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1 // 1 // 2 // ******************************************* 2 // ******************************************************************** 3 // * License and Disclaimer 3 // * License and Disclaimer * 4 // * 4 // * * 5 // * The Geant4 software is copyright of th 5 // * The Geant4 software is copyright of the Copyright Holders of * 6 // * the Geant4 Collaboration. It is provided 6 // * the Geant4 Collaboration. It is provided under the terms and * 7 // * conditions of the Geant4 Software License 7 // * conditions of the Geant4 Software License, included in the file * 8 // * LICENSE and available at http://cern.ch/ 8 // * LICENSE and available at http://cern.ch/geant4/license . These * 9 // * include a list of copyright holders. 9 // * include a list of copyright holders. * 10 // * 10 // * * 11 // * Neither the authors of this software syst 11 // * Neither the authors of this software system, nor their employing * 12 // * institutes,nor the agencies providing fin 12 // * institutes,nor the agencies providing financial support for this * 13 // * work make any representation or warran 13 // * work make any representation or warranty, express or implied, * 14 // * regarding this software system or assum 14 // * regarding this software system or assume any liability for its * 15 // * use. Please see the license in the file 15 // * use. Please see the license in the file LICENSE and URL above * 16 // * for the full disclaimer and the limitatio 16 // * for the full disclaimer and the limitation of liability. * 17 // * 17 // * * 18 // * This code implementation is the result 18 // * This code implementation is the result of the scientific and * 19 // * technical work of the GEANT4 collaboratio 19 // * technical work of the GEANT4 collaboration. * 20 // * By using, copying, modifying or distri 20 // * By using, copying, modifying or distributing the software (or * 21 // * any work based on the software) you ag 21 // * any work based on the software) you agree to acknowledge its * 22 // * use in resulting scientific publicati 22 // * use in resulting scientific publications, and indicate your * 23 // * acceptance of all terms of the Geant4 Sof 23 // * acceptance of all terms of the Geant4 Software license. * 24 // ******************************************* 24 // ******************************************************************** 25 // 25 // 26 // G4Integrator inline methods implementation << 27 // 26 // 28 // Author: V.Grichine, 04.09.1999 - First impl << 27 // $Id: G4Integrator.icc,v 1.13 2006/06/29 18:59:47 gunter Exp $ 29 // G4SimpleIntegration class with H.P. << 28 // GEANT4 tag $Name: geant4-09-00 $ 30 // E.TCherniaev advises << 29 // 31 // ------------------------------------------- << 30 // Implementation of G4Integrator methods. >> 31 // >> 32 // 32 33 33 ////////////////////////////////////////////// 34 ///////////////////////////////////////////////////////////////////// 34 // 35 // 35 // Sympson integration method 36 // Sympson integration method 36 // 37 // 37 ////////////////////////////////////////////// 38 ///////////////////////////////////////////////////////////////////// 38 // 39 // 39 // Integration of class member functions T::f << 40 // Integration of class member functions T::f by Simpson method. 40 41 41 template <class T, class F> << 42 template <class T, class F> 42 G4double G4Integrator<T, F>::Simpson(T& typeT, << 43 G4double G4Integrator<T,F>::Simpson( T& typeT, 43 G4double << 44 F f, 44 { << 45 G4double xInitial, 45 G4int i; << 46 G4double xFinal, 46 G4double step = (xFinal - xInitial) / itera << 47 G4int iterationNumber ) 47 G4double x = xInitial; << 48 { 48 G4double xPlus = xInitial + 0.5 * step; << 49 G4int i ; 49 G4double mean = ((typeT.*f)(xInitial) + (ty << 50 G4double step = (xFinal - xInitial)/iterationNumber ; 50 G4double sum = (typeT.*f)(xPlus); << 51 G4double x = xInitial ; 51 << 52 G4double xPlus = xInitial + 0.5*step ; 52 for(i = 1; i < iterationNumber; ++i) << 53 G4double mean = ( (typeT.*f)(xInitial) + (typeT.*f)(xFinal) )*0.5 ; 53 { << 54 G4double sum = (typeT.*f)(xPlus) ; 54 x += step; << 55 55 xPlus += step; << 56 for(i=1;i<iterationNumber;i++) 56 mean += (typeT.*f)(x); << 57 { 57 sum += (typeT.*f)(xPlus); << 58 x += step ; 58 } << 59 xPlus += step ; 59 mean += 2.0 * sum; << 60 mean += (typeT.*f)(x) ; >> 61 sum += (typeT.*f)(xPlus) ; >> 62 } >> 63 mean += 2.0*sum ; 60 64 61 return mean * step / 3.0; << 65 return mean*step/3.0 ; 62 } 66 } 63 67 64 ////////////////////////////////////////////// 68 ///////////////////////////////////////////////////////////////////// 65 // 69 // 66 // Integration of class member functions T::f 70 // Integration of class member functions T::f by Simpson method. 67 // Convenient to use with 'this' pointer 71 // Convenient to use with 'this' pointer 68 72 69 template <class T, class F> << 73 template <class T, class F> 70 G4double G4Integrator<T, F>::Simpson(T* ptrT, << 74 G4double G4Integrator<T,F>::Simpson( T* ptrT, 71 G4double << 75 F f, 72 { << 76 G4double xInitial, 73 G4int i; << 77 G4double xFinal, 74 G4double step = (xFinal - xInitial) / itera << 78 G4int iterationNumber ) 75 G4double x = xInitial; << 79 { 76 G4double xPlus = xInitial + 0.5 * step; << 80 G4int i ; 77 G4double mean = ((ptrT->*f)(xInitial) + (pt << 81 G4double step = (xFinal - xInitial)/iterationNumber ; 78 G4double sum = (ptrT->*f)(xPlus); << 82 G4double x = xInitial ; >> 83 G4double xPlus = xInitial + 0.5*step ; >> 84 G4double mean = ( (ptrT->*f)(xInitial) + (ptrT->*f)(xFinal) )*0.5 ; >> 85 G4double sum = (ptrT->*f)(xPlus) ; >> 86 >> 87 for(i=1;i<iterationNumber;i++) >> 88 { >> 89 x += step ; >> 90 xPlus += step ; >> 91 mean += (ptrT->*f)(x) ; >> 92 sum += (ptrT->*f)(xPlus) ; >> 93 } >> 94 mean += 2.0*sum ; 79 95 80 for(i = 1; i < iterationNumber; ++i) << 96 return mean*step/3.0 ; 81 { << 82 x += step; << 83 xPlus += step; << 84 mean += (ptrT->*f)(x); << 85 sum += (ptrT->*f)(xPlus); << 86 } << 87 mean += 2.0 * sum; << 88 << 89 return mean * step / 3.0; << 90 } 97 } 91 98 92 ////////////////////////////////////////////// 99 ///////////////////////////////////////////////////////////////////// 93 // 100 // 94 // Integration of class member functions T::f 101 // Integration of class member functions T::f by Simpson method. 95 // Convenient to use, when function f is defin 102 // Convenient to use, when function f is defined in global scope, i.e. in main() 96 // program 103 // program 97 104 98 template <class T, class F> << 105 template <class T, class F> 99 G4double G4Integrator<T, F>::Simpson(G4double << 106 G4double G4Integrator<T,F>::Simpson( G4double (*f)(G4double), 100 G4double << 107 G4double xInitial, 101 { << 108 G4double xFinal, 102 G4int i; << 109 G4int iterationNumber ) 103 G4double step = (xFinal - xInitial) / itera << 110 { 104 G4double x = xInitial; << 111 G4int i ; 105 G4double xPlus = xInitial + 0.5 * step; << 112 G4double step = (xFinal - xInitial)/iterationNumber ; 106 G4double mean = ((*f)(xInitial) + (*f)(xFin << 113 G4double x = xInitial ; 107 G4double sum = (*f)(xPlus); << 114 G4double xPlus = xInitial + 0.5*step ; 108 << 115 G4double mean = ( (*f)(xInitial) + (*f)(xFinal) )*0.5 ; 109 for(i = 1; i < iterationNumber; ++i) << 116 G4double sum = (*f)(xPlus) ; 110 { << 117 111 x += step; << 118 for(i=1;i<iterationNumber;i++) 112 xPlus += step; << 119 { 113 mean += (*f)(x); << 120 x += step ; 114 sum += (*f)(xPlus); << 121 xPlus += step ; 115 } << 122 mean += (*f)(x) ; 116 mean += 2.0 * sum; << 123 sum += (*f)(xPlus) ; >> 124 } >> 125 mean += 2.0*sum ; 117 126 118 return mean * step / 3.0; << 127 return mean*step/3.0 ; 119 } 128 } 120 129 121 ////////////////////////////////////////////// 130 ////////////////////////////////////////////////////////////////////////// 122 // 131 // 123 // Adaptive Gauss method 132 // Adaptive Gauss method 124 // 133 // 125 ////////////////////////////////////////////// 134 ////////////////////////////////////////////////////////////////////////// 126 // 135 // 127 // 136 // 128 137 129 template <class T, class F> << 138 template <class T, class F> 130 G4double G4Integrator<T, F>::Gauss(T& typeT, F << 139 G4double G4Integrator<T,F>::Gauss( T& typeT, F f, 131 G4double xF << 140 G4double xInitial, G4double xFinal ) 132 { << 141 { 133 static const G4double root = 1.0 / std::sqrt << 142 static G4double root = 1.0/std::sqrt(3.0) ; 134 << 143 135 G4double xMean = (xInitial + xFinal) / 2.0; << 144 G4double xMean = (xInitial + xFinal)/2.0 ; 136 G4double Step = (xFinal - xInitial) / 2.0; << 145 G4double Step = (xFinal - xInitial)/2.0 ; 137 G4double delta = Step * root; << 146 G4double delta = Step*root ; 138 G4double sum = ((typeT.*f)(xMean + delta) << 147 G4double sum = ((typeT.*f)(xMean + delta) + 139 << 148 (typeT.*f)(xMean - delta)) ; 140 return sum * Step; << 149 >> 150 return sum*Step ; 141 } 151 } 142 152 143 ////////////////////////////////////////////// 153 ////////////////////////////////////////////////////////////////////// 144 // 154 // 145 // 155 // 146 156 147 template <class T, class F> << 157 template <class T, class F> G4double 148 G4double G4Integrator<T, F>::Gauss(T* ptrT, F << 158 G4Integrator<T,F>::Gauss( T* ptrT, F f, G4double a, G4double b ) 149 { 159 { 150 return Gauss(*ptrT, f, a, b); << 160 return Gauss(*ptrT,f,a,b) ; 151 } 161 } 152 162 153 ////////////////////////////////////////////// 163 /////////////////////////////////////////////////////////////////////// 154 // 164 // 155 // 165 // 156 166 157 template <class T, class F> 167 template <class T, class F> 158 G4double G4Integrator<T, F>::Gauss(G4double (* << 168 G4double G4Integrator<T,F>::Gauss( G4double (*f)(G4double), 159 G4double xF << 169 G4double xInitial, G4double xFinal) 160 { 170 { 161 static const G4double root = 1.0 / std::sqrt << 171 static G4double root = 1.0/std::sqrt(3.0) ; 162 << 172 163 G4double xMean = (xInitial + xFinal) / 2.0; << 173 G4double xMean = (xInitial + xFinal)/2.0 ; 164 G4double Step = (xFinal - xInitial) / 2.0; << 174 G4double Step = (xFinal - xInitial)/2.0 ; 165 G4double delta = Step * root; << 175 G4double delta = Step*root ; 166 G4double sum = ((*f)(xMean + delta) + (*f) << 176 G4double sum = ( (*f)(xMean + delta) + (*f)(xMean - delta) ) ; 167 << 177 168 return sum * Step; << 178 return sum*Step ; 169 } 179 } 170 180 171 ////////////////////////////////////////////// 181 /////////////////////////////////////////////////////////////////////////// 172 // 182 // 173 // 183 // 174 184 175 template <class T, class F> << 185 template <class T, class F> 176 void G4Integrator<T, F>::AdaptGauss(T& typeT, << 186 void G4Integrator<T,F>::AdaptGauss( T& typeT, F f, G4double xInitial, 177 G4double x << 187 G4double xFinal, G4double fTolerance, 178 G4double& << 188 G4double& sum, >> 189 G4int& depth ) >> 190 { >> 191 if(depth > 100) >> 192 { >> 193 G4cout<<"G4Integrator<T,F>::AdaptGauss: WARNING !!!"