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>> 1 // This code implementation is the intellectual property of >> 2 // the GEANT4 collaboration. 1 // 3 // 2 // ******************************************* << 4 // By copying, distributing or modifying the Program (or any work 3 // * License and Disclaimer << 5 // based on the Program) you indicate your acceptance of this statement, 4 // * << 6 // and all its terms. 5 // * The Geant4 software is copyright of th << 7 // 6 // * the Geant4 Collaboration. It is provided << 8 // $Id: G4Integrator.icc,v 1.6 2000/06/15 17:31:23 gcosmo Exp $ 7 // * conditions of the Geant4 Software License << 9 // GEANT4 tag $Name: geant4-02-00 $ 8 // * LICENSE and available at http://cern.ch/ << 10 // 9 // * include a list of copyright holders. << 11 // Implementation of G4Integrator methods. 10 // * << 12 // 11 // * Neither the authors of this software syst << 13 // 12 // * institutes,nor the agencies providing fin << 13 // * work make any representation or warran << 14 // * regarding this software system or assum << 15 // * use. Please see the license in the file << 16 // * for the full disclaimer and the limitatio << 17 // * << 18 // * This code implementation is the result << 19 // * technical work of the GEANT4 collaboratio << 20 // * By using, copying, modifying or distri << 21 // * any work based on the software) you ag << 22 // * use in resulting scientific publicati << 23 // * acceptance of all terms of the Geant4 Sof << 24 // ******************************************* << 25 // << 26 // G4Integrator inline methods implementation << 27 // << 28 // Author: V.Grichine, 04.09.1999 - First impl << 29 // G4SimpleIntegration class with H.P. << 30 // E.TCherniaev advises << 31 // ------------------------------------------- << 32 14 33 ////////////////////////////////////////////// 15 ///////////////////////////////////////////////////////////////////// 34 // 16 // 35 // Sympson integration method 17 // Sympson integration method 36 // 18 // 37 ////////////////////////////////////////////// 19 ///////////////////////////////////////////////////////////////////// 38 // 20 // 39 // Integration of class member functions T::f << 21 // Integration of class member functions T::f by Simpson method. 40 22 41 template <class T, class F> << 23 template <class T, class F> 42 G4double G4Integrator<T, F>::Simpson(T& typeT, << 24 G4double G4Integrator<T,F>::Simpson( T& typeT, 43 G4double << 25 F f, 44 { << 26 G4double xInitial, 45 G4int i; << 27 G4double xFinal, 46 G4double step = (xFinal - xInitial) / itera << 28 G4int iterationNumber ) 47 G4double x = xInitial; << 29 { 48 G4double xPlus = xInitial + 0.5 * step; << 30 G4int i ; 49 G4double mean = ((typeT.*f)(xInitial) + (ty << 31 G4double step = (xFinal - xInitial)/iterationNumber ; 50 G4double sum = (typeT.*f)(xPlus); << 32 G4double x = xInitial ; >> 33 G4double xPlus = xInitial + 0.5*step ; >> 34 G4double mean = ( (typeT.*f)(xInitial) + (typeT.*f)(xFinal) )*0.5 ; >> 35 G4double sum = (typeT.*f)(xPlus) ; >> 36 >> 37 for(i=1;i<iterationNumber;i++) >> 38 { >> 39 x += step ; >> 40 xPlus += step ; >> 41 mean += (typeT.*f)(x) ; >> 42 sum += (typeT.*f)(xPlus) ; >> 43 } >> 44 mean += 2.0*sum ; 51 45 52 for(i = 1; i < iterationNumber; ++i) << 46 return mean*step/3.0 ; 53 { << 54 x += step; << 55 xPlus += step; << 56 mean += (typeT.*f)(x); << 57 sum += (typeT.*f)(xPlus); << 58 } << 59 mean += 2.0 * sum; << 60 << 61 return mean * step / 3.0; << 62 } 47 } 63 48 64 ////////////////////////////////////////////// 49 ///////////////////////////////////////////////////////////////////// 65 // 50 // 66 // Integration of class member functions T::f 51 // Integration of class member functions T::f by Simpson method. 67 // Convenient to use with 'this' pointer 52 // Convenient to use with 'this' pointer 68 53 69 template <class T, class F> << 54 template <class T, class F> 70 G4double G4Integrator<T, F>::Simpson(T* ptrT, << 55 G4double G4Integrator<T,F>::Simpson( T* ptrT, 71 G4double << 56 F f, 72 { << 57 G4double xInitial, 73 G4int i; << 58 G4double xFinal, 74 G4double step = (xFinal - xInitial) / itera << 59 G4int iterationNumber ) 75 G4double x = xInitial; << 60 { 76 G4double xPlus = xInitial + 0.5 * step; << 61 G4int i ; 77 G4double mean = ((ptrT->*f)(xInitial) + (pt << 62 G4double step = (xFinal - xInitial)/iterationNumber ; 78 G4double sum = (ptrT->*f)(xPlus); << 63 G4double x = xInitial ; >> 64 G4double xPlus = xInitial + 0.5*step ; >> 65 G4double mean = ( (ptrT->*f)(xInitial) + (ptrT->*f)(xFinal) )*0.5 ; >> 66 G4double sum = (ptrT->*f)(xPlus) ; >> 67 >> 68 for(i=1;i<iterationNumber;i++) >> 69 { >> 70 x += step ; >> 71 xPlus += step ; >> 72 mean += (ptrT->*f)(x) ; >> 73 sum += (ptrT->*f)(xPlus) ; >> 74 } >> 75 mean += 2.0*sum ; 79 76 80 for(i = 1; i < iterationNumber; ++i) << 77 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 } 78 } 91 79 92 ////////////////////////////////////////////// 80 ///////////////////////////////////////////////////////////////////// 93 // 81 // 94 // Integration of class member functions T::f 82 // Integration of class member functions T::f by Simpson method. 95 // Convenient to use, when function f is defin 83 // Convenient to use, when function f is defined in global scope, i.e. in main() 96 // program 84 // program 97 85 98 template <class T, class F> << 86 template <class T, class F> 99 G4double G4Integrator<T, F>::Simpson(G4double << 87 G4double G4Integrator<T,F>::Simpson( G4double (*f)(G4double), 100 G4double << 88 G4double xInitial, 101 { << 89 G4double xFinal, 102 G4int i; << 90 G4int iterationNumber ) 103 G4double step = (xFinal - xInitial) / itera << 91 { 104 G4double x = xInitial; << 92 G4int i ; 105 G4double xPlus = xInitial + 0.5 * step; << 93 G4double step = (xFinal - xInitial)/iterationNumber ; 106 G4double mean = ((*f)(xInitial) + (*f)(xFin << 94 G4double x = xInitial ; 107 G4double sum = (*f)(xPlus); << 95 G4double xPlus = xInitial + 0.5*step ; 108 << 96 G4double mean = ( (*f)(xInitial) + (*f)(xFinal) )*0.5 ; 109 for(i = 1; i < iterationNumber; ++i) << 97 G4double sum = (*f)(xPlus) ; 110 { << 98 111 x += step; << 99 for(i=1;i<iterationNumber;i++) 112 xPlus += step; << 100 { 113 mean += (*f)(x); << 101 x += step ; 114 sum += (*f)(xPlus); << 102 xPlus += step ; 115 } << 103 mean += (*f)(x) ; 116 mean += 2.0 * sum; << 104 sum += (*f)(xPlus) ; >> 105 } >> 106 mean += 2.0*sum ; 117 107 118 return mean * step / 3.0; << 108 return mean*step/3.0 ; 119 } 109 } 120 110 121 ////////////////////////////////////////////// 111 ////////////////////////////////////////////////////////////////////////// 122 // 112 // 123 // Adaptive Gauss method 113 // Adaptive Gauss method 124 // 114 // 125 ////////////////////////////////////////////// 115 ////////////////////////////////////////////////////////////////////////// 126 // 116 // 127 // 117 // 128 118 129 template <class T, class F> << 119 template <class T, class F> 130 G4double G4Integrator<T, F>::Gauss(T& typeT, F << 120 G4double G4Integrator<T,F>::Gauss( T& typeT, F f, 131 G4double xF << 121 G4double xInitial, G4double xFinal ) 132 { << 122 { 133 static const G4double root = 1.0 / std::sqrt << 123 static G4double root = 1.0/sqrt(3.0) ; 134 << 124 135 G4double xMean = (xInitial + xFinal) / 2.0; << 125 G4double xMean = (xInitial + xFinal)/2.0 ; 136 G4double Step = (xFinal - xInitial) / 2.0; << 126 G4double Step = (xFinal - xInitial)/2.0 ; 137 G4double delta = Step * root; << 127 G4double delta = Step*root ; 138 G4double sum = ((typeT.*f)(xMean + delta) << 128 G4double sum = ((typeT.*f)(xMean + delta) + 139 << 129 (typeT.*f)(xMean - delta)) ; 140 return sum * Step; << 130 >> 131 return sum*Step ; 141 } 132 } 142 133 143 ////////////////////////////////////////////// 134 ////////////////////////////////////////////////////////////////////// 144 // 135 // 145 // 136 // 146 137 147 template <class T, class F> << 138 template <class T, class F> G4double 148 G4double G4Integrator<T, F>::Gauss(T* ptrT, F << 139 G4Integrator<T,F>::Gauss( T* ptrT, F f, G4double a, G4double b ) 149 { 140 { 150 return Gauss(*ptrT, f, a, b); << 141 return Gauss(*ptrT,f,a,b) ; 151 } 142 } 152 143 153 ////////////////////////////////////////////// 144 /////////////////////////////////////////////////////////////////////// 154 // 145 // 155 // 146 // 156 147 157 template <class T, class F> 148 template <class T, class F> 158 G4double G4Integrator<T, F>::Gauss(G4double (* << 149 G4double G4Integrator<T,F>::Gauss( G4double (*f)(G4double), 159 G4double xF << 150 G4double xInitial, G4double xFinal) 160 { 151 { 161 static const G4double root = 1.0 / std::sqrt << 152 static G4double root = 1.0/sqrt(3.0) ; 162 << 153 163 G4double xMean = (xInitial + xFinal) / 2.0; << 154 G4double xMean = (xInitial + xFinal)/2.0 ; 164 G4double Step = (xFinal - xInitial) / 2.0; << 155 G4double Step = (xFinal - xInitial)/2.0 ; 165 G4double delta = Step * root; << 156 G4double delta = Step*root ; 166 G4double sum = ((*f)(xMean + delta) + (*f) << 157 G4double sum = ( (*f)(xMean + delta) + (*f)(xMean - delta) ) ; 167 << 158 168 return sum * Step; << 159 return sum*Step ; 169 } 160 } 170 161 171 ////////////////////////////////////////////// 162 /////////////////////////////////////////////////////////////////////////// 172 // 163 // 173 // 164 // 174 165 175 template <class T, class F> << 166 template <class T, class F> 176 void G4Integrator<T, F>::AdaptGauss(T& typeT, << 167 void G4Integrator<T,F>::AdaptGauss( T& typeT, F f, G4double xInitial, 177 G4double x << 168 G4double xFinal, G4double fTolerance, 178 G4double& << 169 G4double& sum, 179 { << 170 G4int& depth ) 180 if(depth > 100) << 171 { 181 { << 172 if(depth > 100) 182 G4cout << "G4Integrator<T,F>::AdaptGauss: << 173 { 183 G4cout << "Function varies too rapidly to << 174 G4cout<<"G4Integrator<T,F>::AdaptGauss: WARNING !!!"