| 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: G4GaussLegendreQ.cc,v 1.8 2007/11/13 17:35:06 gcosmo Exp $
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| 28 | // GEANT4 tag $Name: HEAD $
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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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