<<G4endl ; >> 194 G4cout<<"Function varies too rapidly to get stated accuracy in 100 steps " >> 195 <<G4endl ; >> 196 >> 197 return ; >> 198 } >> 199 G4double xMean = (xInitial + xFinal)/2.0 ; >> 200 G4double leftHalf = Gauss(typeT,f,xInitial,xMean) ; >> 201 G4double rightHalf = Gauss(typeT,f,xMean,xFinal) ; >> 202 G4double full = Gauss(typeT,f,xInitial,xFinal) ; >> 203 if(std::fabs(leftHalf+rightHalf-full) < fTolerance) >> 204 { >> 205 sum += full ; >> 206 } >> 207 else >> 208 { >> 209 depth++ ; >> 210 AdaptGauss(typeT,f,xInitial,xMean,fTolerance,sum,depth) ; >> 211 AdaptGauss(typeT,f,xMean,xFinal,fTolerance,sum,depth) ; >> 212 } >> 213 } >> 214 >> 215 template <class T, class F> >> 216 void G4Integrator<T,F>::AdaptGauss( T* ptrT, F f, G4double xInitial, >> 217 G4double xFinal, G4double fTolerance, >> 218 G4double& sum, >> 219 G4int& depth ) 179 { 220 { 180 if(depth > 100) << 221 AdaptGauss(*ptrT,f,xInitial,xFinal,fTolerance,sum,depth) ; 181 { << 182 G4cout << "G4Integrator<T,F>::AdaptGauss: << 183 G4cout << "Function varies too rapidly to << 184 << G4endl; << 185 << 186 return; << 187 } << 188 G4double xMean = (xInitial + xFinal) / 2 << 189 G4double leftHalf = Gauss(typeT, f, xInitia << 190 G4double rightHalf = Gauss(typeT, f, xMean, << 191 G4double full = Gauss(typeT, f, xInitia << 192 if(std::fabs(leftHalf + rightHalf - full) < << 193 { << 194 sum += full; << 195 } << 196 else << 197 { << 198 ++depth; << 199 AdaptGauss(typeT, f, xInitial, xMean, fTol << 200 AdaptGauss(typeT, f, xMean, xFinal, fToler << 201 } << 202 } << 203 << 204 template <class T, class F> << 205 void G4Integrator<T, F>::AdaptGauss(T* ptrT, F << 206 G4double x << 207 G4double& << 208 { << 209 AdaptGauss(*ptrT, f, xInitial, xFinal, fTole << 210 } 222 } 211 223 212 ////////////////////////////////////////////// 224 ///////////////////////////////////////////////////////////////////////// 213 // 225 // 214 // 226 // 215 template <class T, class F> 227 template <class T, class F> 216 void G4Integrator<T, F>::AdaptGauss(G4double ( << 228 void G4Integrator<T,F>::AdaptGauss( G4double (*f)(G4double), 217 G4double x << 229 G4double xInitial, G4double xFinal, 218 G4double& << 230 G4double fTolerance, G4double& sum, 219 { << 231 G4int& depth ) 220 if(depth > 100) << 232 { 221 { << 233 if(depth > 100) 222 G4cout << "G4SimpleIntegration::AdaptGauss << 234 { 223 G4cout << "Function varies too rapidly to << 235 G4cout<<"G4SimpleIntegration::AdaptGauss: WARNING !!!"<<G4endl ; 224 << G4endl; << 236 G4cout<<"Function varies too rapidly to get stated accuracy in 100 steps " 225 << 237 <<G4endl ; 226 return; << 238 227 } << 239 return ; 228 G4double xMean = (xInitial + xFinal) / 2 << 240 } 229 G4double leftHalf = Gauss(f, xInitial, xMea << 241 G4double xMean = (xInitial + xFinal)/2.0 ; 230 G4double rightHalf = Gauss(f, xMean, xFinal) << 242 G4double leftHalf = Gauss(f,xInitial,xMean) ; 231 G4double full = Gauss(f, xInitial, xFin << 243 G4double rightHalf = Gauss(f,xMean,xFinal) ; 232 if(std::fabs(leftHalf + rightHalf - full) < << 244 G4double full = Gauss(f,xInitial,xFinal) ; 233 { << 245 if(std::fabs(leftHalf+rightHalf-full) < fTolerance) 234 sum += full; << 246 { 235 } << 247 sum += full ; 236 else << 248 } 237 { << 249 else 238 ++depth; << 250 { 239 AdaptGauss(f, xInitial, xMean, fTolerance, << 251 depth++ ; 240 AdaptGauss(f, xMean, xFinal, fTolerance, s << 252 AdaptGauss(f,xInitial,xMean,fTolerance,sum,depth) ; 241 } << 253 AdaptGauss(f,xMean,xFinal,fTolerance,sum,depth) ; >> 254 } 242 } 255 } 243 256 244 ////////////////////////////////////////////// 257 //////////////////////////////////////////////////////////////////////// 245 // 258 // 246 // Adaptive Gauss integration with accuracy 'e 259 // Adaptive Gauss integration with accuracy 'e' 247 // Convenient for using with class object type 260 // Convenient for using with class object typeT 248 << 261 249 template <class T, class F> << 262 template<class T, class F> 250 G4double G4Integrator<T, F>::AdaptiveGauss(T& << 263 G4double G4Integrator<T,F>::AdaptiveGauss( T& typeT, F f, G4double xInitial, 251 G4d << 264 G4double xFinal, G4double e ) 252 { << 265 { 253 G4int depth = 0; << 266 G4int depth = 0 ; 254 G4double sum = 0.0; << 267 G4double sum = 0.0 ; 255 AdaptGauss(typeT, f, xInitial, xFinal, e, su << 268 AdaptGauss(typeT,f,xInitial,xFinal,e,sum,depth) ; 256 return sum; << 269 return sum ; 257 } 270 } 258 271 259 ////////////////////////////////////////////// 272 //////////////////////////////////////////////////////////////////////// 260 // 273 // 261 // Adaptive Gauss integration with accuracy 'e 274 // Adaptive Gauss integration with accuracy 'e' 262 // Convenient for using with 'this' pointer 275 // Convenient for using with 'this' pointer 263 << 276 264 template <class T, class F> << 277 template<class T, class F> 265 G4double G4Integrator<T, F>::AdaptiveGauss(T* << 278 G4double G4Integrator<T,F>::AdaptiveGauss( T* ptrT, F f, G4double xInitial, 266 G4d << 279 G4double xFinal, G4double e ) 267 { 280 { 268 return AdaptiveGauss(*ptrT, f, xInitial, xFi << 281 return AdaptiveGauss(*ptrT,f,xInitial,xFinal,e) ; 269 } 282 } 270 283 271 ////////////////////////////////////////////// 284 //////////////////////////////////////////////////////////////////////// 272 // 285 // 273 // Adaptive Gauss integration with accuracy 'e 286 // Adaptive Gauss integration with accuracy 'e' 274 // Convenient for using with global scope func 287 // Convenient for using with global scope function f 275 << 288 276 template <class T, class F> 289 template <class T, class F> 277 G4double G4Integrator<T, F>::AdaptiveGauss(G4d << 290 G4double G4Integrator<T,F>::AdaptiveGauss( G4double (*f)(G4double), 278 G4d << 291 G4double xInitial, G4double xFinal, G4double e ) 279 G4d << 292 { 280 { << 293 G4int depth = 0 ; 281 G4int depth = 0; << 294 G4double sum = 0.0 ; 282 G4double sum = 0.0; << 295 AdaptGauss(f,xInitial,xFinal,e,sum,depth) ; 283 AdaptGauss(f, xInitial, xFinal, e, sum, dept << 296 return sum ; 284 return sum; << 285 } 297 } 286 298 287 ////////////////////////////////////////////// 299 //////////////////////////////////////////////////////////////////////////// 288 // Gauss integration methods involving ortogon 300 // Gauss integration methods involving ortogonal polynomials 289 ////////////////////////////////////////////// 301 //////////////////////////////////////////////////////////////////////////// 290 // 302 // 291 // Methods involving Legendre polynomials << 303 // Methods involving Legendre polynomials 292 // 304 // 293 ////////////////////////////////////////////// 305 ///////////////////////////////////////////////////////////////////////// 294 // 306 // 295 // The value nLegendre set the accuracy requir 307 // The value nLegendre set the accuracy required, i.e the number of points 296 // where the function pFunction will be evalua 308 // where the function pFunction will be evaluated during integration. 297 // The function creates the arrays for absciss << 309 // The function creates the arrays for abscissas and weights that used 298 // in Gauss-Legendre quadrature method. << 310 // in Gauss-Legendre quadrature method. 299 // The values a and b are the limits of integr 311 // The values a and b are the limits of integration of the function f . 300 // nLegendre MUST BE EVEN !!! 312 // nLegendre MUST BE EVEN !!! 301 // Returns the integral of the function f betw << 313 // Returns the integral of the function f between a and b, by 2*fNumber point 302 // Gauss-Legendre integration: the function is 314 // Gauss-Legendre integration: the function is evaluated exactly 303 // 2*fNumber times at interior points in the r << 315 // 2*fNumber times at interior points in the range of integration. 304 // Since the weights and abscissas are, in thi << 316 // Since the weights and abscissas are, in this case, symmetric around 305 // the midpoint of the range of integration, t << 317 // the midpoint of the range of integration, there are actually only 306 // fNumber distinct values of each. 318 // fNumber distinct values of each. 307 // Convenient for using with some class object 319 // Convenient for using with some class object dataT 308 320 309 template <class T, class F> 321 template <class T, class F> 310 G4double G4Integrator<T, F>::Legendre(T& typeT << 322 G4double G4Integrator<T,F>::Legendre( T& typeT, F f, G4double a, G4double b, 311 G4int nL << 323 G4int nLegendre ) 312 { 324 { 313 G4double nwt, nwt1, temp1, temp2, temp3, tem << 325 G4double newton, newton1, temp1, temp2, temp3, temp ; 314 G4double xDiff, xMean, dx, integral; << 326 G4double xDiff, xMean, dx, integral ; 315 327 316 const G4double tolerance = 1.6e-10; << 328 const G4double tolerance = 1.6e-10 ; 317 G4int i, j, k = nLegendre; << 329 G4int i, j, k = nLegendre ; 318 G4int fNumber = (nLegendre + 1) / 2; << 330 G4int fNumber = (nLegendre + 1)/2 ; 319 << 331 320 if(2 * fNumber != k) << 332 if(2*fNumber != k) 321 { << 333 { 322 G4Exception("G4Integrator<T,F>::Legendre(T << 334 G4Exception("G4Integrator<T,F>::Legendre(T&,F, ...)", "InvalidCall", 323 FatalException, "Invalid (odd) << 335 FatalException, "Invalid (odd) nLegendre in constructor."); 324 } << 336 } 325 << 337 326 G4double* fAbscissa = new G4double[fNumber]; << 338 G4double* fAbscissa = new G4double[fNumber] ; 327 G4double* fWeight = new G4double[fNumber]; << 339 G4double* fWeight = new G4double[fNumber] ; 328 << 340 329 for(i = 1; i <= fNumber; ++i) // Loop over << 341 for(i=1;i<=fNumber;i++) // Loop over the desired roots 330 { << 342 { 331 nwt = std::cos(CLHEP::pi * (i - 0.25) / << 343 newton = std::cos(pi*(i - 0.25)/(k + 0.5)) ; // Initial root approximation 332 (k + 0.5)); // Initial roo << 344 333 << 345 do // loop of Newton's method 334 do // loop of Newton's method << 346 { 335 { << 347 temp1 = 1.0 ; 336 temp1 = 1.0; << 348 temp2 = 0.0 ; 337 temp2 = 0.0; << 349 for(j=1;j<=k;j++) 338 for(j = 1; j <= k; ++j) << 350 { 339 { << 351 temp3 = temp2 ; 340 temp3 = temp2; << 352 temp2 = temp1 ; 341 temp2 = temp1; << 353 temp1 = ((2.0*j - 1.0)*newton*temp2 - (j - 1.0)*temp3)/j ; 342 temp1 = ((2.0 * j - 1.0) * nwt * temp2 << 354 } 343 } << 355 temp = k*(newton*temp1 - temp2)/(newton*newton - 1.0) ; 344 temp = k * (nwt * temp1 - temp2) / (nwt << 356 newton1 = newton ; 345 nwt1 = nwt; << 357 newton = newton1 - temp1/temp ; // Newton's method 346 nwt = nwt1 - temp1 / temp; // Newton's << 358 } 347 } while(std::fabs(nwt - nwt1) > tolerance) << 359 while(std::fabs(newton - newton1) > tolerance) ; 348 << 360 349 fAbscissa[fNumber - i] = nwt; << 361 fAbscissa[fNumber-i] = newton ; 350 fWeight[fNumber - i] = 2.0 / ((1.0 - nwt << 362 fWeight[fNumber-i] = 2.0/((1.0 - newton*newton)*temp*temp) ; 351 } << 363 } 352 << 364 353 // << 365 // 354 // Now we ready to get integral << 366 // Now we ready to get integral 355 // << 367 // 356 << 368 357 xMean = 0.5 * (a + b); << 369 xMean = 0.5*(a + b) ; 358 xDiff = 0.5 * (b - a); << 370 xDiff = 0.5*(b - a) ; 359 integral = 0.0; << 371 integral = 0.0 ; 360 for(i = 0; i < fNumber; ++i) << 372 for(i=0;i<fNumber;i++) 361 { << 373 { 362 dx = xDiff * fAbscissa[i]; << 374 dx = xDiff*fAbscissa[i] ; 363 integral += fWeight[i] * ((typeT.