---|
| 1238 | G4double G4Integrator<T,F>::Jacobi( T& typeT, F f, G4double alpha, |
---|
| 1239 | G4double beta, G4int nJacobi) |
---|
| 1240 | { |
---|
| 1241 | const G4double tolerance = 1.0e-12 ; |
---|
| 1242 | const G4double maxNumber = 12 ; |
---|
| 1243 | G4int i, k, j ; |
---|
| 1244 | G4double alphaBeta, alphaReduced, betaReduced, root1=0., root2=0., root3=0. ; |
---|
| 1245 | G4double a, b, c, newton1, newton2, newton3, newton, temp, root=0., rootTemp ; |
---|
| 1246 | |
---|
| 1247 | G4int fNumber = nJacobi ; |
---|
| 1248 | G4double* fAbscissa = new G4double[fNumber] ; |
---|
| 1249 | G4double* fWeight = new G4double[fNumber] ; |
---|
| 1250 | |
---|
| 1251 | for (i=1;i<=nJacobi;i++) |
---|
| 1252 | { |
---|
| 1253 | if (i == 1) |
---|
| 1254 | { |
---|
| 1255 | alphaReduced = alpha/nJacobi ; |
---|
| 1256 | betaReduced = beta/nJacobi ; |
---|
| 1257 | root1 = (1.0+alpha)*(2.78002/(4.0+nJacobi*nJacobi)+ |
---|
| 1258 | 0.767999*alphaReduced/nJacobi) ; |
---|
| 1259 | root2 = 1.0+1.48*alphaReduced+0.96002*betaReduced + |
---|
| 1260 | 0.451998*alphaReduced*alphaReduced + |
---|
| 1261 | 0.83001*alphaReduced*betaReduced ; |
---|
| 1262 | root = 1.0-root1/root2 ; |
---|
| 1263 | } |
---|
| 1264 | else if (i == 2) |
---|
| 1265 | { |
---|
| 1266 | root1=(4.1002+alpha)/((1.0+alpha)*(1.0+0.155998*alpha)) ; |
---|
| 1267 | root2=1.0+0.06*(nJacobi-8.0)*(1.0+0.12*alpha)/nJacobi ; |
---|
| 1268 | root3=1.0+0.012002*beta*(1.0+0.24997*std::fabs(alpha))/nJacobi ; |
---|
| 1269 | root -= (1.0-root)*root1*root2*root3 ; |
---|
| 1270 | } |
---|
| 1271 | else if (i == 3) |
---|
| 1272 | { |
---|
| 1273 | root1=(1.67001+0.27998*alpha)/(1.0+0.37002*alpha) ; |
---|
| 1274 | root2=1.0+0.22*(nJacobi-8.0)/nJacobi ; |
---|
| 1275 | root3=1.0+8.0*beta/((6.28001+beta)*nJacobi*nJacobi) ; |
---|
| 1276 | root -= (fAbscissa[0]-root)*root1*root2*root3 ; |
---|
| 1277 | } |
---|
| 1278 | else if (i == nJacobi-1) |
---|
| 1279 | { |
---|
| 1280 | root1=(1.0+0.235002*beta)/(0.766001+0.118998*beta) ; |
---|
| 1281 | root2=1.0/(1.0+0.639002*(nJacobi-4.0)/(1.0+0.71001*(nJacobi-4.0))) ; |
---|
| 1282 | root3=1.0/(1.0+20.0*alpha/((7.5+alpha)*nJacobi*nJacobi)) ; |
---|
| 1283 | root += (root-fAbscissa[nJacobi-4])*root1*root2*root3 ; |
---|
| 1284 | } |
---|
| 1285 | else if (i == nJacobi) |
---|
| 1286 | { |
---|
| 1287 | root1 = (1.0+0.37002*beta)/(1.67001+0.27998*beta) ; |
---|
| 1288 | root2 = 1.0/(1.0+0.22*(nJacobi-8.0)/nJacobi) ; |
---|
| 1289 | root3 = 1.0/(1.0+8.0*alpha/((6.28002+alpha)*nJacobi*nJacobi)) ; |
---|
| 1290 | root += (root-fAbscissa[nJacobi-3])*root1*root2*root3 ; |
---|
| 1291 | } |
---|