<<G4endl ; 184 << G4endl; << 175 G4cout 185 << 176 <<"Function varies too rapidly to get stated accuracy in 100 steps "<<G4endl ; 186 return; << 177 187 } << 178 return ; 188 G4double xMean = (xInitial + xFinal) / 2 << 179 } 189 G4double leftHalf = Gauss(typeT, f, xInitia << 180 G4double xMean = (xInitial + xFinal)/2.0 ; 190 G4double rightHalf = Gauss(typeT, f, xMean, << 181 G4double leftHalf = Gauss(typeT,f,xInitial,xMean) ; 191 G4double full = Gauss(typeT, f, xInitia << 182 G4double rightHalf = Gauss(typeT,f,xMean,xFinal) ; 192 if(std::fabs(leftHalf + rightHalf - full) < << 183 G4double full = Gauss(typeT,f,xInitial,xFinal) ; 193 { << 184 if(fabs(leftHalf+rightHalf-full) < fTolerance) 194 sum += full; << 185 { 195 } << 186 sum += full ; 196 else << 187 } 197 { << 188 else 198 ++depth; << 189 { 199 AdaptGauss(typeT, f, xInitial, xMean, fTol << 190 depth++ ; 200 AdaptGauss(typeT, f, xMean, xFinal, fToler << 191 AdaptGauss(typeT,f,xInitial,xMean,fTolerance,sum,depth) ; 201 } << 192 AdaptGauss(typeT,f,xMean,xFinal,fTolerance,sum,depth) ; 202 } << 193 } 203 << 194 } 204 template <class T, class F> << 195 205 void G4Integrator<T, F>::AdaptGauss(T* ptrT, F << 196 template <class T, class F> 206 G4double x << 197 void G4Integrator<T,F>::AdaptGauss( T* ptrT, F f, G4double xInitial, 207 G4double& << 198 G4double xFinal, G4double fTolerance, >> 199 G4double& sum, >> 200 G4int& depth ) 208 { 201 { 209 AdaptGauss(*ptrT, f, xInitial, xFinal, fTole << 202 AdaptGauss(*ptrT,f,xInitial,xFinal,fTolerance,sum,depth) ; 210 } 203 } 211 204 212 ////////////////////////////////////////////// 205 ///////////////////////////////////////////////////////////////////////// 213 // 206 // 214 // 207 // 215 template <class T, class F> 208 template <class T, class F> 216 void G4Integrator<T, F>::AdaptGauss(G4double ( << 209 void G4Integrator<T,F>::AdaptGauss( G4double (*f)(G4double), 217 G4double x << 210 G4double xInitial, G4double xFinal, 218 G4double& << 211 G4double fTolerance, G4double& sum, 219 { << 212 G4int& depth ) 220 if(depth > 100) << 213 { 221 { << 214 if(depth > 100) 222 G4cout << "G4SimpleIntegration::AdaptGauss << 215 { 223 G4cout << "Function varies too rapidly to << 216 G4cout<<"G4SimpleIntegration::AdaptGauss: WARNING !!!"<<G4endl ; 224 << G4endl; << 217 G4cout<<"Function varies too rapidly to get stated accuracy in 100 steps " 225 << 218 <<G4endl ; 226 return; << 219 227 } << 220 return ; 228 G4double xMean = (xInitial + xFinal) / 2 << 221 } 229 G4double leftHalf = Gauss(f, xInitial, xMea << 222 G4double xMean = (xInitial + xFinal)/2.0 ; 230 G4double rightHalf = Gauss(f, xMean, xFinal) << 223 G4double leftHalf = Gauss(f,xInitial,xMean) ; 231 G4double full = Gauss(f, xInitial, xFin << 224 G4double rightHalf = Gauss(f,xMean,xFinal) ; 232 if(std::fabs(leftHalf + rightHalf - full) < << 225 G4double full = Gauss(f,xInitial,xFinal) ; 233 { << 226 if(fabs(leftHalf+rightHalf-full) < fTolerance) 234 sum += full; << 227 { 235 } << 228 sum += full ; 236 else << 229 } 237 { << 230 else 238 ++depth; << 231 { 239 AdaptGauss(f, xInitial, xMean, fTolerance, << 232 depth++ ; 240 AdaptGauss(f, xMean, xFinal, fTolerance, s << 233 AdaptGauss(f,xInitial,xMean,fTolerance,sum,depth) ; 241 } << 234 AdaptGauss(f,xMean,xFinal,fTolerance,sum,depth) ; >> 235 } 242 } 236 } 243 237 >> 238 >> 239 >> 240 244 ////////////////////////////////////////////// 241 //////////////////////////////////////////////////////////////////////// 245 // 242 // 246 // Adaptive Gauss integration with accuracy 'e 243 // Adaptive Gauss integration with accuracy 'e' 247 // Convenient for using with class object type 244 // Convenient for using with class object typeT 248 << 245 249 template <class T, class F> << 246 template<class T, class F> G4double 250 G4double G4Integrator<T, F>::AdaptiveGauss(T& << 247 G4Integrator<T,F>::AdaptiveGauss( T& typeT, F f, G4double xInitial, 251 G4d << 248 G4double xFinal, G4double e ) 252 { << 249 { 253 G4int depth = 0; << 250 G4int depth = 0 ; 254 G4double sum = 0.0; << 251 G4double sum = 0.0 ; 255 AdaptGauss(typeT, f, xInitial, xFinal, e, su << 252 AdaptGauss(typeT,f,xInitial,xFinal,e,sum,depth) ; 256 return sum; << 253 return sum ; 257 } 254 } 258 255 259 ////////////////////////////////////////////// 256 //////////////////////////////////////////////////////////////////////// 260 // 257 // 261 // Adaptive Gauss integration with accuracy 'e 258 // Adaptive Gauss integration with accuracy 'e' 262 // Convenient for using with 'this' pointer 259 // Convenient for using with 'this' pointer 263 << 260 264 template <class T, class F> << 261 template<class T, class F> G4double 265 G4double G4Integrator<T, F>::AdaptiveGauss(T* << 262 G4Integrator<T,F>::AdaptiveGauss( T* ptrT, F f, G4double xInitial, 266 G4d << 263 G4double xFinal, G4double e ) 267 { 264 { 268 return AdaptiveGauss(*ptrT, f, xInitial, xFi << 265 return AdaptiveGauss(*ptrT,f,xInitial,xFinal,e) ; 269 } 266 } 270 267 271 ////////////////////////////////////////////// 268 //////////////////////////////////////////////////////////////////////// 272 // 269 // 273 // Adaptive Gauss integration with accuracy 'e 270 // Adaptive Gauss integration with accuracy 'e' 274 // Convenient for using with global scope func 271 // Convenient for using with global scope function f 275 << 272 276 template <class T, class F> << 273 template <class T, class F> G4double 277 G4double G4Integrator<T, F>::AdaptiveGauss(G4d << 274 G4Integrator<T,F>::AdaptiveGauss( G4double (*f)(G4double), 278 G4d << 275 G4double xInitial, G4double xFinal, G4double e ) 279 G4d << 276 { 280 { << 277 G4int depth = 0 ; 281 G4int depth = 0; << 278 G4double sum = 0.0 ; 282 G4double sum = 0.0; << 279 AdaptGauss(f,xInitial,xFinal,e,sum,depth) ; 283 AdaptGauss(f, xInitial, xFinal, e, sum, dept << 280 return sum ; 284 return sum; << 285 } 281 } 286 282 287 ////////////////////////////////////////////// 283 //////////////////////////////////////////////////////////////////////////// 288 // Gauss integration methods involving ortogon 284 // Gauss integration methods involving ortogonal polynomials 289 ////////////////////////////////////////////// 285 //////////////////////////////////////////////////////////////////////////// 290 // 286 // 291 // Methods involving Legendre polynomials << 287 // Methods involving Legendre polynomials 292 // 288 // 293 ////////////////////////////////////////////// 289 ///////////////////////////////////////////////////////////////////////// 294 // 290 // 295 // The value nLegendre set the accuracy requir 291 // The value nLegendre set the accuracy required, i.e the number of points 296 // where the function pFunction will be evalua 292 // where the function pFunction will be evaluated during integration. 297 // The function creates the arrays for absciss << 293 // The function creates the arrays for abscissas and weights that used 298 // in Gauss-Legendre quadrature method. << 294 // in Gauss-Legendre quadrature method. 299 // The values a and b are the limits of integr 295 // The values a and b are the limits of integration of the function f . 300 // nLegendre MUST BE EVEN !!! 296 // nLegendre MUST BE EVEN !!! 301 // Returns the integral of the function f betw << 297 // Returns the integral of the function f between a and b, by 2*fNumber point 302 // Gauss-Legendre integration: the function is 298 // Gauss-Legendre integration: the function is evaluated exactly 303 // 2*fNumber times at interior points in the r << 299 // 2*fNumber times at interior points in the range of integration. 304 // Since the weights and abscissas are, in thi << 300 // Since the weights and abscissas are, in this case, symmetric around 305 // the midpoint of the range of integration, t << 301 // the midpoint of the range of integration, there are actually only 306 // fNumber distinct values of each. 302 // fNumber distinct values of each. 307 // Convenient for using with some class object 303 // Convenient for using with some class object dataT 308 304 309 template <class T, class F> << 305 template <class T, class F> G4double 310 G4double G4Integrator<T, F>::Legendre(T& typeT << 306 G4Integrator<T,F>::Legendre( T& typeT, F f, G4double a, G4double b, G4int nLegendre) 311 G4int nL << 312 { 307 { 313 G4double nwt, nwt1, temp1, temp2, temp3, tem << 308 G4double newton, newton1, temp1, temp2, temp3, temp ; 314 G4double xDiff, xMean, dx, integral; << 309 G4double xDiff, xMean, dx, integral ; 315 310 316 const G4double tolerance = 1.6e-10; << 311 const G4double tolerance = 1.6e-10 ; 317 G4int i, j, k = nLegendre; << 312 G4int i, j, k = nLegendre ; 318 G4int fNumber = (nLegendre + 1) / 2; << 313 G4int fNumber = (nLegendre + 1)/2 ; 319 << 314 320 if(2 * fNumber != k) << 315 if(2*fNumber != k) 321 { << 316 { 322 G4Exception("G4Integrator<T,F>::Legendre(T << 317 G4Exception("Invalid (odd) n Legendre in G4Integrator<T,F>::Legendre") ; 323 FatalException, "Invalid (odd) << 318 } 324 } << 319 325 << 320 G4double* fAbscissa = new G4double[fNumber] ; 326 G4double* fAbscissa = new G4double[fNumber]; << 321 G4double* fWeight = new G4double[fNumber] ; 327 G4double* fWeight = new G4double[fNumber]; << 322 328 << 323 for(i=1;i<=fNumber;i++) // Loop over the desired roots 329 for(i = 1; i <= fNumber; ++i) // Loop over << 324 { 330 { << 325 newton = cos(pi*(i - 0.25)/(k + 0.5)) ; // Initial root approximation 331 nwt = std::cos(CLHEP::pi * (i - 0.25) / << 326 332 (k + 0.5)); // Initial roo << 327 do // loop of Newton's method 333 << 328 { 334 do // loop of Newton's method << 329 temp1 = 1.0 ; 335 { << 330 temp2 = 0.0 ; 336 temp1 = 1.0; << 331 for(j=1;j<=k;j++) 337 temp2 = 0.0; << 332 { 338 for(j = 1; j <= k; ++j) << 333 temp3 = temp2 ; 339 { << 334 temp2 = temp1 ; 340 temp3 = temp2; << 335 temp1 = ((2.0*j - 1.0)*newton*temp2 - (j - 1.0)*temp3)/j ; 341 temp2 = temp1; << 336 } 342 temp1 = ((2.0 * j - 1.0) * nwt * temp2 << 337 temp = k*(newton*temp1 - temp2)/(newton*newton - 1.0) ; 343 } << 338 newton1 = newton ; 344 temp = k * (nwt * temp1 - temp2) / (nwt << 339 newton = newton1 - temp1/temp ; // Newton's method 345 nwt1 = nwt; << 340 } 346 nwt = nwt1 - temp1 / temp; // Newton's << 341 while(fabs(newton - newton1) > tolerance) ; 347 } while(std::fabs(nwt - nwt1) > tolerance) << 342 348 << 343 fAbscissa[fNumber-i] = newton ; 349 fAbscissa[fNumber - i] = nwt; << 344 fWeight[fNumber-i] = 2.0/((1.0 - newton*newton)*temp*temp) ; 350 fWeight[fNumber - i] = 2.0 / ((1.0 - nwt << 345 } 351 } << 346 // 352 << 347 // Now we ready to get integral 353 // << 348 // 354 // Now we ready to get integral << 349 355 // << 350 xMean = 0.5*(a + b) ; 356 << 351 xDiff = 0.5*(b - a) ; 357 xMean = 0.5 * (a + b); << 352 integral = 0.0 ; 358 xDiff = 0.5 * (b - a); << 353 for(i=0;i<fNumber;i++) 359 integral = 0.0; << 354 { 360 for(i = 0; i < fNumber; ++i) << 355 dx = xDiff*fAbscissa[i] ; 361 { << 356 integral += fWeight[i]*( (typeT.