*f)(xMean << 375 integral += fWeight[i]*( (typeT.*f)(xMean + dx) + 364 } << 376 (typeT.*f)(xMean - dx) ) ; 365 delete[] fAbscissa; << 377 } 366 delete[] fWeight; << 378 delete[] fAbscissa; 367 return integral *= xDiff; << 379 delete[] fWeight; 368 } << 380 return integral *= xDiff ; >> 381 } 369 382 370 ////////////////////////////////////////////// 383 /////////////////////////////////////////////////////////////////////// 371 // 384 // 372 // Convenient for using with the pointer 'this 385 // Convenient for using with the pointer 'this' 373 386 374 template <class T, class F> 387 template <class T, class F> 375 G4double G4Integrator<T, F>::Legendre(T* ptrT, << 388 G4double G4Integrator<T,F>::Legendre( T* ptrT, F f, G4double a, 376 G4int nL << 389 G4double b, G4int nLegendre ) 377 { 390 { 378 return Legendre(*ptrT, f, a, b, nLegendre); << 391 return Legendre(*ptrT,f,a,b,nLegendre) ; 379 } 392 } 380 393 381 ////////////////////////////////////////////// 394 /////////////////////////////////////////////////////////////////////// 382 // 395 // 383 // Convenient for using with global scope func 396 // Convenient for using with global scope function f 384 397 385 template <class T, class F> 398 template <class T, class F> 386 G4double G4Integrator<T, F>::Legendre(G4double << 399 G4double G4Integrator<T,F>::Legendre( G4double (*f)(G4double), 387 G4double << 400 G4double a, G4double b, G4int nLegendre) 388 { 401 { 389 G4double nwt, nwt1, temp1, temp2, temp3, tem << 402 G4double newton, newton1, temp1, temp2, temp3, temp ; 390 G4double xDiff, xMean, dx, integral; << 403 G4double xDiff, xMean, dx, integral ; 391 404 392 const G4double tolerance = 1.6e-10; << 405 const G4double tolerance = 1.6e-10 ; 393 G4int i, j, k = nLegendre; << 406 G4int i, j, k = nLegendre ; 394 G4int fNumber = (nLegendre + 1) / 2; << 407 G4int fNumber = (nLegendre + 1)/2 ; 395 << 408 396 if(2 * fNumber != k) << 409 if(2*fNumber != k) 397 { << 410 { 398 G4Exception("G4Integrator<T,F>::Legendre(. << 411 G4Exception("G4Integrator<T,F>::Legendre(...)", "InvalidCall", 399 FatalException, "Invalid (odd) << 412 FatalException, "Invalid (odd) nLegendre in constructor."); 400 } << 413 } 401 << 414 402 G4double* fAbscissa = new G4double[fNumber]; << 415 G4double* fAbscissa = new G4double[fNumber] ; 403 G4double* fWeight = new G4double[fNumber]; << 416 G4double* fWeight = new G4double[fNumber] ; 404 << 417 405 for(i = 1; i <= fNumber; i++) // Loop over << 418 for(i=1;i<=fNumber;i++) // Loop over the desired roots 406 { << 419 { 407 nwt = std::cos(CLHEP::pi * (i - 0.25) / << 420 newton = std::cos(pi*(i - 0.25)/(k + 0.5)) ; // Initial root approximation 408 (k + 0.5)); // Initial roo << 421 409 << 422 do // loop of Newton's method 410 do // loop of Newton's method << 423 { 411 { << 424 temp1 = 1.0 ; 412 temp1 = 1.0; << 425 temp2 = 0.0 ; 413 temp2 = 0.0; << 426 for(j=1;j<=k;j++) 414 for(j = 1; j <= k; ++j) << 427 { 415 { << 428 temp3 = temp2 ; 416 temp3 = temp2; << 429 temp2 = temp1 ; 417 temp2 = temp1; << 430 temp1 = ((2.0*j - 1.0)*newton*temp2 - (j - 1.0)*temp3)/j ; 418 temp1 = ((2.0 * j - 1.0) * nwt * temp2 << 431 } 419 } << 432 temp = k*(newton*temp1 - temp2)/(newton*newton - 1.0) ; 420 temp = k * (nwt * temp1 - temp2) / (nwt << 433 newton1 = newton ; 421 nwt1 = nwt; << 434 newton = newton1 - temp1/temp ; // Newton's method 422 nwt = nwt1 - temp1 / temp; // Newton's << 435 } 423 } while(std::fabs(nwt - nwt1) > tolerance) << 436 while(std::fabs(newton - newton1) > tolerance) ; 424 << 437 425 fAbscissa[fNumber - i] = nwt; << 438 fAbscissa[fNumber-i] = newton ; 426 fWeight[fNumber - i] = 2.0 / ((1.0 - nwt << 439 fWeight[fNumber-i] = 2.0/((1.0 - newton*newton)*temp*temp) ; 427 } << 440 } 428 << 441 429 // << 442 // 430 // Now we ready to get integral << 443 // Now we ready to get integral 431 // << 444 // 432 << 445 433 xMean = 0.5 * (a + b); << 446 xMean = 0.5*(a + b) ; 434 xDiff = 0.5 * (b - a); << 447 xDiff = 0.5*(b - a) ; 435 integral = 0.0; << 448 integral = 0.0 ; 436 for(i = 0; i < fNumber; ++i) << 449 for(i=0;i<fNumber;i++) 437 { << 450 { 438 dx = xDiff * fAbscissa[i]; << 451 dx = xDiff*fAbscissa[i] ; 439 integral += fWeight[i] * ((*f)(xMean + dx) << 452 integral += fWeight[i]*( (*f)(xMean + dx) + (*f)(xMean - dx) ) ; 440 } << 453 } 441 delete[] fAbscissa; << 454 delete[] fAbscissa; 442 delete[] fWeight; << 455 delete[] fWeight; 443 456 444 return integral *= xDiff; << 457 return integral *= xDiff ; 445 } << 458 } 446 459 447 ////////////////////////////////////////////// 460 //////////////////////////////////////////////////////////////////////////// 448 // 461 // 449 // Returns the integral of the function to be 462 // Returns the integral of the function to be pointed by T::f between a and b, 450 // by ten point Gauss-Legendre integration: th 463 // by ten point Gauss-Legendre integration: the function is evaluated exactly 451 // ten times at interior points in the range o 464 // ten times at interior points in the range of integration. Since the weights 452 // and abscissas are, in this case, symmetric << 465 // and abscissas are, in this case, symmetric around the midpoint of the 453 // range of integration, there are actually on 466 // range of integration, there are actually only five distinct values of each 454 // Convenient for using with class object type 467 // Convenient for using with class object typeT 455 468 456 template <class T, class F> << 469 template <class T, class F> 457 G4double G4Integrator<T, F>::Legendre10(T& typ << 470 G4double G4Integrator<T,F>::Legendre10( T& typeT, F f,G4double a, G4double b) 458 { 471 { 459 G4int i; << 472 G4int i ; 460 G4double xDiff, xMean, dx, integral; << 473 G4double xDiff, xMean, dx, integral ; 461 << 474 462 // From Abramowitz M., Stegan I.A. 1964 , Ha << 475 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 916 463 << 476 464 static const G4double abscissa[] = { 0.14887 << 477 static G4double abscissa[] = { 0.148874338981631, 0.433395394129247, 465 0.67940 << 478 0.679409568299024, 0.865063366688985, 466 0.97390 << 479 0.973906528517172 } ; 467 << 480 468 static const G4double weight[] = { 0.2955242 << 481 static G4double weight[] = { 0.295524224714753, 0.269266719309996, 469 0.2190863 << 482 0.219086362515982, 0.149451349150581, 470 0.0666713 << 483 0.066671344308688 } ; 471 xMean = 0.5 * (a + << 484 xMean = 0.5*(a + b) ; 472 xDiff = 0.5 * (b - << 485 xDiff = 0.5*(b - a) ; 473 integral = 0.0; << 486 integral = 0.0 ; 474 for(i = 0; i < 5; ++i) << 487 for(i=0;i<5;i++) 475 { << 488 { 476 dx = xDiff * abscissa[i]; << 489 dx = xDiff*abscissa[i] ; 477 integral += weight[i] * ((typeT.*f)(xMean << 490 integral += weight[i]*( (typeT.*f)(xMean + dx) + (typeT.*f)(xMean - dx)) ; 478 } << 491 } 479 return integral *= xDiff; << 492 return integral *= xDiff ; 480 } 493 } 481 494 482 ////////////////////////////////////////////// 495 /////////////////////////////////////////////////////////////////////////// 483 // 496 // 484 // Convenient for using with the pointer 'this 497 // Convenient for using with the pointer 'this' 485 498 486 template <class T, class F> << 499 template <class T, class F> 487 G4double G4Integrator<T, F>::Legendre10(T* ptr << 500 G4double G4Integrator<T,F>::Legendre10( T* ptrT, F f,G4double a, G4double b) 488 { 501 { 489 return Legendre10(*ptrT, f, a, b); << 502 return Legendre10(*ptrT,f,a,b) ; 490 } << 503 } 491 504 492 ////////////////////////////////////////////// 505 ////////////////////////////////////////////////////////////////////////// 493 // 506 // 494 // Convenient for using with global scope func 507 // Convenient for using with global scope functions 495 508 496 template <class T, class F> 509 template <class T, class F> 497 G4double G4Integrator<T, F>::Legendre10(G4doub << 510 G4double G4Integrator<T,F>::Legendre10( G4double (*f)(G4double), 498 G4doub << 511 G4double a, G4double b ) 499 { 512 { 500 G4int i; << 513 G4int i ; 501 G4double xDiff, xMean, dx, integral; << 514 G4double xDiff, xMean, dx, integral ; 502 << 515 503 // From Abramowitz M., Stegan I.A. 1964 , Ha << 516 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 916 504 << 517 505 static const G4double abscissa[] = { 0.14887 << 518 static G4double abscissa[] = { 0.148874338981631, 0.433395394129247, 506 0.67940 << 519 0.679409568299024, 0.865063366688985, 507 0.97390 << 520 0.973906528517172 } ; 508 << 521 509 static const G4double weight[] = { 0.2955242 << 522 static G4double weight[] = { 0.295524224714753, 0.269266719309996, 510 0.2190863 << 523 0.219086362515982, 0.149451349150581, 511 0.0666713 << 524 0.066671344308688 } ; 512 xMean = 0.5 * (a + << 525 xMean = 0.5*(a + b) ; 513 xDiff = 0.5 * (b - << 526 xDiff = 0.5*(b - a) ; 514 integral = 0.0; << 527 integral = 0.0 ; 515 for(i = 0; i < 5; ++i) << 528 for(i=0;i<5;i++) 516 { << 529 { 517 dx = xDiff * abscissa[i]; << 530 dx = xDiff*abscissa[i] ; 518 integral += weight[i] * ((*f)(xMean + dx) << 531 integral += weight[i]*( (*f)(xMean + dx) + (*f)(xMean - dx)) ; 519 } << 532 } 520 return integral *= xDiff; << 533 return integral *= xDiff ; 521 } 534 } 522 535 523 ////////////////////////////////////////////// 536 /////////////////////////////////////////////////////////////////////// 524 // 537 // 525 // Returns the integral of the function to be 538 // Returns the integral of the function to be pointed by T::f between a and b, 526 // by 96 point Gauss-Legendre integration: the 539 // by 96 point Gauss-Legendre integration: the function is evaluated exactly 527 // ten Times at interior points in the range o 540 // ten Times at interior points in the range of integration. Since the weights 528 // and abscissas are, in this case, symmetric << 541 // and abscissas are, in this case, symmetric around the midpoint of the 529 // range of integration, there are actually on 542 // range of integration, there are actually only five distinct values of each 530 // Convenient for using with some class object 543 // Convenient for using with some class object typeT 531 544 532 template <class T, class F> << 545 template <class T, class F> 533 G4double G4Integrator<T, F>::Legendre96(T& typ << 546 G4double G4Integrator<T,F>::Legendre96( T& typeT, F f,G4double a, G4double b) 534 { 547 { 535 G4int i; << 548 G4int i ; 536 G4double xDiff, xMean, dx, integral; << 549 G4double xDiff, xMean, dx, integral ; 537 << 550 538 // From Abramowitz M., Stegan I.A. 1964 , Ha << 551 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 919 539 << 552 540 static const G4double abscissa[] = { << 553 static G4double 541 0.016276744849602969579, 0.048812985136049 << 554 abscissa[] = { 542 0.081297495464425558994, 0.113695850110665 << 555 0.016276744849602969579, 0.048812985136049731112, 543 0.145973714654896941989, 0.178096882367618 << 556 0.081297495464425558994, 0.113695850110665920911, 544 << 557 0.145973714654896941989, 0.178096882367618602759, // 6 545 0.210031310460567203603, 0.241743156163840 << 558 546 0.273198812591049141487, 0.304364944354496 << 559 0.210031310460567203603, 0.241743156163840012328, 547 0.335208522892625422616, 0.365696861472313 << 560 0.273198812591049141487, 0.304364944354496353024, 548 << 561 0.335208522892625422616, 0.365696861472313635031, // 12 549 0.395797649828908603285, 0.425478988407300 << 562 550 0.454709422167743008636, 0.483457973920596 << 563 0.395797649828908603285, 0.425478988407300545365, 551 0.511694177154667673586, 0.539388108324357 << 564 0.454709422167743008636, 0.483457973920596359768, 552 << 565 0.511694177154667673586, 0.539388108324357436227, // 18 553 0.566510418561397168404, 0.593032364777572 << 566 554 0.618925840125468570386, 0.644163403784967 << 567 0.566510418561397168404, 0.593032364777572080684, 555 0.668718310043916153953, 0.692564536642171 << 568 0.618925840125468570386, 0.644163403784967106798, 556 << 569 0.668718310043916153953, 0.692564536642171561344, // 24 557 0.715676812348967626225, 0.738030643744400 << 570 558 0.759602341176647498703, 0.780369043867433 << 571 0.715676812348967626225, 0.738030643744400132851, 559 0.800308744139140817229, 0.819400310737931 << 572 0.759602341176647498703, 0.780369043867433217604, 560 << 573 0.800308744139140817229, 0.819400310737931675539, // 30 561 0.837623511228187121494, 0.854959033434601 << 574 562 0.871388505909296502874, 0.886894517402420 << 575 0.837623511228187121494, 0.854959033434601455463, 563 0.901460635315852341319, 0.915071423120898 << 576 0.871388505909296502874, 0.886894517402420416057, 564 << 577 0.901460635315852341319, 0.915071423120898074206, // 36 565 0.927712456722308690965, 0.939370339752755 << 578 566 0.950032717784437635756, 0.959688291448742 << 579 0.927712456722308690965, 0.939370339752755216932, 567 0.968326828463264212174, 0.975939174585136 << 580 0.950032717784437635756, 0.959688291448742539300, 568 << 581 0.968326828463264212174, 0.975939174585136466453, // 42 569 0.982517263563014677447, 0.988054126329623 << 582 570 0.992543900323762624572, 0.995981842987209 << 583 0.982517263563014677447, 0.988054126329623799481, 571 0.998364375863181677724, 0.999689503883230 << 584 0.992543900323762624572, 0.995981842987209290650, 572 }; << 585 0.998364375863181677724, 0.999689503883230766828 // 48 573 << 586 } ; 574 static const G4double weight[] = { << 587 575 0.032550614492363166242, 0.032516118713868 << 588 static G4double 576 0.032447163714064269364, 0.032343822568575 << 589 weight[] = { 577 0.032206204794030250669, 0.032034456231992 << 590 0.032550614492363166242, 0.032516118713868835987, 578 << 591 0.032447163714064269364, 0.032343822568575928429, 579 0.031828758894411006535, 0.031589330770727 << 592 0.032206204794030250669, 0.032034456231992663218, // 6 580 0.031316425596862355813, 0.031010332586313 << 593 581 0.030671376123669149014, 0.030299915420827 << 594 0.031828758894411006535, 0.031589330770727168558, 582 << 595 0.031316425596862355813, 0.031010332586313837423, 583 0.029896344136328385984, 0.029461089958167 << 596 0.030671376123669149014, 0.030299915420827593794, // 12 584 0.028994614150555236543, 0.028497411065085 << 597 585 0.027970007616848334440, 0.027412962726029 << 598 0.029896344136328385984, 0.029461089958167905970, 586 << 599 0.028994614150555236543, 0.028497411065085385646, 587 0.026826866725591762198, 0.026212340735672 << 600 0.027970007616848334440, 0.027412962726029242823, // 18 588 0.025570036005349361499, 0.024900633222483 << 601 589 0.024204841792364691282, 0.023483399085926 << 602 0.026826866725591762198, 0.026212340735672413913, 590 << 603 0.025570036005349361499, 0.024900633222483610288, 591 0.022737069658329374001, 0.021966644438744 << 604 0.024204841792364691282, 0.023483399085926219842, // 24 592 0.021172939892191298988, 0.020356797154333 << 605 593 0.019519081140145022410, 0.018660679627411 << 606 0.022737069658329374001, 0.021966644438744349195, 594 << 607 0.021172939892191298988, 0.020356797154333324595, 595 0.017782502316045260838, 0.016885479864245 << 608 0.019519081140145022410, 0.018660679627411467385, // 30 596 0.015970562902562291381, 0.015038721026994 << 609 597 0.014090941772314860916, 0.013128229566961 << 610 0.017782502316045260838, 0.016885479864245172450, 598 << 611 0.015970562902562291381, 0.015038721026994938006, 599 0.012151604671088319635, 0.011162102099838 << 612 0.014090941772314860916, 0.013128229566961572637, // 36 600 0.010160770535008415758, 0.009148671230783 << 613 601 0.008126876925698759217, 0.007096470791153 << 614 0.012151604671088319635, 0.011162102099838498591, 602 << 615 0.010160770535008415758, 0.009148671230783386633, 603 0.006058545504235961683, 0.005014202742927 << 616 0.008126876925698759217, 0.007096470791153865269, // 42 604 0.003964554338444686674, 0.002910731817934 << 617 605 0.001853960788946921732, 0.000796792065552 << 618 0.006058545504235961683, 0.005014202742927517693, 606 }; << 619 0.003964554338444686674, 0.002910731817934946408, 607 xMean = 0.5 * (a + b); << 620 0.001853960788946921732, 0.000796792065552012429 // 48 608 xDiff = 0.5 * (b - a); << 621 } ; 609 integral = 0.0; << 622 xMean = 0.5*(a + b) ; 610 for(i = 0; i < 48; ++i) << 623 xDiff = 0.5*(b - a) ; 611 { << 624 integral = 0.0 ; 612 dx = xDiff * abscissa[i]; << 625 for(i=0;i<48;i++) 613 integral += weight[i] * ((typeT.*f)(xMean << 626 { 614 } << 627 dx = xDiff*abscissa[i] ; 615 return integral *= xDiff; << 628 integral += weight[i]*((typeT.*f)(xMean + dx) + (typeT.*f)(xMean - dx)) ; >> 629 } >> 630 return integral *= xDiff ; 616 } 631 } 617 632 618 ////////////////////////////////////////////// 633 /////////////////////////////////////////////////////////////////////// 619 // 634 // 620 // Convenient for using with the pointer 'this 635 // Convenient for using with the pointer 'this' 621 636 622 template <class T, class F> << 637 template <class T, class F> 623 G4double G4Integrator<T, F>::Legendre96(T* ptr << 638 G4double G4Integrator<T,F>::Legendre96( T* ptrT, F f,G4double a, G4double b) 624 { 639 { 625 return Legendre96(*ptrT, f, a, b); << 640 return Legendre96(*ptrT,f,a,b) ; 626 } << 641 } 627 642 628 ////////////////////////////////////////////// 643 /////////////////////////////////////////////////////////////////////// 629 // 644 // 630 // Convenient for using with global scope func << 645 // Convenient for using with global scope function f 631 646 632 template <class T, class F> 647 template <class T, class F> 633 G4double G4Integrator<T, F>::Legendre96(G4doub << 648 G4double G4Integrator<T,F>::Legendre96( G4double (*f)(G4double), 634 G4doub << 649 G4double a, G4double b ) 635 { 650 { 636 G4int i; << 651 G4int i ; 637 G4double xDiff, xMean, dx, integral; << 652 G4double xDiff, xMean, dx, integral ; 638 << 653 639 // From Abramowitz M., Stegan I.A. 1964 , Ha << 654 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 919 640 << 655 641 static const G4double abscissa[] = { << 656 static G4double 642 0.016276744849602969579, 0.048812985136049 << 657 abscissa[] = { 643 0.081297495464425558994, 0.113695850110665 << 658 0.016276744849602969579, 0.048812985136049731112, 644 0.145973714654896941989, 0.178096882367618 << 659 0.081297495464425558994, 0.113695850110665920911, 645 << 660 0.145973714654896941989, 0.178096882367618602759, // 6 646 0.210031310460567203603, 0.241743156163840 << 661 647 0.273198812591049141487, 0.304364944354496 << 662 0.210031310460567203603, 0.241743156163840012328, 648 0.335208522892625422616, 0.365696861472313 << 663 0.273198812591049141487, 0.304364944354496353024, 649 << 664 0.335208522892625422616, 0.365696861472313635031, // 12 650 0.395797649828908603285, 0.425478988407300 << 665 651 0.454709422167743008636, 0.483457973920596 << 666 0.395797649828908603285, 0.425478988407300545365, 652 0.511694177154667673586, 0.539388108324357 << 667 0.454709422167743008636, 0.483457973920596359768, 653 << 668 0.511694177154667673586, 0.539388108324357436227, // 18 654 0.566510418561397168404, 0.593032364777572 << 669 655 0.618925840125468570386, 0.644163403784967 << 670 0.566510418561397168404, 0.593032364777572080684, 656 0.668718310043916153953, 0.692564536642171 << 671 0.618925840125468570386, 0.644163403784967106798, 657 << 672 0.668718310043916153953, 0.692564536642171561344, // 24 658 0.715676812348967626225, 0.738030643744400 << 673 659 0.759602341176647498703, 0.780369043867433 << 674 0.715676812348967626225, 0.738030643744400132851, 660 0.800308744139140817229, 0.819400310737931 << 675 0.759602341176647498703, 0.780369043867433217604, 661 << 676 0.800308744139140817229, 0.819400310737931675539, // 30 662 0.837623511228187121494, 0.854959033434601 << 677 663 0.871388505909296502874, 0.886894517402420 << 678 0.837623511228187121494, 0.854959033434601455463, 664 0.901460635315852341319, 0.915071423120898 << 679 0.871388505909296502874, 0.886894517402420416057, 665 << 680 0.901460635315852341319, 0.915071423120898074206, // 36 666 0.927712456722308690965, 0.939370339752755 << 681 667 0.950032717784437635756, 0.959688291448742 << 682 0.927712456722308690965, 0.939370339752755216932, 668 0.968326828463264212174, 0.975939174585136 << 683 0.950032717784437635756, 0.959688291448742539300, 669 << 684 0.968326828463264212174, 0.975939174585136466453, // 42 670 0.982517263563014677447, 0.988054126329623 << 685 671 0.992543900323762624572, 0.995981842987209 << 686 0.982517263563014677447, 0.988054126329623799481, 672 0.998364375863181677724, 0.999689503883230 << 687 0.992543900323762624572, 