| 1292 | else |
---|
| 1293 | { |
---|
| 1294 | root = 3.0*fAbscissa[i-2]-3.0*fAbscissa[i-3]+fAbscissa[i-4] ; |
---|
| 1295 | } |
---|
| 1296 | alphaBeta = alpha + beta ; |
---|
| 1297 | for (k=1;k<=maxNumber;k++) |
---|
| 1298 | { |
---|
| 1299 | temp = 2.0 + alphaBeta ; |
---|
| 1300 | newton1 = (alpha-beta+temp*root)/2.0 ; |
---|
| 1301 | newton2 = 1.0 ; |
---|
| 1302 | for (j=2;j<=nJacobi;j++) |
---|
| 1303 | { |
---|
| 1304 | newton3 = newton2 ; |
---|
| 1305 | newton2 = newton1 ; |
---|
| 1306 | temp = 2*j+alphaBeta ; |
---|
| 1307 | a = 2*j*(j+alphaBeta)*(temp-2.0) ; |
---|
| 1308 | b = (temp-1.0)*(alpha*alpha-beta*beta+temp*(temp-2.0)*root) ; |
---|
| 1309 | c = 2.0*(j-1+alpha)*(j-1+beta)*temp ; |
---|
| 1310 | newton1 = (b*newton2-c*newton3)/a ; |
---|
| 1311 | } |
---|
| 1312 | newton = (nJacobi*(alpha - beta - temp*root)*newton1 + |
---|
| 1313 | 2.0*(nJacobi + alpha)*(nJacobi + beta)*newton2)/ |
---|
| 1314 | (temp*(1.0 - root*root)) ; |
---|
| 1315 | rootTemp = root ; |
---|
| 1316 | root = rootTemp - newton1/newton ; |
---|
| 1317 | if (std::fabs(root-rootTemp) <= tolerance) |
---|
| 1318 | { |
---|
| 1319 | break ; |
---|
| 1320 | } |
---|
| 1321 | } |
---|
| 1322 | if (k > maxNumber) |
---|
| 1323 | { |
---|
| 1324 | G4Exception("G4Integrator<T,F>::Jacobi(T,F, ...)", "Error", |
---|
| 1325 | FatalException, "Too many (>12) iterations."); |
---|
| 1326 | } |
---|
| 1327 | fAbscissa[i-1] = root ; |
---|
| 1328 | fWeight[i-1] = std::exp(GammaLogarithm((G4double)(alpha+nJacobi)) + |
---|
| 1329 | GammaLogarithm((G4double)(beta+nJacobi)) - |
---|
| 1330 | GammaLogarithm((G4double)(nJacobi+1.0)) - |
---|
| 1331 | GammaLogarithm((G4double)(nJacobi + alphaBeta + 1.0))) |
---|
| 1332 | *temp*std::pow(2.0,alphaBeta)/(newton*newton2) ; |
---|
| 1333 | } |
---|
| 1334 | |
---|
| 1335 | // |
---|
| 1336 | // Calculation of the integral |
---|
| 1337 | // |
---|
| 1338 | |
---|
| 1339 | G4double integral = 0.0 ; |
---|
| 1340 | for(i=0;i<fNumber;i++) |
---|
| 1341 | { |
---|
| 1342 | integral += fWeight[i]*(typeT.*f)(fAbscissa[i]) ; |
---|
| 1343 | } |
---|
| 1344 | delete[] fAbscissa; |
---|
| 1345 | delete[] fWeight; |
---|
| 1346 | return integral ; |
---|
| 1347 | } |
---|
| 1348 | |
---|
| 1349 | |
---|
| 1350 | ///////////////////////////////////////////////////////////////////////// |
---|
| 1351 | // |
---|
| 1352 | // For use with 'this' pointer |
---|
| 1353 | |
---|
| 1354 | template <class T, class F> |
---|
| 1355 | G4double G4Integrator<T,F>::Jacobi( T* ptrT, F f, G4double alpha, |
---|
| 1356 | G4double beta, G4int n) |
---|
| 1357 | { |
---|
| 1358 | return Jacobi(*ptrT,f,alpha,beta,n) ; |