*f)(xMean + dx) + 362 dx = xDiff * fAbscissa[i]; << 357 (typeT.*f)(xMean - dx) ) ; 363 integral += fWeight[i] * ((typeT.*f)(xMean << 358 } 364 } << 359 return integral *= xDiff ; 365 delete[] fAbscissa; << 360 } 366 delete[] fWeight; << 367 return integral *= xDiff; << 368 } << 369 361 370 ////////////////////////////////////////////// 362 /////////////////////////////////////////////////////////////////////// 371 // 363 // 372 // Convenient for using with the pointer 'this 364 // Convenient for using with the pointer 'this' 373 365 374 template <class T, class F> << 366 template <class T, class F> G4double 375 G4double G4Integrator<T, F>::Legendre(T* ptrT, << 367 G4Integrator<T,F>::Legendre( T* ptrT, F f, G4double a, G4double b, G4int nLegendre) 376 G4int nL << 377 { 368 { 378 return Legendre(*ptrT, f, a, b, nLegendre); << 369 return Legendre(*ptrT,f,a,b,nLegendre) ; 379 } 370 } 380 371 381 ////////////////////////////////////////////// 372 /////////////////////////////////////////////////////////////////////// 382 // 373 // 383 // Convenient for using with global scope func 374 // Convenient for using with global scope function f 384 375 385 template <class T, class F> 376 template <class T, class F> 386 G4double G4Integrator<T, F>::Legendre(G4double << 377 G4double G4Integrator<T,F>:: 387 G4double << 378 Legendre( G4double (*f)(G4double), G4double a, G4double b, G4int nLegendre) 388 { 379 { 389 G4double nwt, nwt1, temp1, temp2, temp3, tem << 380 G4double newton, newton1, temp1, temp2, temp3, temp ; 390 G4double xDiff, xMean, dx, integral; << 381 G4double xDiff, xMean, dx, integral ; 391 << 392 const G4double tolerance = 1.6e-10; << 393 G4int i, j, k = nLegendre; << 394 G4int fNumber = (nLegendre + 1) / 2; << 395 << 396 if(2 * fNumber != k) << 397 { << 398 G4Exception("G4Integrator<T,F>::Legendre(. << 399 FatalException, "Invalid (odd) << 400 } << 401 << 402 G4double* fAbscissa = new G4double[fNumber]; << 403 G4double* fWeight = new G4double[fNumber]; << 404 << 405 for(i = 1; i <= fNumber; i++) // Loop over << 406 { << 407 nwt = std::cos(CLHEP::pi * (i - 0.25) / << 408 (k + 0.5)); // Initial roo << 409 382 410 do // loop of Newton's method << 383 const G4double tolerance = 1.6e-10 ; 411 { << 384 G4int i, j, k = nLegendre ; 412 temp1 = 1.0; << 385 G4int fNumber = (nLegendre + 1)/2 ; 413 temp2 = 0.0; << 386 414 for(j = 1; j <= k; ++j) << 387 if(2*fNumber != k) 415 { << 388 { 416 temp3 = temp2; << 389 G4Exception("Invalid (odd) n Legendre in G4Integrator<T,F>::Legendre") ; 417 temp2 = temp1; << 390 } 418 temp1 = ((2.0 * j - 1.0) * nwt * temp2 << 391 419 } << 392 G4double* fAbscissa = new G4double[fNumber] ; 420 temp = k * (nwt * temp1 - temp2) / (nwt << 393 G4double* fWeight = new G4double[fNumber] ; 421 nwt1 = nwt; << 394 422 nwt = nwt1 - temp1 / temp; // Newton's << 395 for(i=1;i<=fNumber;i++) // Loop over the desired roots 423 } while(std::fabs(nwt - nwt1) > tolerance) << 396 { 424 << 397 newton = cos(pi*(i - 0.25)/(k + 0.5)) ; // Initial root approximation 425 fAbscissa[fNumber - i] = nwt; << 398 426 fWeight[fNumber - i] = 2.0 / ((1.0 - nwt << 399 do // loop of Newton's method 427 } << 400 { 428 << 401 temp1 = 1.0 ; 429 // << 402 temp2 = 0.0 ; 430 // Now we ready to get integral << 403 for(j=1;j<=k;j++) 431 // << 404 { 432 << 405 temp3 = temp2 ; 433 xMean = 0.5 * (a + b); << 406 temp2 = temp1 ; 434 xDiff = 0.5 * (b - a); << 407 temp1 = ((2.0*j - 1.0)*newton*temp2 - (j - 1.0)*temp3)/j ; 435 integral = 0.0; << 408 } 436 for(i = 0; i < fNumber; ++i) << 409 temp = k*(newton*temp1 - temp2)/(newton*newton - 1.0) ; 437 { << 410 newton1 = newton ; 438 dx = xDiff * fAbscissa[i]; << 411 newton = newton1 - temp1/temp ; // Newton's method 439 integral += fWeight[i] * ((*f)(xMean + dx) << 412 } 440 } << 413 while(fabs(newton - newton1) > tolerance) ; 441 delete[] fAbscissa; << 414 442 delete[] fWeight; << 415 fAbscissa[fNumber-i] = newton ; 443 << 416 fWeight[fNumber-i] = 2.0/((1.0 - newton*newton)*temp*temp) ; 444 return integral *= xDiff; << 417 } 445 } << 418 // >> 419 // Now we ready to get integral >> 420 // >> 421 >> 422 xMean = 0.5*(a + b) ; >> 423 xDiff = 0.5*(b - a) ; >> 424 integral = 0.0 ; >> 425 for(i=0;i<fNumber;i++) >> 426 { >> 427 dx = xDiff*fAbscissa[i] ; >> 428 integral += fWeight[i]*( (*f)(xMean + dx) + (*f)(xMean - dx) ) ; >> 429 } >> 430 return integral *= xDiff ; >> 431 } 446 432 447 ////////////////////////////////////////////// 433 //////////////////////////////////////////////////////////////////////////// 448 // 434 // 449 // Returns the integral of the function to be 435 // Returns the integral of the function to be pointed by T::f between a and b, 450 // by ten point Gauss-Legendre integration: th 436 // by ten point Gauss-Legendre integration: the function is evaluated exactly 451 // ten times at interior points in the range o 437 // ten times at interior points in the range of integration. Since the weights 452 // and abscissas are, in this case, symmetric << 438 // and abscissas are, in this case, symmetric around the midpoint of the 453 // range of integration, there are actually on 439 // range of integration, there are actually only five distinct values of each 454 // Convenient for using with class object type 440 // Convenient for using with class object typeT 455 441 456 template <class T, class F> << 442 template <class T, class F> 457 G4double G4Integrator<T, F>::Legendre10(T& typ << 443 G4double G4Integrator<T,F>::Legendre10( T& typeT, F f,G4double a, G4double b) 458 { 444 { 459 G4int i; << 445 G4int i ; 460 G4double xDiff, xMean, dx, integral; << 446 G4double xDiff, xMean, dx, integral ; 461 << 447 462 // From Abramowitz M., Stegan I.A. 1964 , Ha << 448 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 916 463 << 449 464 static const G4double abscissa[] = { 0.14887 << 450 static G4double abscissa[] = { 0.148874338981631, 0.433395394129247, 465 0.67940 << 451 0.679409568299024, 0.865063366688985, 466 0.97390 << 452 0.973906528517172 } ; 467 << 453 468 static const G4double weight[] = { 0.2955242 << 454 static G4double weight[] = { 0.295524224714753, 0.269266719309996, 469 0.2190863 << 455 0.219086362515982, 0.149451349150581, 470 0.0666713 << 456 0.066671344308688 } ; 471 xMean = 0.5 * (a + << 457 xMean = 0.5*(a + b) ; 472 xDiff = 0.5 * (b - << 458 xDiff = 0.5*(b - a) ; 473 integral = 0.0; << 459 integral = 0.0 ; 474 for(i = 0; i < 5; ++i) << 460 for(i=0;i<5;i++) 475 { << 461 { 476 dx = xDiff * abscissa[i]; << 462 dx = xDiff*abscissa[i] ; 477 integral += weight[i] * ((typeT.*f)(xMean << 463 integral += weight[i]*( (typeT.*f)(xMean + dx) + (typeT.*f)(xMean - dx)) ; 478 } << 464 } 479 return integral *= xDiff; << 465 return integral *= xDiff ; 480 } 466 } 481 467 482 ////////////////////////////////////////////// 468 /////////////////////////////////////////////////////////////////////////// 483 // 469 // 484 // Convenient for using with the pointer 'this 470 // Convenient for using with the pointer 'this' 485 471 486 template <class T, class F> << 472 template <class T, class F> 487 G4double G4Integrator<T, F>::Legendre10(T* ptr << 473 G4double G4Integrator<T,F>::Legendre10( T* ptrT, F f,G4double a, G4double b) 488 { 474 { 489 return Legendre10(*ptrT, f, a, b); << 475 return Legendre10(*ptrT,f,a,b) ; 490 } << 476 } 491 477 492 ////////////////////////////////////////////// 478 ////////////////////////////////////////////////////////////////////////// 493 // 479 // 494 // Convenient for using with global scope func 480 // Convenient for using with global scope functions 495 481 496 template <class T, class F> << 482 template <class T, class F> G4double 497 G4double G4Integrator<T, F>::Legendre10(G4doub << 483 G4Integrator<T,F>::Legendre10( G4double (*f)(G4double), G4double a, G4double b) 498 G4doub << 499 { 484 { 500 G4int i; << 485 G4int i ; 501 G4double xDiff, xMean, dx, integral; << 486 G4double xDiff, xMean, dx, integral ; 502 << 487 503 // From Abramowitz M., Stegan I.A. 1964 , Ha << 488 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 916 504 << 489 505 static const G4double abscissa[] = { 0.14887 << 490 static G4double abscissa[] = { 0.148874338981631, 0.433395394129247, 506 0.67940 << 491 0.679409568299024, 0.865063366688985, 507 0.97390 << 492 0.973906528517172 } ; 508 << 493 509 static const G4double weight[] = { 0.2955242 << 494 static G4double weight[] = { 0.295524224714753, 0.269266719309996, 510 0.2190863 << 495 0.219086362515982, 0.149451349150581, 511 0.0666713 << 496 0.066671344308688 } ; 512 xMean = 0.5 * (a + << 497 xMean = 0.5*(a + b) ; 513 xDiff = 0.5 * (b - << 498 xDiff = 0.5*(b - a) ; 514 integral = 0.0; << 499 integral = 0.0 ; 515 for(i = 0; i < 5; ++i) << 500 for(i=0;i<5;i++) 516 { << 501 { 517 dx = xDiff * abscissa[i]; << 502 dx = xDiff*abscissa[i] ; 518 integral += weight[i] * ((*f)(xMean + dx) << 503 integral += weight[i]*( (*f)(xMean + dx) + (*f)(xMean - dx)) ; 519 } << 504 } 520 return integral *= xDiff; << 505 return integral *= xDiff ; 521 } 506 } 522 507 523 ////////////////////////////////////////////// 508 /////////////////////////////////////////////////////////////////////// 524 // 509 // 525 // Returns the integral of the function to be 510 // Returns the integral of the function to be pointed by T::f between a and b, 526 // by 96 point Gauss-Legendre integration: the 511 // by 96 point Gauss-Legendre integration: the function is evaluated exactly 527 // ten Times at interior points in the range o 512 // ten Times at interior points in the range of integration. Since the weights 528 // and abscissas are, in this case, symmetric << 513 // and abscissas are, in this case, symmetric around the midpoint of the 529 // range of integration, there are actually on 514 // range of integration, there are actually only five distinct values of each 530 // Convenient for using with some class object 515 // Convenient for using with some class object typeT 531 516 532 template <class T, class F> << 517 template <class T, class F> 533 G4double G4Integrator<T, F>::Legendre96(T& typ << 518 G4double G4Integrator<T,F>::Legendre96( T& typeT, F f,G4double a, G4double b) 534 { 519 { 535 G4int i; << 520 G4int i ; 536 G4double xDiff, xMean, dx, integral; << 521 G4double xDiff, xMean, dx, integral ; 537 << 522 538 // From Abramowitz M., Stegan I.A. 1964 , Ha << 523 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 919 539 << 524 540 static const G4double abscissa[] = { << 525 static G4double 541 0.016276744849602969579, 0.048812985136049 << 526 abscissa[] = { 542 0.081297495464425558994, 0.113695850110665 << 527 0.016276744849602969579, 0.048812985136049731112, 543 0.145973714654896941989, 0.178096882367618 << 528 0.081297495464425558994, 0.113695850110665920911, 544 << 529 0.145973714654896941989, 0.178096882367618602759, // 6 545 0.210031310460567203603, 0.241743156163840 << 530 546 0.273198812591049141487, 0.304364944354496 << 531 0.210031310460567203603, 0.241743156163840012328, 547 0.335208522892625422616, 0.365696861472313 << 532 0.273198812591049141487, 0.304364944354496353024, 548 << 533 0.335208522892625422616, 0.365696861472313635031, // 12 549 0.395797649828908603285, 0.425478988407300 << 534 550 0.454709422167743008636, 0.483457973920596 << 535 0.395797649828908603285, 0.425478988407300545365, 551 0.511694177154667673586, 0.539388108324357 << 536 0.454709422167743008636, 0.483457973920596359768, 552 << 537 0.511694177154667673586, 0.539388108324357436227, // 18 553 0.566510418561397168404, 0.593032364777572 << 538 554 0.618925840125468570386, 0.644163403784967 << 539 0.566510418561397168404, 0.593032364777572080684, 555 0.668718310043916153953, 0.692564536642171 << 540 0.618925840125468570386, 0.644163403784967106798, 556 << 541 0.668718310043916153953, 0.692564536642171561344, // 24 557 0.715676812348967626225, 0.738030643744400 << 542 558 0.759602341176647498703, 0.780369043867433 << 543 0.715676812348967626225, 0.738030643744400132851, 559 0.800308744139140817229, 0.819400310737931 << 544 0.759602341176647498703, 0.780369043867433217604, 560 << 545 0.800308744139140817229, 0.819400310737931675539, // 30 561 0.837623511228187121494, 0.854959033434601 << 546 562 0.871388505909296502874, 0.886894517402420 << 547 0.837623511228187121494, 0.854959033434601455463, 563 0.901460635315852341319, 0.915071423120898 << 548 0.871388505909296502874, 0.886894517402420416057, 564 << 549 0.901460635315852341319, 0.915071423120898074206, // 36 565 0.927712456722308690965, 0.939370339752755 << 550 566 0.950032717784437635756, 0.959688291448742 << 551 0.927712456722308690965, 0.939370339752755216932, 567 0.968326828463264212174, 0.975939174585136 << 552 0.950032717784437635756, 0.959688291448742539300, 568 << 553 0.968326828463264212174, 0.975939174585136466453, // 42 569 0.982517263563014677447, 0.988054126329623 << 554 570 0.992543900323762624572, 0.995981842987209 << 555 0.982517263563014677447, 0.988054126329623799481, 571 0.998364375863181677724, 0.999689503883230 << 556 0.992543900323762624572, 0.995981842987209290650, 572 }; << 557 0.998364375863181677724, 0.999689503883230766828 // 48 573 << 558 } ; 574 static const G4double weight[] = { << 559 575 0.032550614492363166242, 0.032516118713868 << 560 static G4double 576 0.032447163714064269364, 0.032343822568575 << 561 weight[] = { 577 0.032206204794030250669, 0.032034456231992 << 562 0.032550614492363166242, 0.032516118713868835987, 578 << 563 0.032447163714064269364, 0.032343822568575928429, 579 0.031828758894411006535, 0.031589330770727 << 564 0.032206204794030250669, 0.032034456231992663218, // 6 580 0.031316425596862355813, 0.031010332586313 << 565 581 0.030671376123669149014, 0.030299915420827 << 566 0.031828758894411006535, 0.031589330770727168558, 582 << 567 0.031316425596862355813, 0.031010332586313837423, 583 0.029896344136328385984, 0.029461089958167 << 568 0.030671376123669149014, 0.030299915420827593794, // 12 584 0.028994614150555236543, 0.028497411065085 << 569 585 0.027970007616848334440, 0.027412962726029 << 570 0.029896344136328385984, 0.029461089958167905970, 586 << 571 0.028994614150555236543, 0.028497411065085385646, 587 0.026826866725591762198, 0.026212340735672 << 572 0.027970007616848334440, 0.027412962726029242823, // 18 588 0.025570036005349361499, 0.024900633222483 << 573 589 0.024204841792364691282, 0.023483399085926 << 574 0.026826866725591762198, 0.026212340735672413913, 590 << 575 0.025570036005349361499, 0.024900633222483610288, 591 0.022737069658329374001, 0.021966644438744 << 576 0.024204841792364691282, 0.023483399085926219842, // 24 592 0.021172939892191298988, 0.020356797154333 << 577 593 0.019519081140145022410, 0.018660679627411 << 578 0.022737069658329374001, 0.021966644438744349195, 594 << 579 0.021172939892191298988, 0.020356797154333324595, 595 0.017782502316045260838, 0.016885479864245 << 580 0.019519081140145022410, 0.018660679627411467385, // 30 596 0.015970562902562291381, 0.015038721026994 << 581 597 0.014090941772314860916, 0.013128229566961 << 582 0.017782502316045260838, 0.016885479864245172450, 598 << 583 0.015970562902562291381, 0.015038721026994938006, 599 0.012151604671088319635, 0.011162102099838 << 584 0.014090941772314860916, 0.013128229566961572637, // 36 600 0.010160770535008415758, 0.009148671230783 << 585 601 0.008126876925698759217, 0.007096470791153 << 586 0.012151604671088319635, 0.011162102099838498591, 602 << 587 0.010160770535008415758, 0.009148671230783386633, 603 0.006058545504235961683, 0.005014202742927 << 588 0.008126876925698759217, 0.007096470791153865269, // 42 604 0.003964554338444686674, 0.002910731817934 << 589 605 0.001853960788946921732, 0.000796792065552 << 590 0.006058545504235961683, 0.005014202742927517693, 606 }; << 591 0.003964554338444686674, 0.002910731817934946408, 607 xMean = 0.5 * (a + b); << 592 0.001853960788946921732, 0.000796792065552012429 // 48 608 xDiff = 0.5 * (b - a); << 593 } ; 609 integral = 0.0; << 594 xMean = 0.5*(a + b) ; 610 for(i = 0; i < 48; ++i) << 595 xDiff = 0.5*(b - a) ; 611 { << 596 integral = 0.0 ; 612 dx = xDiff * abscissa[i]; << 597 for(i=0;i<48;i++) 613 integral += weight[i] * ((typeT.*f)(xMean << 598 { 614 } << 599 dx = xDiff*abscissa[i] ; 615 return integral *= xDiff; << 600 integral += weight[i]*((typeT.*f)(xMean + dx) + (typeT.*f)(xMean - dx)) ; >> 601 } >> 602 return integral *= xDiff ; 616 } 603 } 617 604 618 ////////////////////////////////////////////// 605 /////////////////////////////////////////////////////////////////////// 619 // 606 // 620 // Convenient for using with the pointer 'this 607 // Convenient for using with the pointer 'this' 621 608 622 template <class T, class F> << 609 template <class T, class F> 623 G4double G4Integrator<T, F>::Legendre96(T* ptr << 610 G4double G4Integrator<T,F>::Legendre96( T* ptrT, F f,G4double a, G4double b) 624 { 611 { 625 return Legendre96(*ptrT, f, a, b); << 612 return Legendre96(*ptrT,f,a,b) ; 626 } << 613 } 627 614 628 ////////////////////////////////////////////// 615 /////////////////////////////////////////////////////////////////////// 629 // 616 // 630 // Convenient for using with global scope func << 617 // Convenient for using with global scope function f 631 618 632 template <class T, class F> << 619 template <class T, class F> G4double 633 G4double G4Integrator<T, F>::Legendre96(G4doub << 620 G4Integrator<T,F>::Legendre96( G4double (*f)(G4double), G4double a, G4double b) 634 G4doub << 635 { 621 { 636 G4int i; << 622 G4int i ; 637 G4double xDiff, xMean, dx, integral; << 623 G4double xDiff, xMean, dx, integral ; 638 << 624 639 // From Abramowitz M., Stegan I.A. 1964 , Ha << 625 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 919 640 << 626 641 static const G4double abscissa[] = { << 627 static G4double 642 0.016276744849602969579, 0.048812985136049 << 628 abscissa[] = { 643 0.081297495464425558994, 0.113695850110665 << 629 0.016276744849602969579, 0.048812985136049731112, 644 0.145973714654896941989, 0.178096882367618 << 630 0.081297495464425558994, 0.113695850110665920911, 645 << 631 0.145973714654896941989, 0.178096882367618602759, // 6 646 0.210031310460567203603, 0.241743156163840 << 632 647 0.273198812591049141487, 0.304364944354496 << 633 0.210031310460567203603, 0.241743156163840012328, 648 0.335208522892625422616, 0.365696861472313 << 634 0.273198812591049141487, 0.304364944354496353024, 649 << 635 0.335208522892625422616, 0.365696861472313635031, // 12 650 0.395797649828908603285, 0.425478988407300 << 636 651 0.454709422167743008636, 0.483457973920596 << 637 0.395797649828908603285, 0.425478988407300545365, 652 0.511694177154667673586, 0.539388108324357 << 638 0.454709422167743008636, 0.483457973920596359768, 653 << 639 0.511694177154667673586, 0.539388108324357436227, // 18 654 0.566510418561397168404, 0.593032364777572 << 640 655 0.618925840125468570386, 0.644163403784967 << 641 0.566510418561397168404, 0.593032364777572080684, 656 0.668718310043916153953, 0.692564536642171 << 642 0.618925840125468570386, 0.644163403784967106798, 657 << 643 0.668718310043916153953, 0.692564536642171561344, // 24 658 0.715676812348967626225, 0.738030643744400 << 644 659 0.759602341176647498703, 0.780369043867433 << 645 0.715676812348967626225, 0.738030643744400132851, 660 0.800308744139140817229, 0.819400310737931 << 646 0.759602341176647498703, 0.780369043867433217604, 661 << 647 