0.995981842987209290650, 673 }; << 688 0.998364375863181677724, 0.999689503883230766828 // 48 674 << 689 } ; 675 static const G4double weight[] = { << 690 676 0.032550614492363166242, 0.032516118713868 << 691 static G4double 677 0.032447163714064269364, 0.032343822568575 << 692 weight[] = { 678 0.032206204794030250669, 0.032034456231992 << 693 0.032550614492363166242, 0.032516118713868835987, 679 << 694 0.032447163714064269364, 0.032343822568575928429, 680 0.031828758894411006535, 0.031589330770727 << 695 0.032206204794030250669, 0.032034456231992663218, // 6 681 0.031316425596862355813, 0.031010332586313 << 696 682 0.030671376123669149014, 0.030299915420827 << 697 0.031828758894411006535, 0.031589330770727168558, 683 << 698 0.031316425596862355813, 0.031010332586313837423, 684 0.029896344136328385984, 0.029461089958167 << 699 0.030671376123669149014, 0.030299915420827593794, // 12 685 0.028994614150555236543, 0.028497411065085 << 700 686 0.027970007616848334440, 0.027412962726029 << 701 0.029896344136328385984, 0.029461089958167905970, 687 << 702 0.028994614150555236543, 0.028497411065085385646, 688 0.026826866725591762198, 0.026212340735672 << 703 0.027970007616848334440, 0.027412962726029242823, // 18 689 0.025570036005349361499, 0.024900633222483 << 704 690 0.024204841792364691282, 0.023483399085926 << 705 0.026826866725591762198, 0.026212340735672413913, 691 << 706 0.025570036005349361499, 0.024900633222483610288, 692 0.022737069658329374001, 0.021966644438744 << 707 0.024204841792364691282, 0.023483399085926219842, // 24 693 0.021172939892191298988, 0.020356797154333 << 708 694 0.019519081140145022410, 0.018660679627411 << 709 0.022737069658329374001, 0.021966644438744349195, 695 << 710 0.021172939892191298988, 0.020356797154333324595, 696 0.017782502316045260838, 0.016885479864245 << 711 0.019519081140145022410, 0.018660679627411467385, // 30 697 0.015970562902562291381, 0.015038721026994 << 712 698 0.014090941772314860916, 0.013128229566961 << 713 0.017782502316045260838, 0.016885479864245172450, 699 << 714 0.015970562902562291381, 0.015038721026994938006, 700 0.012151604671088319635, 0.011162102099838 << 715 0.014090941772314860916, 0.013128229566961572637, // 36 701 0.010160770535008415758, 0.009148671230783 << 716 702 0.008126876925698759217, 0.007096470791153 << 717 0.012151604671088319635, 0.011162102099838498591, 703 << 718 0.010160770535008415758, 0.009148671230783386633, 704 0.006058545504235961683, 0.005014202742927 << 719 0.008126876925698759217, 0.007096470791153865269, // 42 705 0.003964554338444686674, 0.002910731817934 << 720 706 0.001853960788946921732, 0.000796792065552 << 721 0.006058545504235961683, 0.005014202742927517693, 707 }; << 722 0.003964554338444686674, 0.002910731817934946408, 708 xMean = 0.5 * (a + b); << 723 0.001853960788946921732, 0.000796792065552012429 // 48 709 xDiff = 0.5 * (b - a); << 724 } ; 710 integral = 0.0; << 725 xMean = 0.5*(a + b) ; 711 for(i = 0; i < 48; ++i) << 726 xDiff = 0.5*(b - a) ; 712 { << 727 integral = 0.0 ; 713 dx = xDiff * abscissa[i]; << 728 for(i=0;i<48;i++) 714 integral += weight[i] * ((*f)(xMean + dx) << 729 { 715 } << 730 dx = xDiff*abscissa[i] ; 716 return integral *= xDiff; << 731 integral += weight[i]*((*f)(xMean + dx) + (*f)(xMean - dx)) ; >> 732 } >> 733 return integral *= xDiff ; 717 } 734 } 718 735 719 ////////////////////////////////////////////// 736 ////////////////////////////////////////////////////////////////////////////// 720 // 737 // 721 // Methods involving Chebyshev polynomials << 738 // Methods involving Chebyshev polynomials 722 // 739 // 723 ////////////////////////////////////////////// 740 /////////////////////////////////////////////////////////////////////////// 724 // 741 // 725 // Integrates function pointed by T::f from a << 742 // Integrates function pointed by T::f from a to b by Gauss-Chebyshev 726 // quadrature method. 743 // quadrature method. 727 // Convenient for using with class object type 744 // Convenient for using with class object typeT 728 745 729 template <class T, class F> 746 template <class T, class F> 730 G4double G4Integrator<T, F>::Chebyshev(T& type << 747 G4double G4Integrator<T,F>::Chebyshev( T& typeT, F f, G4double a, 731 G4int n << 748 G4double b, G4int nChebyshev ) 732 { 749 { 733 G4int i; << 750 G4int i ; 734 G4double xDiff, xMean, dx, integral = 0.0; << 751 G4double xDiff, xMean, dx, integral = 0.0 ; 735 << 752 736 G4int fNumber = nChebyshev; // Try to << 753 G4int fNumber = nChebyshev ; // Try to reduce fNumber twice ?? 737 G4double cof = CLHEP::pi / fNumber; << 754 G4double cof = pi/fNumber ; 738 G4double* fAbscissa = new G4double[fNumber]; << 755 G4double* fAbscissa = new G4double[fNumber] ; 739 G4double* fWeight = new G4double[fNumber]; << 756 G4double* fWeight = new G4double[fNumber] ; 740 for(i = 0; i < fNumber; ++i) << 757 for(i=0;i<fNumber;i++) 741 { << 758 { 742 fAbscissa[i] = std::cos(cof * (i + 0.5)); << 759 fAbscissa[i] = std::cos(cof*(i + 0.5)) ; 743 fWeight[i] = cof * std::sqrt(1 - fAbscis << 760 fWeight[i] = cof*std::sqrt(1 - fAbscissa[i]*fAbscissa[i]) ; 744 } << 761 } 745 << 762 746 // << 763 // 747 // Now we ready to estimate the integral << 764 // Now we ready to estimate the integral 748 // << 765 // 749 << 766 750 xMean = 0.5 * (a + b); << 767 xMean = 0.5*(a + b) ; 751 xDiff = 0.5 * (b - a); << 768 xDiff = 0.5*(b - a) ; 752 for(i = 0; i < fNumber; ++i) << 769 for(i=0;i<fNumber;i++) 753 { << 770 { 754 dx = xDiff * fAbscissa[i]; << 771 dx = xDiff*fAbscissa[i] ; 755 integral += fWeight[i] * (typeT.*f)(xMean << 772 integral += fWeight[i]*(typeT.*f)(xMean + dx) ; 756 } << 773 } 757 delete[] fAbscissa; << 774 delete[] fAbscissa; 758 delete[] fWeight; << 775 delete[] fWeight; 759 return integral *= xDiff; << 776 return integral *= xDiff ; 760 } 777 } 761 778 762 ////////////////////////////////////////////// 779 /////////////////////////////////////////////////////////////////////// 763 // 780 // 764 // Convenient for using with 'this' pointer 781 // Convenient for using with 'this' pointer 765 782 766 template <class T, class F> 783 template <class T, class F> 767 G4double G4Integrator<T, F>::Chebyshev(T* ptrT << 784 G4double G4Integrator<T,F>::Chebyshev( T* ptrT, F f, G4double a, 768 G4int n << 785 G4double b, G4int n ) 769 { 786 { 770 return Chebyshev(*ptrT, f, a, b, n); << 787 return Chebyshev(*ptrT,f,a,b,n) ; 771 } << 788 } 772 789 773 ////////////////////////////////////////////// 790 //////////////////////////////////////////////////////////////////////// 774 // 791 // 775 // For use with global scope functions f << 792 // For use with global scope functions f 776 793 777 template <class T, class F> 794 template <class T, class F> 778 G4double G4Integrator<T, F>::Chebyshev(G4doubl << 795 G4double G4Integrator<T,F>::Chebyshev( G4double (*f)(G4double), 779 G4doubl << 796 G4double a, G4double b, G4int nChebyshev ) 780 { 797 { 781 G4int i; << 798 G4int i ; 782 G4double xDiff, xMean, dx, integral = 0.0; << 799 G4double xDiff, xMean, dx, integral = 0.0 ; 783 << 800 784 G4int fNumber = nChebyshev; // Try to << 801 G4int fNumber = nChebyshev ; // Try to reduce fNumber twice ?? 785 G4double cof = CLHEP::pi / fNumber; << 802 G4double cof = pi/fNumber ; 786 G4double* fAbscissa = new G4double[fNumber]; << 803 G4double* fAbscissa = new G4double[fNumber] ; 787 G4double* fWeight = new G4double[fNumber]; << 804 G4double* fWeight = new G4double[fNumber] ; 788 for(i = 0; i < fNumber; ++i) << 805 for(i=0;i<fNumber;i++) 789 { << 806 { 790 fAbscissa[i] = std::cos(cof * (i + 0.5)); << 807 fAbscissa[i] = std::cos(cof*(i + 0.5)) ; 791 fWeight[i] = cof * std::sqrt(1 - fAbscis << 808 fWeight[i] = cof*std::sqrt(1 - fAbscissa[i]*fAbscissa[i]) ; 792 } << 809 } 793 << 810 794 // << 811 // 795 // Now we ready to estimate the integral << 812 // Now we ready to estimate the integral 796 // << 813 // 797 << 814 798 xMean = 0.5 * (a + b); << 815 xMean = 0.5*(a + b) ; 799 xDiff = 0.5 * (b - a); << 816 xDiff = 0.5*(b - a) ; 800 for(i = 0; i < fNumber; ++i) << 817 for(i=0;i<fNumber;i++) 801 { << 818 { 802 dx = xDiff * fAbscissa[i]; << 819 dx = xDiff*fAbscissa[i] ; 803 integral += fWeight[i] * (*f)(xMean + dx); << 820 integral += fWeight[i]*(*f)(xMean + dx) ; 804 } << 821 } 805 delete[] fAbscissa; << 822 delete[] fAbscissa; 806 delete[] fWeight; << 823 delete[] fWeight; 807 return integral *= xDiff; << 824 return integral *= xDiff ; 808 } 825 } 809 826 810 ////////////////////////////////////////////// 827 ////////////////////////////////////////////////////////////////////// 811 // 828 // 812 // Method involving Laguerre polynomials 829 // Method involving Laguerre polynomials 813 // 830 // 814 ////////////////////////////////////////////// 831 ////////////////////////////////////////////////////////////////////// 815 // 832 // 816 // Integral from zero to infinity of std::pow( << 833 // Integral from zero to infinity of std::pow(x,alpha)*std::exp(-x)*f(x). 817 // The value of nLaguerre sets the accuracy. 834 // The value of nLaguerre sets the accuracy. 818 // The function creates arrays fAbscissa[0,.., << 835 // The function creates arrays fAbscissa[0,..,nLaguerre-1] and 819 // fWeight[0,..,nLaguerre-1] . << 836 // fWeight[0,..,nLaguerre-1] . 820 // Convenient for using with class object 'typ 837 // Convenient for using with class object 'typeT' and (typeT.*f) function 821 // (T::f) 838 // (T::f) 822 839 823 template <class T, class F> 840 template <class T, class F> 824 G4double G4Integrator<T, F>::Laguerre(T& typeT << 841 G4double G4Integrator<T,F>::Laguerre( T& typeT, F f, G4double alpha, 825 G4int nL << 842 G4int nLaguerre ) 826 { 843 { 827 const G4double tolerance = 1.0e-10; << 844 const G4double tolerance = 1.0e-10 ; 828 const G4int maxNumber = 12; << 845 const G4int maxNumber = 12 ; 829 G4int i, j, k; << 846 G4int i, j, k ; 830 G4double nwt = 0., nwt1, temp1, temp2, << 847 G4double newton=0., newton1, temp1, temp2, temp3, temp, cofi ; 831 G4double integral = 0.0; << 848 G4double integral = 0.0 ; 832 << 849 833 G4int fNumber = nLaguerre; << 850 G4int fNumber = nLaguerre ; 834 G4double* fAbscissa = new G4double[fNumber]; << 851 G4double* fAbscissa = new G4double[fNumber] ; 835 G4double* fWeight = 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 ; 836 876 837 for(i = 1; i <= fNumber; ++i) // Loop over << 877 for(j=1;j<=fNumber;j++) 838 { << 878 { 839 if(i == 1) << 879 temp3 = temp2 ; 840 { << 880 temp2 = temp1 ; 841 nwt = (1.0 + alpha) * (3.0 + 0.92 * alph << 881 temp1 = ((2*j - 1 + alpha - newton)*temp2 - (j - 1 + alpha)*temp3)/j ; 842 (1.0 + 2.4 * fNumber + 1.8 * alpha << 882 } 843 } << 883 temp = (fNumber*temp1 - (fNumber +alpha)*temp2)/newton ; 844 else if(i == 2) << 884 newton1 = newton ; 845 { << 885 newton = newton1 - temp1/temp ; 846 nwt += (15.0 + 6.25 * alpha) / (1.0 + 0. << 886 847 } << 887 if(std::fabs(newton - newton1) <= tolerance) 848 else << 888 { 849 { << 889 break ; 850 cofi = i - 2; << 890 } 851 nwt += ((1.0 + 2.55 * cofi) / (1.9 * cof << 891 } 852 1.26 * cofi * alpha / (1.0 + 3.5 << 892 if(k > maxNumber) 853 (nwt - fAbscissa[i - 3]) / (1.0 + << 893 { 854 } << 894 G4Exception("G4Integrator<T,F>::Laguerre(T,F, ...)", "Error", 855 for(k = 1; k <= maxNumber; ++k) << 895 FatalException, "Too many (>12) iterations."); 856 { << 896 } 857 temp1 = 1.0; << 897 858 temp2 = 0.0; << 898 fAbscissa[i-1] = newton ; 859 << 899 fWeight[i-1] = -std::exp(GammaLogarithm(alpha + fNumber) - 860 for(j = 1; j <= fNumber; ++j) << 900 GammaLogarithm((G4double)fNumber))/(temp*fNumber*temp2) ; 861 { << 901 } 862 temp3 = temp2; << 902 863 temp2 = temp1; << 903 // 864 temp1 = << 904 // Integral evaluation 865 ((2 * j - 1 + alpha - nwt) * temp2 - << 905 // 866 } << 906 867 temp = (fNumber * temp1 - (fNumber + alp << 907 for(i=0;i<fNumber;i++) 868 nwt1 = nwt; << 908 { 869 nwt = nwt1 - temp1 / temp; << 909 integral += fWeight[i]*(typeT.