---|
| 1359 | } |
---|
| 1360 | |
---|
| 1361 | ///////////////////////////////////////////////////////////////////////// |
---|
| 1362 | // |
---|
| 1363 | // For use with global scope f |
---|
| 1364 | |
---|
| 1365 | template <class T, class F> |
---|
| 1366 | G4double G4Integrator<T,F>::Jacobi( G4double (*f)(G4double), G4double alpha, |
---|
| 1367 | G4double beta, G4int nJacobi) |
---|
| 1368 | { |
---|
| 1369 | const G4double tolerance = 1.0e-12 ; |
---|
| 1370 | const G4double maxNumber = 12 ; |
---|
| 1371 | G4int i, k, j ; |
---|
| 1372 | G4double alphaBeta, alphaReduced, betaReduced, root1=0., root2=0., root3=0. ; |
---|
| 1373 | G4double a, b, c, newton1, newton2, newton3, newton, temp, root=0., rootTemp ; |
---|
| 1374 | |
---|
| 1375 | G4int fNumber = nJacobi ; |
---|
| 1376 | G4double* fAbscissa = new G4double[fNumber] ; |
---|
| 1377 | G4double* fWeight = new G4double[fNumber] ; |
---|
| 1378 | |
---|
| 1379 | for (i=1;i<=nJacobi;i++) |
---|
| 1380 | { |
---|
| 1381 | if (i == 1) |
---|
| 1382 | { |
---|
| 1383 | alphaReduced = alpha/nJacobi ; |
---|
| 1384 | betaReduced = beta/nJacobi ; |
---|
| 1385 | root1 = (1.0+alpha)*(2.78002/(4.0+nJacobi*nJacobi)+ |
---|
| 1386 | 0.767999*alphaReduced/nJacobi) ; |
---|
| 1387 | root2 = 1.0+1.48*alphaReduced+0.96002*betaReduced + |
---|
| 1388 | 0.451998*alphaReduced*alphaReduced + |
---|
| 1389 | 0.83001*alphaReduced*betaReduced ; |
---|
| 1390 | root = 1.0-root1/root2 ; |
---|
| 1391 | } |
---|
| 1392 | else if (i == 2) |
---|
| 1393 | { |
---|
| 1394 | root1=(4.1002+alpha)/((1.0+alpha)*(1.0+0.155998*alpha)) ; |
---|
| 1395 | root2=1.0+0.06*(nJacobi-8.0)*(1.0+0.12*alpha)/nJacobi ; |
---|
| 1396 | root3=1.0+0.012002*beta*(1.0+0.24997*std::fabs(alpha))/nJacobi ; |
---|
| 1397 | root -= (1.0-root)*root1*root2*root3 ; |
---|
| 1398 | } |
---|
| 1399 | else if (i == 3) |
---|
| 1400 | { |
---|
| 1401 | root1=(1.67001+0.27998*alpha)/(1.0+0.37002*alpha) ; |
---|
| 1402 | root2=1.0+0.22*(nJacobi-8.0)/nJacobi ; |
---|
| 1403 | root3=1.0+8.0*beta/((6.28001+beta)*nJacobi*nJacobi) ; |
---|
| 1404 | root -= (fAbscissa[0]-root)*root1*root2*root3 ; |
---|
| 1405 | } |
---|
| 1406 | else if (i == nJacobi-1) |
---|
| 1407 | { |
---|
| 1408 | root1=(1.0+0.235002*beta)/(0.766001+0.118998*beta) ; |
---|
| 1409 | root2=1.0/(1.0+0.639002*(nJacobi-4.0)/(1.0+0.71001*(nJacobi-4.0))) ; |
---|
| 1410 | root3=1.0/(1.0+20.0*alpha/((7.5+alpha)*nJacobi*nJacobi)) ; |
---|
| 1411 | root += (root-fAbscissa[nJacobi-4])*root1*root2*root3 ; |
---|
| 1412 | } |
---|
| 1413 | else if (i == nJacobi) |
---|
| 1414 | { |
---|