0.800308744139140817229, 0.819400310737931675539, // 30 662 0.837623511228187121494, 0.854959033434601 << 648 663 0.871388505909296502874, 0.886894517402420 << 649 0.837623511228187121494, 0.854959033434601455463, 664 0.901460635315852341319, 0.915071423120898 << 650 0.871388505909296502874, 0.886894517402420416057, 665 << 651 0.901460635315852341319, 0.915071423120898074206, // 36 666 0.927712456722308690965, 0.939370339752755 << 652 667 0.950032717784437635756, 0.959688291448742 << 653 0.927712456722308690965, 0.939370339752755216932, 668 0.968326828463264212174, 0.975939174585136 << 654 0.950032717784437635756, 0.959688291448742539300, 669 << 655 0.968326828463264212174, 0.975939174585136466453, // 42 670 0.982517263563014677447, 0.988054126329623 << 656 671 0.992543900323762624572, 0.995981842987209 << 657 0.982517263563014677447, 0.988054126329623799481, 672 0.998364375863181677724, 0.999689503883230 << 658 0.992543900323762624572, 0.995981842987209290650, 673 }; << 659 0.998364375863181677724, 0.999689503883230766828 // 48 674 << 660 } ; 675 static const G4double weight[] = { << 661 676 0.032550614492363166242, 0.032516118713868 << 662 static G4double 677 0.032447163714064269364, 0.032343822568575 << 663 weight[] = { 678 0.032206204794030250669, 0.032034456231992 << 664 0.032550614492363166242, 0.032516118713868835987, 679 << 665 0.032447163714064269364, 0.032343822568575928429, 680 0.031828758894411006535, 0.031589330770727 << 666 0.032206204794030250669, 0.032034456231992663218, // 6 681 0.031316425596862355813, 0.031010332586313 << 667 682 0.030671376123669149014, 0.030299915420827 << 668 0.031828758894411006535, 0.031589330770727168558, 683 << 669 0.031316425596862355813, 0.031010332586313837423, 684 0.029896344136328385984, 0.029461089958167 << 670 0.030671376123669149014, 0.030299915420827593794, // 12 685 0.028994614150555236543, 0.028497411065085 << 671 686 0.027970007616848334440, 0.027412962726029 << 672 0.029896344136328385984, 0.029461089958167905970, 687 << 673 0.028994614150555236543, 0.028497411065085385646, 688 0.026826866725591762198, 0.026212340735672 << 674 0.027970007616848334440, 0.027412962726029242823, // 18 689 0.025570036005349361499, 0.024900633222483 << 675 690 0.024204841792364691282, 0.023483399085926 << 676 0.026826866725591762198, 0.026212340735672413913, 691 << 677 0.025570036005349361499, 0.024900633222483610288, 692 0.022737069658329374001, 0.021966644438744 << 678 0.024204841792364691282, 0.023483399085926219842, // 24 693 0.021172939892191298988, 0.020356797154333 << 679 694 0.019519081140145022410, 0.018660679627411 << 680 0.022737069658329374001, 0.021966644438744349195, 695 << 681 0.021172939892191298988, 0.020356797154333324595, 696 0.017782502316045260838, 0.016885479864245 << 682 0.019519081140145022410, 0.018660679627411467385, // 30 697 0.015970562902562291381, 0.015038721026994 << 683 698 0.014090941772314860916, 0.013128229566961 << 684 0.017782502316045260838, 0.016885479864245172450, 699 << 685 0.015970562902562291381, 0.015038721026994938006, 700 0.012151604671088319635, 0.011162102099838 << 686 0.014090941772314860916, 0.013128229566961572637, // 36 701 0.010160770535008415758, 0.009148671230783 << 687 702 0.008126876925698759217, 0.007096470791153 << 688 0.012151604671088319635, 0.011162102099838498591, 703 << 689 0.010160770535008415758, 0.009148671230783386633, 704 0.006058545504235961683, 0.005014202742927 << 690 0.008126876925698759217, 0.007096470791153865269, // 42 705 0.003964554338444686674, 0.002910731817934 << 691 706 0.001853960788946921732, 0.000796792065552 << 692 0.006058545504235961683, 0.005014202742927517693, 707 }; << 693 0.003964554338444686674, 0.002910731817934946408, 708 xMean = 0.5 * (a + b); << 694 0.001853960788946921732, 0.000796792065552012429 // 48 709 xDiff = 0.5 * (b - a); << 695 } ; 710 integral = 0.0; << 696 xMean = 0.5*(a + b) ; 711 for(i = 0; i < 48; ++i) << 697 xDiff = 0.5*(b - a) ; 712 { << 698 integral = 0.0 ; 713 dx = xDiff * abscissa[i]; << 699 for(i=0;i<48;i++) 714 integral += weight[i] * ((*f)(xMean + dx) << 700 { 715 } << 701 dx = xDiff*abscissa[i] ; 716 return integral *= xDiff; << 702 integral += weight[i]*((*f)(xMean + dx) + (*f)(xMean - dx)) ; >> 703 } >> 704 return integral *= xDiff ; 717 } 705 } 718 706 719 ////////////////////////////////////////////// 707 ////////////////////////////////////////////////////////////////////////////// 720 // 708 // 721 // Methods involving Chebyshev polynomials << 709 // Methods involving Chebyshev polynomials 722 // 710 // 723 ////////////////////////////////////////////// 711 /////////////////////////////////////////////////////////////////////////// 724 // 712 // 725 // Integrates function pointed by T::f from a << 713 // Integrates function pointed by T::f from a to b by Gauss-Chebyshev 726 // quadrature method. 714 // quadrature method. 727 // Convenient for using with class object type 715 // Convenient for using with class object typeT 728 716 729 template <class T, class F> << 717 template <class T, class F> G4double 730 G4double G4Integrator<T, F>::Chebyshev(T& type << 718 G4Integrator<T,F>::Chebyshev( T& typeT, F f, G4double a, 731 G4int n << 719 G4double b, G4int nChebyshev ) 732 { << 720 { 733 G4int i; << 721 G4int i ; 734 G4double xDiff, xMean, dx, integral = 0.0; << 722 G4double xDiff, xMean, dx, integral = 0.0 ; 735 << 723 736 G4int fNumber = nChebyshev; // Try to << 724 G4int fNumber = nChebyshev ; // Try to reduce fNumber twice ?? 737 G4double cof = CLHEP::pi / fNumber; << 725 G4double cof = pi/fNumber ; 738 G4double* fAbscissa = new G4double[fNumber]; << 726 G4double* fAbscissa = new G4double[fNumber] ; 739 G4double* fWeight = new G4double[fNumber]; << 727 G4double* fWeight = new G4double[fNumber] ; 740 for(i = 0; i < fNumber; ++i) << 728 for(i=0;i<fNumber;i++) 741 { << 729 { 742 fAbscissa[i] = std::cos(cof * (i + 0.5)); << 730 fAbscissa[i] = cos(cof*(i + 0.5)) ; 743 fWeight[i] = cof * std::sqrt(1 - fAbscis << 731 fWeight[i] = cof*sqrt(1 - fAbscissa[i]*fAbscissa[i]) ; 744 } << 732 } 745 << 733 // 746 // << 734 // Now we ready to estimate the integral 747 // Now we ready to estimate the integral << 735 // 748 // << 736 xMean = 0.5*(a + b) ; 749 << 737 xDiff = 0.5*(b - a) ; 750 xMean = 0.5 * (a + b); << 738 for(i=0;i<fNumber;i++) 751 xDiff = 0.5 * (b - a); << 739 { 752 for(i = 0; i < fNumber; ++i) << 740 dx = xDiff*fAbscissa[i] ; 753 { << 741 integral += fWeight[i]*(typeT.*f)(xMean + dx) ; 754 dx = xDiff * fAbscissa[i]; << 742 } 755 integral += fWeight[i] * (typeT.*f)(xMean << 743 return integral *= xDiff ; 756 } << 757 delete[] fAbscissa; << 758 delete[] fWeight; << 759 return integral *= xDiff; << 760 } 744 } 761 745 762 ////////////////////////////////////////////// 746 /////////////////////////////////////////////////////////////////////// 763 // 747 // 764 // Convenient for using with 'this' pointer 748 // Convenient for using with 'this' pointer 765 749 766 template <class T, class F> << 750 template <class T, class F> G4double 767 G4double G4Integrator<T, F>::Chebyshev(T* ptrT << 751 G4Integrator<T,F>::Chebyshev( T* ptrT, F f, G4double a, G4double b, G4int n) 768 G4int n << 769 { 752 { 770 return Chebyshev(*ptrT, f, a, b, n); << 753 return Chebyshev(*ptrT,f,a,b,n) ; 771 } << 754 } 772 755 773 ////////////////////////////////////////////// 756 //////////////////////////////////////////////////////////////////////// 774 // 757 // 775 // For use with global scope functions f << 758 // For use with global scope functions f 776 << 777 template <class T, class F> << 778 G4double G4Integrator<T, F>::Chebyshev(G4doubl << 779 G4doubl << 780 { << 781 G4int i; << 782 G4double xDiff, xMean, dx, integral = 0.0; << 783 759 784 G4int fNumber = nChebyshev; // Try to << 760 template <class T, class F> G4double 785 G4double cof = CLHEP::pi / fNumber; << 761 G4Integrator<T,F>::Chebyshev( G4double (*f)(G4double), 786 G4double* fAbscissa = new G4double[fNumber]; << 762 G4double a, G4double b, G4int nChebyshev) 787 G4double* fWeight = new G4double[fNumber]; << 763 { 788 for(i = 0; i < fNumber; ++i) << 764 G4int i ; 789 { << 765 G4double xDiff, xMean, dx, integral = 0.0 ; 790 fAbscissa[i] = std::cos(cof * (i + 0.5)); << 766 791 fWeight[i] = cof * std::sqrt(1 - fAbscis << 767 G4int fNumber = nChebyshev ; // Try to reduce fNumber twice ?? 792 } << 768 G4double cof = pi/fNumber ; 793 << 769 G4double* fAbscissa = new G4double[fNumber] ; 794 // << 770 G4double* fWeight = new G4double[fNumber] ; 795 // Now we ready to estimate the integral << 771 for(i=0;i<fNumber;i++) 796 // << 772 { 797 << 773 fAbscissa[i] = cos(cof*(i + 0.5)) ; 798 xMean = 0.5 * (a + b); << 774 fWeight[i] = cof*sqrt(1 - fAbscissa[i]*fAbscissa[i]) ; 799 xDiff = 0.5 * (b - a); << 775 } 800 for(i = 0; i < fNumber; ++i) << 776 // 801 { << 777 // Now we ready to estimate the integral 802 dx = xDiff * fAbscissa[i]; << 778 // 803 integral += fWeight[i] * (*f)(xMean + dx); << 779 xMean = 0.5*(a + b) ; 804 } << 780 xDiff = 0.5*(b - a) ; 805 delete[] fAbscissa; << 781 for(i=0;i<fNumber;i++) 806 delete[] fWeight; << 782 { 807 return integral *= xDiff; << 783 dx = xDiff*fAbscissa[i] ; >> 784 integral += fWeight[i]*(*f)(xMean + dx) ; >> 785 } >> 786 return integral *= xDiff ; 808 } 787 } 809 788 810 ////////////////////////////////////////////// 789 ////////////////////////////////////////////////////////////////////// 811 // 790 // 812 // Method involving Laguerre polynomials 791 // Method involving Laguerre polynomials 813 // 792 // 814 ////////////////////////////////////////////// 793 ////////////////////////////////////////////////////////////////////// 815 // 794 // 816 // Integral from zero to infinity of std::pow( << 795 // Integral from zero to infinity of pow(x,alpha)*exp(-x)*f(x). 817 // The value of nLaguerre sets the accuracy. 796 // The value of nLaguerre sets the accuracy. 818 // The function creates arrays fAbscissa[0,.., << 797 // The function creates arrays fAbscissa[0,..,nLaguerre-1] and 819 // fWeight[0,..,nLaguerre-1] . << 798 // fWeight[0,..,nLaguerre-1] . 