*f)(fAbscissa[i]) ; 870 << 910 } 871 if(std::fabs(nwt - nwt1) <= tolerance) << 911 delete[] fAbscissa; 872 { << 912 delete[] fWeight; 873 break; << 913 return integral ; 874 } << 914 } 875 } << 876 if(k > maxNumber) << 877 { << 878 G4Exception("G4Integrator<T,F>::Laguerre << 879 FatalException, "Too many (> << 880 } << 881 << 882 fAbscissa[i - 1] = nwt; << 883 fWeight[i - 1] = -std::exp(GammaLogarith << 884 GammaLogarithm( << 885 (temp * fNumber * temp2); << 886 } << 887 915 888 // << 889 // Integral evaluation << 890 // << 891 916 892 for(i = 0; i < fNumber; ++i) << 893 { << 894 integral += fWeight[i] * (typeT.*f)(fAbsci << 895 } << 896 delete[] fAbscissa; << 897 delete[] fWeight; << 898 return integral; << 899 } << 900 917 901 ////////////////////////////////////////////// 918 ////////////////////////////////////////////////////////////////////// 902 // 919 // 903 // 920 // 904 921 905 template <class T, class F> << 922 template <class T, class F> G4double 906 G4double G4Integrator<T, F>::Laguerre(T* ptrT, << 923 G4Integrator<T,F>::Laguerre( T* ptrT, F f, G4double alpha, G4int nLaguerre ) 907 G4int nL << 908 { 924 { 909 return Laguerre(*ptrT, f, alpha, nLaguerre); << 925 return Laguerre(*ptrT,f,alpha,nLaguerre) ; 910 } 926 } 911 927 912 ////////////////////////////////////////////// 928 //////////////////////////////////////////////////////////////////////// 913 // 929 // 914 // For use with global scope functions f << 930 // For use with global scope functions f 915 931 916 template <class T, class F> << 932 template <class T, class F> G4double 917 G4double G4Integrator<T, F>::Laguerre(G4double << 933 G4Integrator<T,F>::Laguerre( G4double (*f)(G4double), 918 G4int nL << 934 G4double alpha, G4int nLaguerre ) 919 { << 935 { 920 const G4double tolerance = 1.0e-10; << 936 const G4double tolerance = 1.0e-10 ; 921 const G4int maxNumber = 12; << 937 const G4int maxNumber = 12 ; 922 G4int i, j, k; << 938 G4int i, j, k ; 923 G4double nwt = 0., nwt1, temp1, temp2, << 939 G4double newton=0., newton1, temp1, temp2, temp3, temp, cofi ; 924 G4double integral = 0.0; << 940 G4double integral = 0.0 ; 925 << 941 926 G4int fNumber = nLaguerre; << 942 G4int fNumber = nLaguerre ; 927 G4double* fAbscissa = new G4double[fNumber]; << 943 G4double* fAbscissa = new G4double[fNumber] ; 928 G4double* fWeight = new G4double[fNumber]; << 944 G4double* fWeight = new G4double[fNumber] ; 929 << 945 930 for(i = 1; i <= fNumber; ++i) // Loop over << 946 for(i=1;i<=fNumber;i++) // Loop over the desired roots 931 { << 947 { 932 if(i == 1) << 948 if(i == 1) 933 { << 949 { 934 nwt = (1.0 + alpha) * (3.0 + 0.92 * alph << 950 newton = (1.0 + alpha)*(3.0 + 0.92*alpha) 935 (1.0 + 2.4 * fNumber + 1.8 * alpha << 951 / (1.0 + 2.4*fNumber + 1.8*alpha) ; 936 } << 952 } 937 else if(i == 2) << 953 else if(i == 2) 938 { << 954 { 939 nwt += (15.0 + 6.25 * alpha) / (1.0 + 0. << 955 newton += (15.0 + 6.25*alpha)/(1.0 + 0.9*alpha + 2.5*fNumber) ; 940 } << 956 } 941 else << 957 else 942 { << 958 { 943 cofi = i - 2; << 959 cofi = i - 2 ; 944 nwt += ((1.0 + 2.55 * cofi) / (1.9 * cof << 960 newton += ((1.0+2.55*cofi)/(1.9*cofi) 945 1.26 * cofi * alpha / (1.0 + 3.5 << 961 + 1.26*cofi*alpha/(1.0+3.5*cofi)) 946 (nwt - fAbscissa[i - 3]) / (1.0 + << 962 * (newton - fAbscissa[i-3])/(1.0 + 0.3*alpha) ; 947 } << 963 } 948 for(k = 1; k <= maxNumber; ++k) << 964 for(k=1;k<=maxNumber;k++) 949 { << 965 { 950 temp1 = 1.0; << 966 temp1 = 1.0 ; 951 temp2 = 0.0; << 967 temp2 = 0.0 ; 952 << 953 for(j = 1; j <= fNumber; ++j) << 954 { << 955 temp3 = temp2; << 956 temp2 = temp1; << 957 temp1 = << 958 ((2 * j - 1 + alpha - nwt) * temp2 - << 959 } << 960 temp = (fNumber * temp1 - (fNumber + alp << 961 nwt1 = nwt; << 962 nwt = nwt1 - temp1 / temp; << 963 << 964 if(std::fabs(nwt - nwt1) <= tolerance) << 965 { << 966 break; << 967 } << 968 } << 969 if(k > maxNumber) << 970 { << 971 G4Exception("G4Integrator<T,F>::Laguerre << 972 "Too many (>12) iterations." << 973 } << 974 << 975 fAbscissa[i - 1] = nwt; << 976 fWeight[i - 1] = -std::exp(GammaLogarith << 977 GammaLogarithm( << 978 (temp * fNumber * temp2); << 979 } << 980 << 981 // << 982 // Integral evaluation << 983 // << 984 968 985 for(i = 0; i < fNumber; i++) << 969 for(j=1;j<=fNumber;j++) 986 { << 970 { 987 integral += fWeight[i] * (*f)(fAbscissa[i] << 971 temp3 = temp2 ; 988 } << 972 temp2 = temp1 ; 989 delete[] fAbscissa; << 973 temp1 = ((2*j - 1 + alpha - newton)*temp2 - (j - 1 + alpha)*temp3)/j ; 990 delete[] fWeight; << 974 } 991 return integral; << 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 ; 992 } 1006 } 993 1007 994 ////////////////////////////////////////////// 1008 /////////////////////////////////////////////////////////////////////// 995 // 1009 // 996 // Auxiliary function which returns the value 1010 // Auxiliary function which returns the value of std::log(gamma-function(x)) 997 // Returns the value ln(Gamma(xx) for xx > 0. << 1011 // Returns the value ln(Gamma(xx) for xx > 0. Full accuracy is obtained for 998 // xx > 1. For 0 < xx < 1. the reflection form 1012 // xx > 1. For 0 < xx < 1. the reflection formula (6.1.4) can be used first. 999 // (Adapted from Numerical Recipes in C) 1013 // (Adapted from Numerical Recipes in C) 1000 // 1014 // 1001 1015 1002 template <class T, class F> 1016 template <class T, class F> 1003 G4double G4Integrator<T, F>::GammaLogarithm(G << 1017 G4double G4Integrator<T,F>::GammaLogarithm(G4double xx) 1004 { 1018 { 1005 static const G4double cof[6] = { 76.1800917 << 1019 static G4double cof[6] = { 76.18009172947146, -86.50532032941677, 1006 24.0140982 << 1020 24.01409824083091, -1.231739572450155, 1007 0.12086509 << 1021 0.1208650973866179e-2, -0.5395239384953e-5 } ; 1008 G4int j; << 1022 register G4int j; 1009 G4double x = xx - 1.0; << 1023 G4double x = xx - 1.0 ; 1010 G4double tmp = x + 5.5; << 1024 G4double tmp = x + 5.5 ; 1011 tmp -= (x + 0.5) * std::log(tmp); << 1025 tmp -= (x + 0.5) * std::log(tmp) ; 1012 G4double ser = 1.000000000190015; << 1026 G4double ser = 1.000000000190015 ; 1013 1027 1014 for(j = 0; j <= 5; ++j) << 1028 for ( j = 0; j <= 5; j++ ) 1015 { 1029 { 1016 x += 1.0; << 1030 x += 1.0 ; 1017 ser += cof[j] / x; << 1031 ser += cof[j]/x ; 1018 } 1032 } 1019 return -tmp + std::log(2.5066282746310005 * << 1033 return -tmp + std::log(2.5066282746310005*ser) ; 1020 } 1034 } 1021 1035 1022 ///////////////////////////////////////////// 1036 /////////////////////////////////////////////////////////////////////// 1023 // 1037 // 1024 // Method involving Hermite polynomials 1038 // Method involving Hermite polynomials 1025 // 1039 // 1026 ///////////////////////////////////////////// 1040 /////////////////////////////////////////////////////////////////////// 1027 // 1041 // 1028 // 1042 // 1029 // Gauss-Hermite method for integration of st << 1043 // Gauss-Hermite method for integration of std::exp(-x*x)*f(x) 1030 // from minus infinity to plus infinity . << 1044 // from minus infinity to plus infinity . 1031 // 1045 // 1032 1046 1033 template <class T, class F> << 1047 template <class T, class F> 1034 G4double G4Integrator<T, F>::Hermite(T& typeT << 1048 G4double G4Integrator<T,F>::Hermite( T& typeT, F f, G4int nHermite ) 1035 { 1049 { 1036 const G4double tolerance = 1.0e-12; << 1050 const G4double tolerance = 1.0e-12 ; 1037 const G4int maxNumber = 12; << 1051 const G4int maxNumber = 12 ; 1038 << 1052 1039 G4int i, j, k; << 1053 G4int i, j, k ; 1040 G4double integral = 0.0; << 1054 G4double integral = 0.0 ; 1041 G4double nwt = 0., nwt1, temp1, temp2, << 1055 G4double newton=0., newton1, temp1, temp2, temp3, temp ; 1042 1056 1043 G4double piInMinusQ = << 1057 G4double piInMinusQ = std::pow(pi,-0.25) ; // 1.0/std::sqrt(std::sqrt(pi)) ?? 1044 std::pow(CLHEP::pi, -0.25); // 1.0/std:: << 1045 1058 1046 G4int fNumber = (nHermite + 1) / 2; << 1059 G4int fNumber = (nHermite +1)/2 ; 1047 G4double* fAbscissa = new G4double[fNumber] << 1060 G4double* fAbscissa = new G4double[fNumber] ; 1048 G4double* fWeight = new G4double[fNumber] << 1061 G4double* fWeight = new G4double[fNumber] ; 1049 1062 1050 for(i = 1; i <= fNumber; ++i) << 1063 for(i=1;i<=fNumber;i++) 1051 { << 1064 { 1052 if(i == 1) << 1065 if(i == 1) 1053 { << 1066 { 1054 nwt = std::sqrt((G4double)(2 * nHermite << 1067 newton = std::sqrt((G4double)(2*nHermite + 1)) - 1055 1.85575001 * std::pow((G4double)( << 1068 1.85575001*std::pow((G4double)(2*nHermite + 1),-0.16666999) ; 1056 } << 1069 } 1057 else if(i == 2) << 1070 else if(i == 2) 1058 { << 1071 { 1059 nwt -= 1.14001 * std::pow((G4double) nH << 1072 newton -= 1.14001*std::pow((G4double)nHermite,0.425999)/newton ; 1060 } << 1073 } 1061 else if(i == 3) << 1074 else if(i == 3) 1062 { << 1075 { 1063 nwt = 1.86002 * nwt - 0.86002 * fAbscis << 1076 newton = 1.86002*newton - 0.86002*fAbscissa[0] ; 1064 } << 1077 } 1065 else if(i == 4) << 1078 else if(i == 4) 1066 { << 1079 { 1067 nwt = 1.91001 * nwt - 0.91001 * fAbscis << 1080 newton = 1.91001*newton - 0.91001*fAbscissa[1] ; 1068 } << 1081 } 1069 else << 1082 else 1070 { << 1083 { 1071 nwt = 2.0 * nwt - fAbscissa[i - 3]; << 1084 newton = 2.0*newton - fAbscissa[i - 3] ; 1072 } << 1085 } 1073 for(k = 1; k <= maxNumber; ++k) << 1086 for(k=1;k<=maxNumber;k++) 1074 { << 1087 { 1075 temp1 = piInMinusQ; << 1088 temp1 = piInMinusQ ; 1076 temp2 = 0.0; << 1089 temp2 = 0.0 ; 1077 << 1078 for(j = 1; j <= nHermite; ++j) << 1079 { << 1080 temp3 = temp2; << 1081 temp2 = temp1; << 1082 temp1 = nwt * std::sqrt(2.0 / j) * te << 1083 std::sqrt(((G4double)(j - 1)) << 1084 } << 1085 temp = std::sqrt((G4double) 2 * nHermit << 1086 nwt1 = nwt; << 1087 nwt = nwt1 - temp1 / temp; << 1088 << 1089 if(std::fabs(nwt - nwt1) <= tolerance) << 1090 { << 1091 break; << 1092 } << 1093 } << 1094 if(k > maxNumber) << 1095 { << 1096 G4Exception("G4Integrator<T,F>::Hermite << 1097 FatalException, "Too many ( << 1098 } << 1099 fAbscissa[i - 1] = nwt; << 1100 fWeight[i - 1] = 2.0 / (temp * temp); << 1101 } << 1102 << 1103 // << 1104 // Integral calculation << 1105 // << 1106 1090 1107 for(i = 0; i < fNumber; ++i) << 1091 for(j=1;j<=nHermite;j++) 1108 { << 1092 { 1109 integral += << 1093 temp3 = temp2 ; 1110 fWeight[i] * ((typeT.