| 1415 | root1 = (1.0+0.37002*beta)/(1.67001+0.27998*beta) ; |
---|
| 1416 | root2 = 1.0/(1.0+0.22*(nJacobi-8.0)/nJacobi) ; |
---|
| 1417 | root3 = 1.0/(1.0+8.0*alpha/((6.28002+alpha)*nJacobi*nJacobi)) ; |
---|
| 1418 | root += (root-fAbscissa[nJacobi-3])*root1*root2*root3 ; |
---|
| 1419 | } |
---|
| 1420 | else |
---|
| 1421 | { |
---|
| 1422 | root = 3.0*fAbscissa[i-2]-3.0*fAbscissa[i-3]+fAbscissa[i-4] ; |
---|
| 1423 | } |
---|
| 1424 | alphaBeta = alpha + beta ; |
---|
| 1425 | for (k=1;k<=maxNumber;k++) |
---|
| 1426 | { |
---|
| 1427 | temp = 2.0 + alphaBeta ; |
---|
| 1428 | newton1 = (alpha-beta+temp*root)/2.0 ; |
---|
| 1429 | newton2 = 1.0 ; |
---|
| 1430 | for (j=2;j<=nJacobi;j++) |
---|
| 1431 | { |
---|
| 1432 | newton3 = newton2 ; |
---|
| 1433 | newton2 = newton1 ; |
---|
| 1434 | temp = 2*j+alphaBeta ; |
---|
| 1435 | a = 2*j*(j+alphaBeta)*(temp-2.0) ; |
---|
| 1436 | b = (temp-1.0)*(alpha*alpha-beta*beta+temp*(temp-2.0)*root) ; |
---|
| 1437 | c = 2.0*(j-1+alpha)*(j-1+beta)*temp ; |
---|
| 1438 | newton1 = (b*newton2-c*newton3)/a ; |
---|
| 1439 | } |
---|
| 1440 | newton = (nJacobi*(alpha - beta - temp*root)*newton1 + |
---|
| 1441 | 2.0*(nJacobi + alpha)*(nJacobi + beta)*newton2) / |
---|
| 1442 | (temp*(1.0 - root*root)) ; |
---|
| 1443 | rootTemp = root ; |
---|
| 1444 | root = rootTemp - newton1/newton ; |
---|
| 1445 | if (std::fabs(root-rootTemp) <= tolerance) |
---|
| 1446 | { |
---|
| 1447 | break ; |
---|
| 1448 | } |
---|
| 1449 | } |
---|
| 1450 | if (k > maxNumber) |
---|
| 1451 | { |
---|
| 1452 | G4Exception("G4Integrator<T,F>::Jacobi(...)", "Error", |
---|
| 1453 | FatalException, "Too many (>12) iterations."); |
---|
| 1454 | } |
---|
| 1455 | fAbscissa[i-1] = root ; |
---|
| 1456 | fWeight[i-1] = |
---|
| 1457 | std::exp(GammaLogarithm((G4double)(alpha+nJacobi)) + |
---|
| 1458 | GammaLogarithm((G4double)(beta+nJacobi)) - |
---|
| 1459 | GammaLogarithm((G4double)(nJacobi+1.0)) - |
---|
| 1460 | GammaLogarithm((G4double)(nJacobi + alphaBeta + 1.0))) |
---|
| 1461 | *temp*std::pow(2.0,alphaBeta)/(newton*newton2); |
---|
| 1462 | } |
---|
| 1463 | |
---|
| 1464 | // |
---|
| 1465 | // Calculation of the integral |
---|
| 1466 | // |
---|
| 1467 | |
---|
| 1468 | G4double integral = 0.0 ; |
---|
| 1469 | for(i=0;i<fNumber;i++) |
---|
| 1470 | { |
---|
| 1471 | integral += fWeight[i]*(*f)(fAbscissa[i]) ; |
---|
| 1472 | } |
---|
| 1473 | delete[] fAbscissa; |
---|
| 1474 | delete[] fWeight; |
---|
| 1475 | return integral ; |
---|
| 1476 | } |
---|
| 1477 | |
---|
| 1478 | // |
---|
| 1479 | // |
---|
| 1480 | /////////////////////////////////////////////////////////////////// |
---|