820 // Convenient for using with class object 'typ 799 // Convenient for using with class object 'typeT' and (typeT.*f) function 821 // (T::f) 800 // (T::f) 822 801 823 template <class T, class F> << 802 template <class T, class F> G4double 824 G4double G4Integrator<T, F>::Laguerre(T& typeT << 803 G4Integrator<T,F>::Laguerre( T& typeT, F f, G4double alpha, G4int nLaguerre ) 825 G4int nL << 826 { 804 { 827 const G4double tolerance = 1.0e-10; << 805 const G4double tolerance = 1.0e-10 ; 828 const G4int maxNumber = 12; << 806 const G4int maxNumber = 12 ; 829 G4int i, j, k; << 807 G4int i, j, k ; 830 G4double nwt = 0., nwt1, temp1, temp2, << 808 G4double newton, newton1, temp1, temp2, temp3, temp, cofi ; 831 G4double integral = 0.0; << 809 G4double integral = 0.0 ; 832 << 810 833 G4int fNumber = nLaguerre; << 811 G4int fNumber = nLaguerre ; 834 G4double* fAbscissa = new G4double[fNumber]; << 812 G4double* fAbscissa = new G4double[fNumber] ; 835 G4double* fWeight = new G4double[fNumber]; << 813 G4double* fWeight = new G4double[fNumber] ; >> 814 >> 815 for(i=1;i<=fNumber;i++) // Loop over the desired roots >> 816 { >> 817 if(i == 1) >> 818 { >> 819 newton = (1.0 + alpha)*(3.0 + 0.92*alpha)/(1.0 + 2.4*fNumber + 1.8*alpha) ; >> 820 } >> 821 else if(i == 2) >> 822 { >> 823 newton += (15.0 + 6.25*alpha)/(1.0 + 0.9*alpha + 2.5*fNumber) ; >> 824 } >> 825 else >> 826 { >> 827 cofi = i - 2 ; >> 828 newton += ((1.0+2.55*cofi)/(1.9*cofi) + 1.26*cofi*alpha/(1.0+3.5*cofi))* >> 829 (newton - fAbscissa[i-3])/(1.0 + 0.3*alpha) ; >> 830 } >> 831 for(k=1;k<=maxNumber;k++) >> 832 { >> 833 temp1 = 1.0 ; >> 834 temp2 = 0.0 ; 836 835 837 for(i = 1; i <= fNumber; ++i) // Loop over << 836 for(j=1;j<=fNumber;j++) 838 { << 837 { 839 if(i == 1) << 838 temp3 = temp2 ; 840 { << 839 temp2 = temp1 ; 841 nwt = (1.0 + alpha) * (3.0 + 0.92 * alph << 840 temp1 = ((2*j - 1 + alpha - newton)*temp2 - (j - 1 + alpha)*temp3)/j ; 842 (1.0 + 2.4 * fNumber + 1.8 * alpha << 841 } 843 } << 842 temp = (fNumber*temp1 - (fNumber +alpha)*temp2)/newton ; 844 else if(i == 2) << 843 newton1 = newton ; 845 { << 844 newton = newton1 - temp1/temp ; 846 nwt += (15.0 + 6.25 * alpha) / (1.0 + 0. << 845 847 } << 846 if(fabs(newton - newton1) <= tolerance) 848 else << 847 { 849 { << 848 break ; 850 cofi = i - 2; << 849 } 851 nwt += ((1.0 + 2.55 * cofi) / (1.9 * cof << 850 } 852 1.26 * cofi * alpha / (1.0 + 3.5 << 851 if(k > maxNumber) 853 (nwt - fAbscissa[i - 3]) / (1.0 + << 852 { 854 } << 853 G4Exception("Too many (>12) iterations in G4Integration::Laguerre") ; 855 for(k = 1; k <= maxNumber; ++k) << 854 } 856 { << 855 857 temp1 = 1.0; << 856 fAbscissa[i-1] = newton ; 858 temp2 = 0.0; << 857 fWeight[i-1] = -exp(GammaLogarithm(alpha + fNumber) - 859 << 858 GammaLogarithm((G4double)fNumber))/(temp*fNumber*temp2) ; 860 for(j = 1; j <= fNumber; ++j) << 859 } 861 { << 860 // 862 temp3 = temp2; << 861 // Integral evaluation 863 temp2 = temp1; << 862 // 864 temp1 = << 863 for(i=0;i<fNumber;i++) 865 ((2 * j - 1 + alpha - nwt) * temp2 - << 864 { 866 } << 865 integral += fWeight[i]*(typeT.*f)(fAbscissa[i]) ; 867 temp = (fNumber * temp1 - (fNumber + alp << 866 } 868 nwt1 = nwt; << 867 return integral ; 869 nwt = nwt1 - temp1 / temp; << 868 } 870 << 871 if(std::fabs(nwt - nwt1) <= tolerance) << 872 { << 873 break; << 874 } << 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 } << 887 869 888 // << 889 // Integral evaluation << 890 // << 891 870 892 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 } << 900 871 901 ////////////////////////////////////////////// 872 ////////////////////////////////////////////////////////////////////// 902 // 873 // 903 // 874 // 904 875 905 template <class T, class F> << 876 template <class T, class F> G4double 906 G4double G4Integrator<T, F>::Laguerre(T* ptrT, << 877 G4Integrator<T,F>::Laguerre( T* ptrT, F f, G4double alpha, G4int nLaguerre ) 907 G4int nL << 908 { 878 { 909 return Laguerre(*ptrT, f, alpha, nLaguerre); << 879 return Laguerre(*ptrT,f,alpha,nLaguerre) ; 910 } 880 } 911 881 912 ////////////////////////////////////////////// 882 //////////////////////////////////////////////////////////////////////// 913 // 883 // 914 // For use with global scope functions f << 884 // For use with global scope functions f 915 885 916 template <class T, class F> << 886 template <class T, class F> G4double 917 G4double G4Integrator<T, F>::Laguerre(G4double << 887 G4Integrator<T,F>::Laguerre( G4double (*f)(G4double), 918 G4int nL << 888 G4double alpha, G4int nLaguerre) 919 { << 889 { 920 const G4double tolerance = 1.0e-10; << 890 const G4double tolerance = 1.0e-10 ; 921 const G4int maxNumber = 12; << 891 const G4int maxNumber = 12 ; 922 G4int i, j, k; << 892 G4int i, j, k ; 923 G4double nwt = 0., nwt1, temp1, temp2, << 893 G4double newton, newton1, temp1, temp2, temp3, temp, cofi ; 924 G4double integral = 0.0; << 894 G4double integral = 0.0 ; 925 << 895 926 G4int fNumber = nLaguerre; << 896 G4int fNumber = nLaguerre ; 927 G4double* fAbscissa = new G4double[fNumber]; << 897 G4double* fAbscissa = new G4double[fNumber] ; 928 G4double* fWeight = new G4double[fNumber]; << 898 G4double* fWeight = new G4double[fNumber] ; 929 << 899 930 for(i = 1; i <= fNumber; ++i) // Loop over << 900 for(i=1;i<=fNumber;i++) // Loop over the desired roots 931 { << 901 { 932 if(i == 1) << 902 if(i == 1) 933 { << 903 { 934 nwt = (1.0 + alpha) * (3.0 + 0.92 * alph << 904 newton = (1.0 + alpha)*(3.0 + 0.92*alpha)/(1.0 + 2.4*fNumber + 1.8*alpha) ; 935 (1.0 + 2.4 * fNumber + 1.8 * alpha << 905 } 936 } << 906 else if(i == 2) 937 else if(i == 2) << 907 { 938 { << 908 newton += (15.0 + 6.25*alpha)/(1.0 + 0.9*alpha + 2.5*fNumber) ; 939 nwt += (15.0 + 6.25 * alpha) / (1.0 + 0. << 909 } 940 } << 910 else 941 else << 911 { 942 { << 912 cofi = i - 2 ; 943 cofi = i - 2; << 913 newton += ((1.0+2.55*cofi)/(1.9*cofi) + 1.26*cofi*alpha/(1.0+3.5*cofi))* 944 nwt += ((1.0 + 2.55 * cofi) / (1.9 * cof << 914 (newton - fAbscissa[i-3])/(1.0 + 0.3*alpha) ; 945 1.26 * cofi * alpha / (1.0 + 3.5 << 915 } 946 (nwt - fAbscissa[i - 3]) / (1.0 + << 916 for(k=1;k<=maxNumber;k++) 947 } << 917 { 948 for(k = 1; k <= maxNumber; ++k) << 918 temp1 = 1.0 ; 949 { << 919 temp2 = 0.0 ; 950 temp1 = 1.0; << 951 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 // << 984 920 985 for(i = 0; i < fNumber; i++) << 921 for(j=1;j<=fNumber;j++) 986 { << 922 { 987 integral += fWeight[i] * (*f)(fAbscissa[i] << 923 temp3 = temp2 ; 988 } << 924 temp2 = temp1 ; 989 delete[] fAbscissa; << 925 temp1 = ((2*j - 1 + alpha - newton)*temp2 - (j - 1 + alpha)*temp3)/j ; 990 delete[] fWeight; << 926 } 991 return integral; << 927 temp = (fNumber*temp1 - (fNumber +alpha)*temp2)/newton ; >> 928 newton1 = newton ; >> 929 newton = newton1 - temp1/temp ; >> 930 >> 931 if(fabs(newton - newton1) <= tolerance) >> 932 { >> 933 break ; >> 934 } >> 935 } >> 936 if(k > maxNumber) >> 937 { >> 938 G4Exception("Too many (>12) iterations in G4Integration::Laguerre") ; >> 939 } >> 940 >> 941 fAbscissa[i-1] = newton ; >> 942 fWeight[i-1] = -exp(GammaLogarithm(alpha + fNumber) - >> 943 GammaLogarithm((G4double)fNumber))/(temp*fNumber*temp2) ; >> 944 } >> 945 // >> 946 // Integral evaluation >> 947 // >> 948 for(i=0;i<fNumber;i++) >> 949 { >> 950 integral += fWeight[i]*(*f)(fAbscissa[i]) ; >> 951 } >> 952 return integral ; 992 } 953 } 993 954 994 ////////////////////////////////////////////// 955 /////////////////////////////////////////////////////////////////////// 995 // 956 // 996 // Auxiliary function which returns the value << 957 // Auxiliary function which returns the value of log(gamma-function(x)) 997 // Returns the value ln(Gamma(xx) for xx > 0. << 958 // Returns the value ln(Gamma(xx) for xx > 0. Full accuracy is obtained for 998 // xx > 1. For 0 < xx < 1. the reflection form 959 // xx > 1. For 0 < xx < 1. the reflection formula (6.1.4) can be used first. 999 // (Adapted from Numerical Recipes in C) 960 // (Adapted from Numerical Recipes in C) 1000 // 961 // 1001 962 1002 template <class T, class F> 963 template <class T, class F> 1003 G4double G4Integrator<T, F>::GammaLogarithm(G << 964 G4double G4Integrator<T,F>::GammaLogarithm(G4double xx) 1004 { 965 { 1005 static const G4double cof[6] = { 76.1800917 << 966 static G4double cof[6] = { 76.18009172947146, -86.50532032941677, 1006 24.0140982 << 967 24.01409824083091, -1.231739572450155, 1007 0.12086509 << 968 0.1208650973866179e-2, -0.5395239384953e-5 } ; 1008 G4int j; << 969 register HepInt j; 1009 G4double x = xx - 1.0; << 970 G4double x = xx - 1.0 ; 1010 G4double tmp = x + 5.5; << 971 G4double tmp = x + 5.5 ; 1011 tmp -= (x + 0.5) * std::log(tmp); << 972 tmp -= (x + 0.5) * log(tmp) ; 1012 G4double ser = 1.000000000190015; << 973 G4double ser = 1.000000000190015 ; 1013 974 1014 for(j = 0; j <= 5; ++j) << 975 for ( j = 0; j <= 5; j++ ) 1015 { 976 { 1016 x += 1.0; << 977 x += 1.0 ; 1017 ser += cof[j] / x; << 978 ser += cof[j]/x ; 1018 } 979 } 1019 return -tmp + std::log(2.5066282746310005 * << 980 return -tmp + log(2.5066282746310005*ser) ; 1020 } 981 } 1021 982 1022 ///////////////////////////////////////////// 983 /////////////////////////////////////////////////////////////////////// 1023 // 984 // 1024 // Method involving Hermite polynomials 985 // Method involving Hermite polynomials 1025 // 986 // 1026 ///////////////////////////////////////////// 987 /////////////////////////////////////////////////////////////////////// 1027 // 988 // 1028 // 989 // 1029 // Gauss-Hermite method for integration of st << 990 // Gauss-Hermite method for integration of exp(-x*x)*f(x) 1030 // from minus infinity to plus infinity . << 991 // from minus infinity to plus infinity . 