*f)(fAbscissa[i]) << 1094 temp2 = temp1 ; 1111 } << 1095 temp1 = newton*std::sqrt(2.0/j)*temp2 - 1112 delete[] fAbscissa; << 1096 std::sqrt(((G4double)(j - 1))/j)*temp3 ; 1113 delete[] fWeight; << 1097 } 1114 return integral; << 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 ; 1115 } 1128 } 1116 1129 >> 1130 1117 ///////////////////////////////////////////// 1131 //////////////////////////////////////////////////////////////////////// 1118 // 1132 // 1119 // For use with 'this' pointer 1133 // For use with 'this' pointer 1120 1134 1121 template <class T, class F> << 1135 template <class T, class F> 1122 G4double G4Integrator<T, F>::Hermite(T* ptrT, << 1136 G4double G4Integrator<T,F>::Hermite( T* ptrT, F f, G4int n ) 1123 { 1137 { 1124 return Hermite(*ptrT, f, n); << 1138 return Hermite(*ptrT,f,n) ; 1125 } << 1139 } 1126 1140 1127 ///////////////////////////////////////////// 1141 //////////////////////////////////////////////////////////////////////// 1128 // 1142 // 1129 // For use with global scope f 1143 // For use with global scope f 1130 1144 1131 template <class T, class F> 1145 template <class T, class F> 1132 G4double G4Integrator<T, F>::Hermite(G4double << 1146 G4double G4Integrator<T,F>::Hermite( G4double (*f)(G4double), G4int nHermite) 1133 { 1147 { 1134 const G4double tolerance = 1.0e-12; << 1148 const G4double tolerance = 1.0e-12 ; 1135 const G4int maxNumber = 12; << 1149 const G4int maxNumber = 12 ; 1136 << 1150 1137 G4int i, j, k; << 1151 G4int i, j, k ; 1138 G4double integral = 0.0; << 1152 G4double integral = 0.0 ; 1139 G4double nwt = 0., nwt1, temp1, temp2, << 1153 G4double newton=0., newton1, temp1, temp2, temp3, temp ; 1140 << 1154 1141 G4double piInMinusQ = << 1155 G4double piInMinusQ = std::pow(pi,-0.25) ; // 1.0/std::sqrt(std::sqrt(pi)) ?? 1142 std::pow(CLHEP::pi, -0.25); // 1.0/std:: << 1156 1143 << 1157 G4int fNumber = (nHermite +1)/2 ; 1144 G4int fNumber = (nHermite + 1) / 2; << 1158 G4double* fAbscissa = new G4double[fNumber] ; 1145 G4double* fAbscissa = new G4double[fNumber] << 1159 G4double* fWeight = new G4double[fNumber] ; 1146 G4double* fWeight = new G4double[fNumber] << 1160 1147 << 1161 for(i=1;i<=fNumber;i++) 1148 for(i = 1; i <= fNumber; ++i) << 1162 { 1149 { << 1163 if(i == 1) 1150 if(i == 1) << 1164 { 1151 { << 1165 newton = std::sqrt((G4double)(2*nHermite + 1)) - 1152 nwt = std::sqrt((G4double)(2 * nHermite << 1166 1.85575001*std::pow((G4double)(2*nHermite + 1),-0.16666999) ; 1153 1.85575001 * std::pow((G4double)( << 1167 } 1154 } << 1168 else if(i == 2) 1155 else if(i == 2) << 1169 { 1156 { << 1170 newton -= 1.14001*std::pow((G4double)nHermite,0.425999)/newton ; 1157 nwt -= 1.14001 * std::pow((G4double) nH << 1171 } 1158 } << 1172 else if(i == 3) 1159 else if(i == 3) << 1173 { 1160 { << 1174 newton = 1.86002*newton - 0.86002*fAbscissa[0] ; 1161 nwt = 1.86002 * nwt - 0.86002 * fAbscis << 1175 } 1162 } << 1176 else if(i == 4) 1163 else if(i == 4) << 1177 { 1164 { << 1178 newton = 1.91001*newton - 0.91001*fAbscissa[1] ; 1165 nwt = 1.91001 * nwt - 0.91001 * fAbscis << 1179 } 1166 } << 1180 else 1167 else << 1181 { 1168 { << 1182 newton = 2.0*newton - fAbscissa[i - 3] ; 1169 nwt = 2.0 * nwt - fAbscissa[i - 3]; << 1183 } 1170 } << 1184 for(k=1;k<=maxNumber;k++) 1171 for(k = 1; k <= maxNumber; ++k) << 1185 { 1172 { << 1186 temp1 = piInMinusQ ; 1173 temp1 = piInMinusQ; << 1187 temp2 = 0.0 ; 1174 temp2 = 0.0; << 1175 << 1176 for(j = 1; j <= nHermite; ++j) << 1177 { << 1178 temp3 = temp2; << 1179 temp2 = temp1; << 1180 temp1 = nwt * std::sqrt(2.0 / j) * te << 1181 std::sqrt(((G4double)(j - 1)) << 1182 } << 1183 temp = std::sqrt((G4double) 2 * nHermit << 1184 nwt1 = nwt; << 1185 nwt = nwt1 - temp1 / temp; << 1186 << 1187 if(std::fabs(nwt - nwt1) <= tolerance) << 1188 { << 1189 break; << 1190 } << 1191 } << 1192 if(k > maxNumber) << 1193 { << 1194 G4Exception("G4Integrator<T,F>::Hermite << 1195 "Too many (>12) iterations. << 1196 } << 1197 fAbscissa[i - 1] = nwt; << 1198 fWeight[i - 1] = 2.0 / (temp * temp); << 1199 } << 1200 << 1201 // << 1202 // Integral calculation << 1203 // << 1204 1188 1205 for(i = 0; i < fNumber; ++i) << 1189 for(j=1;j<=nHermite;j++) 1206 { << 1190 { 1207 integral += fWeight[i] * ((*f)(fAbscissa[ << 1191 temp3 = temp2 ; 1208 } << 1192 temp2 = temp1 ; 1209 delete[] fAbscissa; << 1193 temp1 = newton*std::sqrt(2.0/j)*temp2 - 1210 delete[] fWeight; << 1194 std::sqrt(((G4double)(j - 1))/j)*temp3 ; 1211 return integral; << 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 ; 1212 } 1225 } 1213 1226 1214 ///////////////////////////////////////////// 1227 //////////////////////////////////////////////////////////////////////////// 1215 // 1228 // 1216 // Method involving Jacobi polynomials 1229 // Method involving Jacobi polynomials 1217 // 1230 // 1218 ///////////////////////////////////////////// 1231 //////////////////////////////////////////////////////////////////////////// 1219 // 1232 // 1220 // Gauss-Jacobi method for integration of ((1 1233 // Gauss-Jacobi method for integration of ((1-x)^alpha)*((1+x)^beta)*f(x) 1221 // from minus unit to plus unit . 1234 // from minus unit to plus unit . 1222 // 1235 // 1223 1236 1224 template <class T, class F> << 1237 template <class T, class F> 1225 G4double G4Integrator<T, F>::Jacobi(T& typeT, << 1238 G4double G4Integrator<T,F>::Jacobi( T& typeT, F f, G4double alpha, 1226 G4double << 1239 G4double beta, G4int nJacobi) 1227 { << 1240 { 1228 const G4double tolerance = 1.0e-12; << 1241 const G4double tolerance = 1.0e-12 ; 1229 const G4double maxNumber = 12; << 1242 const G4double maxNumber = 12 ; 1230 G4int i, k, j; << 1243 G4int i, k, j ; 1231 G4double alphaBeta, alphaReduced, betaReduc << 1244 G4double alphaBeta, alphaReduced, betaReduced, root1=0., root2=0., root3=0. ; 1232 << 1245 G4double a, b, c, newton1, newton2, newton3, newton, temp, root=0., rootTemp ; 1233 G4double a, b, c, nwt1, nwt2, nwt3, nwt, te << 1246 1234 << 1247 G4int fNumber = nJacobi ; 1235 G4int fNumber = nJacobi; << 1248 G4double* fAbscissa = new G4double[fNumber] ; 1236 G4double* fAbscissa = new G4double[fNumber] << 1249 G4double* fWeight = new G4double[fNumber] ; 1237 G4double* fWeight = new G4double[fNumber] << 1250 1238 << 1251 for (i=1;i<=nJacobi;i++) 1239 for(i = 1; i <= nJacobi; ++i) << 1252 { 1240 { << 1253 if (i == 1) 1241 if(i == 1) << 1254 { 1242 { << 1255 alphaReduced = alpha/nJacobi ; 1243 alphaReduced = alpha / nJacobi; << 1256 betaReduced = beta/nJacobi ; 1244 betaReduced = beta / nJacobi; << 1257 root1 = (1.0+alpha)*(2.78002/(4.0+nJacobi*nJacobi)+ 1245 root1 = (1.0 + alpha) * (2.78002 << 1258 0.767999*alphaReduced/nJacobi) ; 1246 0.767999 * alp << 1259 root2 = 1.0+1.48*alphaReduced+0.96002*betaReduced + 1247 root2 = 1.0 + 1.48 * alphaReduce << 1260 0.451998*alphaReduced*alphaReduced + 1248 0.451998 * alphaReduced * alpha << 1261 0.83001*alphaReduced*betaReduced ; 1249 0.83001 * alphaReduced * betaRe << 1262 root = 1.0-root1/root2 ; 1250 root = 1.0 - root1 / root2; << 1263 } 1251 } << 1264 else if (i == 2) 1252 else if(i == 2) << 1265 { 1253 { << 1266 root1=(4.1002+alpha)/((1.0+alpha)*(1.0+0.155998*alpha)) ; 1254 root1 = (4.1002 + alpha) / ((1.0 + alph << 1267 root2=1.0+0.06*(nJacobi-8.0)*(1.0+0.12*alpha)/nJacobi ; 1255 root2 = 1.0 + 0.06 * (nJacobi - 8.0) * << 1268 root3=1.0+0.012002*beta*(1.0+0.24997*std::fabs(alpha))/nJacobi ; 1256 root3 = << 1269 root -= (1.0-root)*root1*root2*root3 ; 1257 1.0 + 0.012002 * beta * (1.0 + 0.2499 << 1270 } 1258 root -= (1.0 - root) * root1 * root2 * << 1271 else if (i == 3) 1259 } << 1272 { 1260 else if(i == 3) << 1273 root1=(1.67001+0.27998*alpha)/(1.0+0.37002*alpha) ; 1261 { << 1274 root2=1.0+0.22*(nJacobi-8.0)/nJacobi ; 1262 root1 = (1.67001 + 0.27998 * alpha) / ( << 1275 root3=1.0+8.0*beta/((6.28001+beta)*nJacobi*nJacobi) ; 1263 root2 = 1.0 + 0.22 * (nJacobi - 8.0) / << 1276 root -= (fAbscissa[0]-root)*root1*root2*root3 ; 1264 root3 = 1.0 + 8.0 * beta / ((6.28001 + << 1277 } 1265 root -= (fAbscissa[0] - root) * root1 * << 1278 else if (i == nJacobi-1) 1266 } << 1279 { 1267 else if(i == nJacobi - 1) << 1280 root1=(1.0+0.235002*beta)/(0.766001+0.118998*beta) ; 1268 { << 1281 root2=1.0/(1.0+0.639002*(nJacobi-4.0)/(1.0+0.71001*(nJacobi-4.0))) ; 1269 root1 = (1.0 + 0.235002 * beta) / (0.76 << 1282 root3=1.0/(1.0+20.0*alpha/((7.5+alpha)*nJacobi*nJacobi)) ; 1270 root2 = 1.0 / (1.0 + 0.639002 * (nJacob << 1283 root += (root-fAbscissa[nJacobi-4])*root1*root2*root3 ; 1271 (1.0 + 0.71001 * << 1284 } 1272 root3 = 1.0 / (1.0 + 20.0 * alpha / ((7 << 1285 else if (i == nJacobi) 1273 root += (root - fAbscissa[nJacobi - 4]) << 1286 { 1274 } << 1287 root1 = (1.0+0.37002*beta)/(1.67001+0.27998*beta) ; 1275 else if(i == nJacobi) << 1288 root2 = 1.0/(1.0+0.22*(nJacobi-8.0)/nJacobi) ; 1276 { << 1289 root3 = 1.0/(1.0+8.0*alpha/((6.28002+alpha)*nJacobi*nJacobi)) ; 1277 root1 = (1.0 + 0.37002 * beta) / (1.670 << 1290 root += (root-fAbscissa[nJacobi-3])*root1*root2*root3 ; 1278 root2 = 1.0 / (1.0 + 0.22 * (nJacobi - << 1291 } 1279 root3 = << 1292 else 1280 1.0 / (1.0 + 8.0 * alpha / ((6.28002 << 1293 { 1281 root += (root - fAbscissa[nJacobi - 3]) << 1294 root = 3.0*fAbscissa[i-2]-3.0*fAbscissa[i-3]+fAbscissa[i-4] ; 1282 } << 1295 } 1283 else << 1296 alphaBeta = alpha + beta ; 1284 { << 1297 for (k=1;k<=maxNumber;k++) 1285 root = 3.0 * fAbscissa[i - 2] - 3.0 * f << 1298 { 1286 } << 1299 temp = 2.0 + alphaBeta ; 1287 alphaBeta = alpha + beta; << 1300 newton1 = (alpha-beta+temp*root)/2.0 ; 1288 for(k = 1; k <= maxNumber; ++k) << 1301 newton2 = 1.0 ; 1289 { << 1302 for (j=2;j<=nJacobi;j++) 1290 temp = 2.0 + alphaBeta; << 1303 { 1291 nwt1 = (alpha - beta + temp * root) / 2 << 1304 newton3 = newton2 ; 1292 nwt2 = 1.0; << 1305 newton2 = newton1 ; 1293 for(j = 2; j <= nJacobi; ++j) << 1306 temp = 2*j+alphaBeta ; 1294 { << 1307 a = 2*j*(j+alphaBeta)*(temp-2.0) ; 