1031 // 992 // 1032 993 1033 template <class T, class F> << 994 template <class T, class F> 1034 G4double G4Integrator<T, F>::Hermite(T& typeT << 995 G4double G4Integrator<T,F>::Hermite( T& typeT, F f, G4int nHermite) 1035 { 996 { 1036 const G4double tolerance = 1.0e-12; << 997 const G4double tolerance = 1.0e-12 ; 1037 const G4int maxNumber = 12; << 998 const G4int maxNumber = 12 ; 1038 << 999 1039 G4int i, j, k; << 1000 G4int i, j, k ; 1040 G4double integral = 0.0; << 1001 G4double integral = 0.0 ; 1041 G4double nwt = 0., nwt1, temp1, temp2, << 1002 G4double newton, newton1, temp1, temp2, temp3, temp ; 1042 1003 1043 G4double piInMinusQ = << 1004 G4double piInMinusQ = pow(pi,-0.25) ; // 1.0/sqrt(sqrt(pi)) ?? 1044 std::pow(CLHEP::pi, -0.25); // 1.0/std:: << 1045 1005 1046 G4int fNumber = (nHermite + 1) / 2; << 1006 G4int fNumber = (nHermite +1)/2 ; 1047 G4double* fAbscissa = new G4double[fNumber] << 1007 G4double* fAbscissa = new G4double[fNumber] ; 1048 G4double* fWeight = new G4double[fNumber] << 1008 G4double* fWeight = new G4double[fNumber] ; 1049 << 1050 for(i = 1; i <= fNumber; ++i) << 1051 { << 1052 if(i == 1) << 1053 { << 1054 nwt = std::sqrt((G4double)(2 * nHermite << 1055 1.85575001 * std::pow((G4double)( << 1056 } << 1057 else if(i == 2) << 1058 { << 1059 nwt -= 1.14001 * std::pow((G4double) nH << 1060 } << 1061 else if(i == 3) << 1062 { << 1063 nwt = 1.86002 * nwt - 0.86002 * fAbscis << 1064 } << 1065 else if(i == 4) << 1066 { << 1067 nwt = 1.91001 * nwt - 0.91001 * fAbscis << 1068 } << 1069 else << 1070 { << 1071 nwt = 2.0 * nwt - fAbscissa[i - 3]; << 1072 } << 1073 for(k = 1; k <= maxNumber; ++k) << 1074 { << 1075 temp1 = piInMinusQ; << 1076 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 1009 1103 // << 1010 for(i=1;i<=fNumber;i++) 1104 // Integral calculation << 1011 { 1105 // << 1012 if(i == 1) >> 1013 { >> 1014 newton = sqrt((G4double)(2*nHermite + 1)) - >> 1015 1.85575001*pow((G4double)(2*nHermite + 1),-0.16666999) ; >> 1016 } >> 1017 else if(i == 2) >> 1018 { >> 1019 newton -= 1.14001*pow((G4double)nHermite,0.425999)/newton ; >> 1020 } >> 1021 else if(i == 3) >> 1022 { >> 1023 newton = 1.86002*newton - 0.86002*fAbscissa[0] ; >> 1024 } >> 1025 else if(i == 4) >> 1026 { >> 1027 newton = 1.91001*newton - 0.91001*fAbscissa[1] ; >> 1028 } >> 1029 else >> 1030 { >> 1031 newton = 2.0*newton - fAbscissa[i - 3] ; >> 1032 } >> 1033 for(k=1;k<=maxNumber;k++) >> 1034 { >> 1035 temp1 = piInMinusQ ; >> 1036 temp2 = 0.0 ; 1106 1037 1107 for(i = 0; i < fNumber; ++i) << 1038 for(j=1;j<=nHermite;j++) 1108 { << 1039 { 1109 integral += << 1040 temp3 = temp2 ; 1110 fWeight[i] * ((typeT.*f)(fAbscissa[i]) << 1041 temp2 = temp1 ; 1111 } << 1042 temp1 = newton*sqrt(2.0/j)*temp2 - 1112 delete[] fAbscissa; << 1043 sqrt(((G4double)(j - 1))/j)*temp3 ; 1113 delete[] fWeight; << 1044 } 1114 return integral; << 1045 temp = sqrt((G4double)2*nHermite)*temp2 ; >> 1046 newton1 = newton ; >> 1047 newton = newton1 - temp1/temp ; >> 1048 >> 1049 if(fabs(newton - newton1) <= tolerance) >> 1050 { >> 1051 break ; >> 1052 } >> 1053 } >> 1054 if(k > maxNumber) >> 1055 { >> 1056 G4Exception("Too many (>12) iterations in G4Integrator<T,F>::Hermite") ; >> 1057 } >> 1058 fAbscissa[i-1] = newton ; >> 1059 fWeight[i-1] = 2.0/(temp*temp) ; >> 1060 } >> 1061 // >> 1062 // Integral calculation >> 1063 // >> 1064 for(i=0;i<fNumber;i++) >> 1065 { >> 1066 integral += fWeight[i]*( (typeT.*f)(fAbscissa[i]) + >> 1067 (typeT.*f)(-fAbscissa[i]) ) ; >> 1068 } >> 1069 return integral ; 1115 } 1070 } 1116 1071 >> 1072 1117 ///////////////////////////////////////////// 1073 //////////////////////////////////////////////////////////////////////// 1118 // 1074 // 1119 // For use with 'this' pointer 1075 // For use with 'this' pointer 1120 1076 1121 template <class T, class F> << 1077 template <class T, class F> 1122 G4double G4Integrator<T, F>::Hermite(T* ptrT, << 1078 G4double G4Integrator<T,F>::Hermite( T* ptrT, F f, G4int n) 1123 { 1079 { 1124 return Hermite(*ptrT, f, n); << 1080 return Hermite(*ptrT,f,n) ; 1125 } << 1081 } 1126 1082 1127 ///////////////////////////////////////////// 1083 //////////////////////////////////////////////////////////////////////// 1128 // 1084 // 1129 // For use with global scope f 1085 // For use with global scope f 1130 1086 1131 template <class T, class F> 1087 template <class T, class F> 1132 G4double G4Integrator<T, F>::Hermite(G4double << 1088 G4double G4Integrator<T,F>::Hermite( G4double (*f)(G4double), G4int nHermite) 1133 { 1089 { 1134 const G4double tolerance = 1.0e-12; << 1090 const G4double tolerance = 1.0e-12 ; 1135 const G4int maxNumber = 12; << 1091 const G4int maxNumber = 12 ; 1136 << 1092 1137 G4int i, j, k; << 1093 G4int i, j, k ; 1138 G4double integral = 0.0; << 1094 G4double integral = 0.0 ; 1139 G4double nwt = 0., nwt1, temp1, temp2, << 1095 G4double newton, newton1, temp1, temp2, temp3, temp ; 1140 << 1096 1141 G4double piInMinusQ = << 1097 G4double piInMinusQ = pow(pi,-0.25) ; // 1.0/sqrt(sqrt(pi)) ?? 1142 std::pow(CLHEP::pi, -0.25); // 1.0/std:: << 1098 1143 << 1099 G4int fNumber = (nHermite +1)/2 ; 1144 G4int fNumber = (nHermite + 1) / 2; << 1100 G4double* fAbscissa = new G4double[fNumber] ; 1145 G4double* fAbscissa = new G4double[fNumber] << 1101 G4double* fWeight = new G4double[fNumber] ; 1146 G4double* fWeight = new G4double[fNumber] << 1102 1147 << 1103 for(i=1;i<=fNumber;i++) 1148 for(i = 1; i <= fNumber; ++i) << 1104 { 1149 { << 1105 if(i == 1) 1150 if(i == 1) << 1106 { 1151 { << 1107 newton = sqrt((G4double)(2*nHermite + 1)) - 1152 nwt = std::sqrt((G4double)(2 * nHermite << 1108 1.85575001*pow((G4double)(2*nHermite + 1),-0.16666999) ; 1153 1.85575001 * std::pow((G4double)( << 1109 } 1154 } << 1110 else if(i == 2) 1155 else if(i == 2) << 1111 { 1156 { << 1112 newton -= 1.14001*pow((G4double)nHermite,0.425999)/newton ; 1157 nwt -= 1.14001 * std::pow((G4double) nH << 1113 } 1158 } << 1114 else if(i == 3) 1159 else if(i == 3) << 1115 { 1160 { << 1116 newton = 1.86002*newton - 0.86002*fAbscissa[0] ; 1161 nwt = 1.86002 * nwt - 0.86002 * fAbscis << 1117 } 1162 } << 1118 else if(i == 4) 1163 else if(i == 4) << 1119 { 1164 { << 1120 newton = 1.91001*newton - 0.91001*fAbscissa[1] ; 1165 nwt = 1.91001 * nwt - 0.91001 * fAbscis << 1121 } 1166 } << 1122 else 1167 else << 1123 { 1168 { << 1124 newton = 2.0*newton - fAbscissa[i - 3] ; 1169 nwt = 2.0 * nwt - fAbscissa[i - 3]; << 1125 } 1170 } << 1126 for(k=1;k<=maxNumber;k++) 1171 for(k = 1; k <= maxNumber; ++k) << 1127 { 1172 { << 1128 temp1 = piInMinusQ ; 1173 temp1 = piInMinusQ; << 1129 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 1130 1205 for(i = 0; i < fNumber; ++i) << 1131 for(j=1;j<=nHermite;j++) 1206 { << 1132 { 1207 integral += fWeight[i] * ((*f)(fAbscissa[ << 1133 temp3 = temp2 ; 1208 } << 1134 temp2 = temp1 ; 1209 delete[] fAbscissa; << 1135 temp1 = newton*sqrt(2.0/j)*temp2 - 1210 delete[] fWeight; << 1136 sqrt(((G4double)(j - 1))/j)*temp3 ; 1211 return integral; << 1137 } >> 1138 temp = sqrt((G4double)2*nHermite)*temp2 ; >> 1139 newton1 = newton ; >> 1140 newton = newton1 - temp1/temp ; >> 1141 >> 1142 if(fabs(newton - newton1) <= tolerance) >> 1143 { >> 1144 break ; >> 1145 } >> 1146 } >> 1147 if(k > maxNumber) >> 1148 { >> 1149 G4Exception("Too many (>12) iterations in G4Integrator<T,F>::Hermite") ; >> 1150 } >> 1151 fAbscissa[i-1] = newton ; >> 1152 fWeight[i-1] = 2.0/(temp*temp) ; >> 1153 } >> 1154 // >> 1155 // Integral calculation >> 1156 // >> 1157 for(i=0;i<fNumber;i++) >> 1158 { >> 1159 integral += fWeight[i]*( (*f)(fAbscissa[i]) + (*f)(-fAbscissa[i]) ) ; >> 1160 } >> 1161 return integral ; 1212 } 1162 } 1213 1163 1214 ///////////////////////////////////////////// 1164 //////////////////////////////////////////////////////////////////////////// 1215 // 1165 // 1216 // Method involving Jacobi polynomials 1166 // Method involving Jacobi polynomials 1217 // 1167 // 1218 ///////////////////////////////////////////// 1168 //////////////////////////////////////////////////////////////////////////// 1219 // 1169 // 1220 // Gauss-Jacobi method for integration of ((1 1170 // Gauss-Jacobi method for integration of ((1-x)^alpha)*((1+x)^beta)*f(x) 1221 // from minus unit to plus unit . 1171 // from minus unit to plus unit . 1222 // 1172 // 1223 1173 1224 template <class T, class F> << 1174 template <class T, class F> 1225 G4double G4Integrator<T, F>::Jacobi(T& typeT, << 1175 G4double G4Integrator<T,F>::Jacobi( T& typeT, F f, G4double alpha, 1226 G4double << 1176 G4double beta, G4int nJacobi) 1227 { << 1177 { 1228 const G4double tolerance = 1.0e-12; << 1178 const G4double tolerance = 1.0e-12 ; 1229 const G4double maxNumber = 12; << 1179 const G4double maxNumber = 12 ; 1230 G4int i, k, j; << 1180 G4int i, k, j ; 1231 G4double alphaBeta, alphaReduced, betaReduc << 1181 G4double alphaBeta, alphaReduced, betaReduced, root1, root2, root3 ; 1232 << 1182 G4double a, b, c, newton1, newton2, newton3, newton, temp, root, rootTemp ; 1233 G4double a, b, c, nwt1, nwt2, nwt3, nwt, te << 1183 1234 << 1184 G4int fNumber = nJacobi ; 1235 G4int fNumber = nJacobi; << 1185 G4double* fAbscissa = new G4double[fNumber] ; 1236 G4double* fAbscissa = new G4double[fNumber] << 1186 G4double* fWeight = new G4double[fNumber] ; 1237 G4double* fWeight = new G4double[fNumber] << 1187 1238 << 1188 for (i=1;i<=nJacobi;i++) 1239 for(i = 1; i <= nJacobi; ++i) << 1189 { 1240 { << 1190 if (i == 1) 1241 if(i == 1) << 1191 { 1242 { << 1192 alphaReduced = alpha/nJacobi ; 1243 alphaReduced = alpha / nJacobi; << 1193 betaReduced = beta/nJacobi ; 1244 betaReduced = beta / nJacobi; << 1194 root1 = (1.0+alpha)*(2.78002/(4.0+nJacobi*nJacobi)+ 1245 root1 = (1.0 + alpha) * (2.78002 << 1195 0.767999*alphaReduced/nJacobi) ; 1246 0.767999 * alp << 1196 root2 = 1.0+1.48*alphaReduced+0.96002*betaReduced + 1247 root2 = 1.0 + 1.48 * alphaReduce << 1197 0.451998*alphaReduced*alphaReduced + 1248 0.451998 * alphaReduced * alpha << 1198 0.83001*alphaReduced*betaReduced ; 1249 0.83001 * alphaReduced * betaRe << 1199 root = 1.0-root1/root2 ; 1250 root = 1.0 - root1 / root2; << 1200 } 1251 } << 1201 else if (i == 2) 1252 else if(i == 2) << 1202 { 1253 { << 1203 root1=(4.1002+alpha)/((1.0+alpha)*(1.0+0.155998*alpha)) ; 1254 root1 = (4.1002 + alpha) / ((1.0 + alph << 1204 root2=1.0+0.06*(nJacobi-8.0)*(1.0+0.12*alpha)/nJacobi ; 