1295 nwt3 = nwt2; << 1308 b = (temp-1.0)*(alpha*alpha-beta*beta+temp*(temp-2.0)*root) ; 1296 nwt2 = nwt1; << 1309 c = 2.0*(j-1+alpha)*(j-1+beta)*temp ; 1297 temp = 2 * j + alphaBeta; << 1310 newton1 = (b*newton2-c*newton3)/a ; 1298 a = 2 * j * (j + alphaBeta) * (tem << 1311 } 1299 b = (temp - 1.0) * << 1312 newton = (nJacobi*(alpha - beta - temp*root)*newton1 + 1300 (alpha * alpha - beta * beta + te << 1313 2.0*(nJacobi + alpha)*(nJacobi + beta)*newton2)/ 1301 c = 2.0 * (j - 1 + alpha) * (j - 1 << 1314 (temp*(1.0 - root*root)) ; 1302 nwt1 = (b * nwt2 - c * nwt3) / a; << 1315 rootTemp = root ; 1303 } << 1316 root = rootTemp - newton1/newton ; 1304 nwt = (nJacobi * (alpha - beta - temp * << 1317 if (std::fabs(root-rootTemp) <= tolerance) 1305 2.0 * (nJacobi + alpha) * (nJaco << 1318 { 1306 (temp * (1.0 - root * root)); << 1319 break ; 1307 rootTemp = root; << 1320 } 1308 root = rootTemp - nwt1 / nwt; << 1321 } 1309 if(std::fabs(root - rootTemp) <= tolera << 1322 if (k > maxNumber) 1310 { << 1323 { 1311 break; << 1324 G4Exception("G4Integrator<T,F>::Jacobi(T,F, ...)", "Error", 1312 } << 1325 FatalException, "Too many (>12) iterations."); 1313 } << 1326 } 1314 if(k > maxNumber) << 1327 fAbscissa[i-1] = root ; 1315 { << 1328 fWeight[i-1] = std::exp(GammaLogarithm((G4double)(alpha+nJacobi)) + 1316 G4Exception("G4Integrator<T,F>::Jacobi( << 1329 GammaLogarithm((G4double)(beta+nJacobi)) - 1317 FatalException, "Too many ( << 1330 GammaLogarithm((G4double)(nJacobi+1.0)) - 1318 } << 1331 GammaLogarithm((G4double)(nJacobi + alphaBeta + 1.0))) 1319 fAbscissa[i - 1] = root; << 1332 *temp*std::pow(2.0,alphaBeta)/(newton*newton2) ; 1320 fWeight[i - 1] = << 1333 } 1321 std::exp(GammaLogarithm((G4double)(alph << 1334 1322 GammaLogarithm((G4double)(beta << 1335 // 1323 GammaLogarithm((G4double)(nJac << 1336 // Calculation of the integral 1324 GammaLogarithm((G4double)(nJac << 1337 // 1325 temp * std::pow(2.0, alphaBeta) / (nwt << 1338 1326 } << 1339 G4double integral = 0.0 ; 1327 << 1340 for(i=0;i<fNumber;i++) 1328 // << 1341 { 1329 // Calculation of the integral << 1342 integral += fWeight[i]*(typeT.*f)(fAbscissa[i]) ; 1330 // << 1343 } 1331 << 1344 delete[] fAbscissa; 1332 G4double integral = 0.0; << 1345 delete[] fWeight; 1333 for(i = 0; i < fNumber; ++i) << 1346 return integral ; 1334 { << 1335 integral += fWeight[i] * (typeT.*f)(fAbsc << 1336 } << 1337 delete[] fAbscissa; << 1338 delete[] fWeight; << 1339 return integral; << 1340 } 1347 } 1341 1348 >> 1349 1342 ///////////////////////////////////////////// 1350 ///////////////////////////////////////////////////////////////////////// 1343 // 1351 // 1344 // For use with 'this' pointer 1352 // For use with 'this' pointer 1345 1353 1346 template <class T, class F> << 1354 template <class T, class F> 1347 G4double G4Integrator<T, F>::Jacobi(T* ptrT, << 1355 G4double G4Integrator<T,F>::Jacobi( T* ptrT, F f, G4double alpha, 1348 G4int n) << 1356 G4double beta, G4int n) 1349 { 1357 { 1350 return Jacobi(*ptrT, f, alpha, beta, n); << 1358 return Jacobi(*ptrT,f,alpha,beta,n) ; 1351 } << 1359 } 1352 1360 1353 ///////////////////////////////////////////// 1361 ///////////////////////////////////////////////////////////////////////// 1354 // 1362 // 1355 // For use with global scope f << 1363 // For use with global scope f 1356 1364 1357 template <class T, class F> 1365 template <class T, class F> 1358 G4double G4Integrator<T, F>::Jacobi(G4double << 1366 G4double G4Integrator<T,F>::Jacobi( G4double (*f)(G4double), G4double alpha, 1359 G4double << 1367 G4double beta, G4int nJacobi) 1360 { 1368 { 1361 const G4double tolerance = 1.0e-12; << 1369 const G4double tolerance = 1.0e-12 ; 1362 const G4double maxNumber = 12; << 1370 const G4double maxNumber = 12 ; 1363 G4int i, k, j; << 1371 G4int i, k, j ; 1364 G4double alphaBeta, alphaReduced, betaReduc << 1372 G4double alphaBeta, alphaReduced, betaReduced, root1=0., root2=0., root3=0. ; 1365 << 1373 G4double a, b, c, newton1, newton2, newton3, newton, temp, root=0., rootTemp ; 1366 G4double a, b, c, nwt1, nwt2, nwt3, nwt, te << 1374 1367 << 1375 G4int fNumber = nJacobi ; 1368 G4int fNumber = nJacobi; << 1376 G4double* fAbscissa = new G4double[fNumber] ; 1369 G4double* fAbscissa = new G4double[fNumber] << 1377 G4double* fWeight = new G4double[fNumber] ; 1370 G4double* fWeight = new G4double[fNumber] << 1378 1371 << 1379 for (i=1;i<=nJacobi;i++) 1372 for(i = 1; i <= nJacobi; ++i) << 1380 { 1373 { << 1381 if (i == 1) 1374 if(i == 1) << 1382 { 1375 { << 1383 alphaReduced = alpha/nJacobi ; 1376 alphaReduced = alpha / nJacobi; << 1384 betaReduced = beta/nJacobi ; 1377 betaReduced = beta / nJacobi; << 1385 root1 = (1.0+alpha)*(2.78002/(4.0+nJacobi*nJacobi)+ 1378 root1 = (1.0 + alpha) * (2.78002 << 1386 0.767999*alphaReduced/nJacobi) ; 1379 0.767999 * alp << 1387 root2 = 1.0+1.48*alphaReduced+0.96002*betaReduced + 1380 root2 = 1.0 + 1.48 * alphaReduce << 1388 0.451998*alphaReduced*alphaReduced + 1381 0.451998 * alphaReduced * alpha << 1389 0.83001*alphaReduced*betaReduced ; 1382 0.83001 * alphaReduced * betaRe << 1390 root = 1.0-root1/root2 ; 1383 root = 1.0 - root1 / root2; << 1391 } 1384 } << 1392 else if (i == 2) 1385 else if(i == 2) << 1393 { 1386 { << 1394 root1=(4.1002+alpha)/((1.0+alpha)*(1.0+0.155998*alpha)) ; 1387 root1 = (4.1002 + alpha) / ((1.0 + alph << 1395 root2=1.0+0.06*(nJacobi-8.0)*(1.0+0.12*alpha)/nJacobi ; 1388 root2 = 1.0 + 0.06 * (nJacobi - 8.0) * << 1396 root3=1.0+0.012002*beta*(1.0+0.24997*std::fabs(alpha))/nJacobi ; 1389 root3 = << 1397 root -= (1.0-root)*root1*root2*root3 ; 1390 1.0 + 0.012002 * beta * (1.0 + 0.2499 << 1398 } 1391 root -= (1.0 - root) * root1 * root2 * << 1399 else if (i == 3) 1392 } << 1400 { 1393 else if(i == 3) << 1401 root1=(1.67001+0.27998*alpha)/(1.0+0.37002*alpha) ; 1394 { << 1402 root2=1.0+0.22*(nJacobi-8.0)/nJacobi ; 1395 root1 = (1.67001 + 0.27998 * alpha) / ( << 1403 root3=1.0+8.0*beta/((6.28001+beta)*nJacobi*nJacobi) ; 1396 root2 = 1.0 + 0.22 * (nJacobi - 8.0) / << 1404 root -= (fAbscissa[0]-root)*root1*root2*root3 ; 1397 root3 = 1.0 + 8.0 * beta / ((6.28001 + << 1405 } 1398 root -= (fAbscissa[0] - root) * root1 * << 1406 else if (i == nJacobi-1) 1399 } << 1407 { 1400 else if(i == nJacobi - 1) << 1408 root1=(1.0+0.235002*beta)/(0.766001+0.118998*beta) ; 1401 { << 1409 root2=1.0/(1.0+0.639002*(nJacobi-4.0)/(1.0+0.71001*(nJacobi-4.0))) ; 1402 root1 = (1.0 + 0.235002 * beta) / (0.76 << 1410 root3=1.0/(1.0+20.0*alpha/((7.5+alpha)*nJacobi*nJacobi)) ; 1403 root2 = 1.0 / (1.0 + 0.639002 * (nJacob << 1411 root += (root-fAbscissa[nJacobi-4])*root1*root2*root3 ; 1404 (1.0 + 0.71001 * << 1412 } 1405 root3 = 1.0 / (1.0 + 20.0 * alpha / ((7 << 1413 else if (i == nJacobi) 1406 root += (root - fAbscissa[nJacobi - 4]) << 1414 { 1407 } << 1415 root1 = (1.0+0.37002*beta)/(1.67001+0.27998*beta) ; 1408 else if(i == nJacobi) << 1416 root2 = 1.0/(1.0+0.22*(nJacobi-8.0)/nJacobi) ; 1409 { << 1417 root3 = 1.0/(1.0+8.0*alpha/((6.28002+alpha)*nJacobi*nJacobi)) ; 1410 root1 = (1.0 + 0.37002 * beta) / (1.670 << 1418 root += (root-fAbscissa[nJacobi-3])*root1*root2*root3 ; 1411 root2 = 1.0 / (1.0 + 0.22 * (nJacobi - << 1419 } 1412 root3 = << 1420 else 1413 1.0 / (1.0 + 8.0 * alpha / ((6.28002 << 1421 { 1414 root += (root - fAbscissa[nJacobi - 3]) << 1422 root = 3.0*fAbscissa[i-2]-3.0*fAbscissa[i-3]+fAbscissa[i-4] ; 1415 } << 1423 } 1416 else << 1424 alphaBeta = alpha + beta ; 1417 { << 1425 for (k=1;k<=maxNumber;k++) 1418 root = 3.0 * fAbscissa[i - 2] - 3.0 * f << 1426 { 1419 } << 1427 temp = 2.0 + alphaBeta ; 1420 alphaBeta = alpha + beta; << 1428 newton1 = (alpha-beta+temp*root)/2.0 ; 1421 for(k = 1; k <= maxNumber; ++k) << 1429 newton2 = 1.0 ; 1422 { << 1430 for (j=2;j<=nJacobi;j++) 1423 temp = 2.0 + alphaBeta; << 1431 { 1424 nwt1 = (alpha - beta + temp * root) / 2 << 1432 newton3 = newton2 ; 1425 nwt2 = 1.0; << 1433 newton2 = newton1 ; 1426 for(j = 2; j <= nJacobi; ++j) << 1434 temp = 2*j+alphaBeta ; 1427 { << 1435 a = 2*j*(j+alphaBeta)*(temp-2.0) ; 1428 nwt3 = nwt2; << 1436 b = (temp-1.0)*(alpha*alpha-beta*beta+temp*(temp-2.0)*root) ; 1429 nwt2 = nwt1; << 1437 c = 2.0*(j-1+alpha)*(j-1+beta)*temp ; 1430 temp = 2 * j + alphaBeta; << 1438 newton1 = (b*newton2-c*newton3)/a ; 1431 a = 2 * j * (j + alphaBeta) * (tem << 1439 } 1432 b = (temp - 1.0) * << 1440 newton = (nJacobi*(alpha - beta - temp*root)*newton1 + 1433 (alpha * alpha - beta * beta + te << 1441 2.0*(nJacobi + alpha)*(nJacobi + beta)*newton2) / 1434 c = 2.0 * (j - 1 + alpha) * (j - 1 << 1442 (temp*(1.0 - root*root)) ; 1435 nwt1 = (b * nwt2 - c * nwt3) / a; << 1443 rootTemp = root ; 1436 } << 1444 root = rootTemp - newton1/newton ; 1437 nwt = (nJacobi * (alpha - beta - temp * << 1445 if (std::fabs(root-rootTemp) <= tolerance) 1438 2.0 * (nJacobi + alpha) * (nJaco << 1446 { 1439 (temp * (1.0 - root * root)); << 1447 break ; 1440 rootTemp = root; << 1448 } 1441 root = rootTemp - nwt1 / nwt; << 1449 } 1442 if(std::fabs(root - rootTemp) <= tolera << 1450 if (k > maxNumber) 1443 { << 1451 { 1444 break; << 1452 G4Exception("G4Integrator<T,F>::Jacobi(...)", "Error", 1445 } << 1453 FatalException, "Too many (>12) iterations."); 1446 } << 1454 } 1447 if(k > maxNumber) << 1455 fAbscissa[i-1] = root ; 1448 { << 1456 fWeight[i-1] = 1449 G4Exception("G4Integrator<T,F>::Jacobi( << 1457 std::exp(GammaLogarithm((G4double)(alpha+nJacobi)) + 1450 "Too many (>12) iterations. << 1458 GammaLogarithm((G4double)(beta+nJacobi)) - 1451 } << 1459 GammaLogarithm((G4double)(nJacobi+1.0)) - 1452 fAbscissa[i - 1] = root; << 1460 GammaLogarithm((G4double)(nJacobi + alphaBeta + 1.0))) 1453 fWeight[i - 1] = << 1461 *temp*std::pow(2.0,alphaBeta)/(newton*newton2); 1454 std::exp(GammaLogarithm((G4double)(alph << 1462 } 1455 GammaLogarithm((G4double)(beta << 1463 1456 GammaLogarithm((G4double)(nJac << 1464 // 1457 GammaLogarithm((G4double)(nJac << 1465 // Calculation of the integral 1458 temp * std::pow(2.0, alphaBeta) / (nwt << 1466 // 1459 } << 1467 1460 << 1468 G4double integral = 0.0 ; 1461 // << 1469 for(i=0;i<fNumber;i++) 1462 // Calculation of the integral << 1470 { 1463 // << 1471 integral += fWeight[i]*(*f)(fAbscissa[i]) ; 1464 << 1472 } 1465 G4double integral = 0.0; << 1473 delete[] fAbscissa; 1466 for(i = 0; i < fNumber; ++i) << 1474 delete[] fWeight; 1467 { << 1475 return integral ; 1468 integral += fWeight[i] * (*f)(fAbscissa[i << 1469 } << 1470 delete[] fAbscissa; << 1471 delete[] fWeight; << 1472 return integral; << 1473 } 1476 } 1474 1477 1475 // 1478 // 1476 // 1479 // 1477 ///////////////////////////////////////////// 1480 /////////////////////////////////////////////////////////////////// 1478 1481