1255 root2 = 1.0 + 0.06 * (nJacobi - 8.0) * << 1205 root3=1.0+0.012002*beta*(1.0+0.24997*fabs(alpha))/nJacobi ; 1256 root3 = << 1206 root -= (1.0-root)*root1*root2*root3 ; 1257 1.0 + 0.012002 * beta * (1.0 + 0.2499 << 1207 } 1258 root -= (1.0 - root) * root1 * root2 * << 1208 else if (i == 3) 1259 } << 1209 { 1260 else if(i == 3) << 1210 root1=(1.67001+0.27998*alpha)/(1.0+0.37002*alpha) ; 1261 { << 1211 root2=1.0+0.22*(nJacobi-8.0)/nJacobi ; 1262 root1 = (1.67001 + 0.27998 * alpha) / ( << 1212 root3=1.0+8.0*beta/((6.28001+beta)*nJacobi*nJacobi) ; 1263 root2 = 1.0 + 0.22 * (nJacobi - 8.0) / << 1213 root -= (fAbscissa[0]-root)*root1*root2*root3 ; 1264 root3 = 1.0 + 8.0 * beta / ((6.28001 + << 1214 } 1265 root -= (fAbscissa[0] - root) * root1 * << 1215 else if (i == nJacobi-1) 1266 } << 1216 { 1267 else if(i == nJacobi - 1) << 1217 root1=(1.0+0.235002*beta)/(0.766001+0.118998*beta) ; 1268 { << 1218 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 << 1219 root3=1.0/(1.0+20.0*alpha/((7.5+alpha)*nJacobi*nJacobi)) ; 1270 root2 = 1.0 / (1.0 + 0.639002 * (nJacob << 1220 root += (root-fAbscissa[nJacobi-4])*root1*root2*root3 ; 1271 (1.0 + 0.71001 * << 1221 } 1272 root3 = 1.0 / (1.0 + 20.0 * alpha / ((7 << 1222 else if (i == nJacobi) 1273 root += (root - fAbscissa[nJacobi - 4]) << 1223 { 1274 } << 1224 root1 = (1.0+0.37002*beta)/(1.67001+0.27998*beta) ; 1275 else if(i == nJacobi) << 1225 root2 = 1.0/(1.0+0.22*(nJacobi-8.0)/nJacobi) ; 1276 { << 1226 root3 = 1.0/(1.0+8.0*alpha/((6.28002+alpha)*nJacobi*nJacobi)) ; 1277 root1 = (1.0 + 0.37002 * beta) / (1.670 << 1227 root += (root-fAbscissa[nJacobi-3])*root1*root2*root3 ; 1278 root2 = 1.0 / (1.0 + 0.22 * (nJacobi - << 1228 } 1279 root3 = << 1229 else 1280 1.0 / (1.0 + 8.0 * alpha / ((6.28002 << 1230 { 1281 root += (root - fAbscissa[nJacobi - 3]) << 1231 root = 3.0*fAbscissa[i-2]-3.0*fAbscissa[i-3]+fAbscissa[i-4] ; 1282 } << 1232 } 1283 else << 1233 alphaBeta = alpha + beta ; 1284 { << 1234 for (k=1;k<=maxNumber;k++) 1285 root = 3.0 * fAbscissa[i - 2] - 3.0 * f << 1235 { 1286 } << 1236 temp = 2.0 + alphaBeta ; 1287 alphaBeta = alpha + beta; << 1237 newton1 = (alpha-beta+temp*root)/2.0 ; 1288 for(k = 1; k <= maxNumber; ++k) << 1238 newton2 = 1.0 ; 1289 { << 1239 for (j=2;j<=nJacobi;j++) 1290 temp = 2.0 + alphaBeta; << 1240 { 1291 nwt1 = (alpha - beta + temp * root) / 2 << 1241 newton3 = newton2 ; 1292 nwt2 = 1.0; << 1242 newton2 = newton1 ; 1293 for(j = 2; j <= nJacobi; ++j) << 1243 temp = 2*j+alphaBeta ; 1294 { << 1244 a = 2*j*(j+alphaBeta)*(temp-2.0) ; 1295 nwt3 = nwt2; << 1245 b = (temp-1.0)*(alpha*alpha-beta*beta+temp*(temp-2.0)*root) ; 1296 nwt2 = nwt1; << 1246 c = 2.0*(j-1+alpha)*(j-1+beta)*temp ; 1297 temp = 2 * j + alphaBeta; << 1247 newton1 = (b*newton2-c*newton3)/a ; 1298 a = 2 * j * (j + alphaBeta) * (tem << 1248 } 1299 b = (temp - 1.0) * << 1249 newton = (nJacobi*(alpha - beta - temp*root)*newton1 + 1300 (alpha * alpha - beta * beta + te << 1250 2.0*(nJacobi + alpha)*(nJacobi + beta)*newton2)/ 1301 c = 2.0 * (j - 1 + alpha) * (j - 1 << 1251 (temp*(1.0 - root*root)) ; 1302 nwt1 = (b * nwt2 - c * nwt3) / a; << 1252 rootTemp = root ; 1303 } << 1253 root = rootTemp - newton1/newton ; 1304 nwt = (nJacobi * (alpha - beta - temp * << 1254 if (fabs(root-rootTemp) <= tolerance) 1305 2.0 * (nJacobi + alpha) * (nJaco << 1255 { 1306 (temp * (1.0 - root * root)); << 1256 break ; 1307 rootTemp = root; << 1257 } 1308 root = rootTemp - nwt1 / nwt; << 1258 } 1309 if(std::fabs(root - rootTemp) <= tolera << 1259 if (k > maxNumber) 1310 { << 1260 { 1311 break; << 1261 G4Exception("Too many iterations (>12) in G4Integrator<T,F>::Jacobi") ; 1312 } << 1262 } 1313 } << 1263 fAbscissa[i-1] = root ; 1314 if(k > maxNumber) << 1264 fWeight[i-1] = exp(GammaLogarithm((G4double)(alpha+nJacobi)) + 1315 { << 1265 GammaLogarithm((G4double)(beta+nJacobi)) - 1316 G4Exception("G4Integrator<T,F>::Jacobi( << 1266 GammaLogarithm((G4double)(nJacobi+1.0)) - 1317 FatalException, "Too many ( << 1267 GammaLogarithm((G4double)(nJacobi + alphaBeta + 1.0))) 1318 } << 1268 *temp*pow(2.0,alphaBeta)/(newton*newton2) ; 1319 fAbscissa[i - 1] = root; << 1269 } 1320 fWeight[i - 1] = << 1270 // 1321 std::exp(GammaLogarithm((G4double)(alph << 1271 // Calculation of the integral 1322 GammaLogarithm((G4double)(beta << 1272 // 1323 GammaLogarithm((G4double)(nJac << 1273 G4double integral = 0.0 ; 1324 GammaLogarithm((G4double)(nJac << 1274 for(i=0;i<fNumber;i++) 1325 temp * std::pow(2.0, alphaBeta) / (nwt << 1275 { 1326 } << 1276 integral += fWeight[i]*(typeT.*f)(fAbscissa[i]) ; 1327 << 1277 } 1328 // << 1278 return integral ; 1329 // Calculation of the integral << 1330 // << 1331 << 1332 G4double integral = 0.0; << 1333 for(i = 0; i < fNumber; ++i) << 1334 { << 1335 integral += fWeight[i] * (typeT.*f)(fAbsc << 1336 } << 1337 delete[] fAbscissa; << 1338 delete[] fWeight; << 1339 return integral; << 1340 } 1279 } 1341 1280 >> 1281 1342 ///////////////////////////////////////////// 1282 ///////////////////////////////////////////////////////////////////////// 1343 // 1283 // 1344 // For use with 'this' pointer 1284 // For use with 'this' pointer 1345 1285 1346 template <class T, class F> << 1286 template <class T, class F> 1347 G4double G4Integrator<T, F>::Jacobi(T* ptrT, << 1287 G4double G4Integrator<T,F>::Jacobi( T* ptrT, F f, G4double alpha, 1348 G4int n) << 1288 G4double beta, G4int n) 1349 { 1289 { 1350 return Jacobi(*ptrT, f, alpha, beta, n); << 1290 return Jacobi(*ptrT,f,alpha,beta,n) ; 1351 } << 1291 } 1352 1292 1353 ///////////////////////////////////////////// 1293 ///////////////////////////////////////////////////////////////////////// 1354 // 1294 // 1355 // For use with global scope f << 1295 // For use with global scope f 1356 1296 1357 template <class T, class F> 1297 template <class T, class F> 1358 G4double G4Integrator<T, F>::Jacobi(G4double << 1298 G4double G4Integrator<T,F>::Jacobi( G4double (*f)(G4double), G4double alpha, 1359 G4double << 1299 G4double beta, G4int nJacobi) 1360 { 1300 { 1361 const G4double tolerance = 1.0e-12; << 1301 const G4double tolerance = 1.0e-12 ; 1362 const G4double maxNumber = 12; << 1302 const G4double maxNumber = 12 ; 1363 G4int i, k, j; << 1303 G4int i, k, j ; 1364 G4double alphaBeta, alphaReduced, betaReduc << 1304 G4double alphaBeta, alphaReduced, betaReduced, root1, root2, root3 ; 1365 << 1305 G4double a, b, c, newton1, newton2, newton3, newton, temp, root, rootTemp ; 1366 G4double a, b, c, nwt1, nwt2, nwt3, nwt, te << 1306 1367 << 1307 G4int fNumber = nJacobi ; 1368 G4int fNumber = nJacobi; << 1308 G4double* fAbscissa = new G4double[fNumber] ; 1369 G4double* fAbscissa = new G4double[fNumber] << 1309 G4double* fWeight = new G4double[fNumber] ; 1370 G4double* fWeight = new G4double[fNumber] << 1310 1371 << 1311 for (i=1;i<=nJacobi;i++) 1372 for(i = 1; i <= nJacobi; ++i) << 1312 { 1373 { << 1313 if (i == 1) 1374 if(i == 1) << 1314 { 1375 { << 1315 alphaReduced = alpha/nJacobi ; 1376 alphaReduced = alpha / nJacobi; << 1316 betaReduced = beta/nJacobi ; 1377 betaReduced = beta / nJacobi; << 1317 root1 = (1.0+alpha)*(2.78002/(4.0+nJacobi*nJacobi)+ 1378 root1 = (1.0 + alpha) * (2.78002 << 1318 0.767999*alphaReduced/nJacobi) ; 1379 0.767999 * alp << 1319 root2 = 1.0+1.48*alphaReduced+0.96002*betaReduced + 1380 root2 = 1.0 + 1.48 * alphaReduce << 1320 0.451998*alphaReduced*alphaReduced + 1381 0.451998 * alphaReduced * alpha << 1321 0.83001*alphaReduced*betaReduced ; 1382 0.83001 * alphaReduced * betaRe << 1322 root = 1.0-root1/root2 ; 1383 root = 1.0 - root1 / root2; << 1323 } 1384 } << 1324 else if (i == 2) 1385 else if(i == 2) << 1325 { 1386 { << 1326 root1=(4.1002+alpha)/((1.0+alpha)*(1.0+0.155998*alpha)) ; 1387 root1 = (4.1002 + alpha) / ((1.0 + alph << 1327 root2=1.0+0.06*(nJacobi-8.0)*(1.0+0.12*alpha)/nJacobi ; 1388 root2 = 1.0 + 0.06 * (nJacobi - 8.0) * << 1328 root3=1.0+0.012002*beta*(1.0+0.24997*fabs(alpha))/nJacobi ; 1389 root3 = << 1329 root -= (1.0-root)*root1*root2*root3 ; 1390 1.0 + 0.012002 * beta * (1.0 + 0.2499 << 1330 } 1391 root -= (1.0 - root) * root1 * root2 * << 1331 else if (i == 3) 1392 } << 1332 { 1393 else if(i == 3) << 1333 root1=(1.67001+0.27998*alpha)/(1.0+0.37002*alpha) ; 1394 { << 1334 root2=1.0+0.22*(nJacobi-8.0)/nJacobi ; 1395 root1 = (1.67001 + 0.27998 * alpha) / ( << 1335 root3=1.0+8.0*beta/((6.28001+beta)*nJacobi*nJacobi) ; 1396 root2 = 1.0 + 0.22 * (nJacobi - 8.0) / << 1336 root -= (fAbscissa[0]-root)*root1*root2*root3 ; 1397 root3 = 1.0 + 8.0 * beta / ((6.28001 + << 1337 } 1398 root -= (fAbscissa[0] - root) * root1 * << 1338 else if (i == nJacobi-1) 1399 } << 1339 { 1400 else if(i == nJacobi - 1) << 1340 root1=(1.0+0.235002*beta)/(0.766001+0.118998*beta) ; 1401 { << 1341 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 << 1342 root3=1.0/(1.0+20.0*alpha/((7.5+alpha)*nJacobi*nJacobi)) ; 1403 root2 = 1.0 / (1.0 + 0.639002 * (nJacob << 1343 root += (root-fAbscissa[nJacobi-4])*root1*root2*root3 ; 1404 (1.0 + 0.71001 * << 1344 } 1405 root3 = 1.0 / (1.0 + 20.0 * alpha / ((7 << 1345 else if (i == nJacobi) 1406 root += (root - fAbscissa[nJacobi - 4]) << 1346 { 1407 } << 1347 root1 = (1.0+0.37002*beta)/(1.67001+0.27998*beta) ; 1408 else if(i == nJacobi) << 1348 root2 = 1.0/(1.0+0.22*(nJacobi-8.0)/nJacobi) ; 1409 { << 1349 root3 = 1.0/(1.0+8.0*alpha/((6.28002+alpha)*nJacobi*nJacobi)) ; 1410 root1 = (1.0 + 0.37002 * beta) / (1.670 << 1350 root += (root-fAbscissa[nJacobi-3])*root1*root2*root3 ; 1411 root2 = 1.0 / (1.0 + 0.22 * (nJacobi - << 1351 } 1412 root3 = << 1352 else 1413 1.0 / (1.0 + 8.0 * alpha / ((6.28002 << 1353 { 1414 root += (root - fAbscissa[nJacobi - 3]) << 1354 root = 3.0*fAbscissa[i-2]-3.0*fAbscissa[i-3]+fAbscissa[i-4] ; 1415 } << 1355 } 1416 else << 1356 alphaBeta = alpha + beta ; 1417 { << 1357 for (k=1;k<=maxNumber;k++) 1418 root = 3.0 * fAbscissa[i - 2] - 3.0 * f << 1358 { 1419 } << 1359 temp = 2.0 + alphaBeta ; 1420 alphaBeta = alpha + beta; << 1360 newton1 = (alpha-beta+temp*root)/2.0 ; 1421 for(k = 1; k <= maxNumber; ++k) << 1361 newton2 = 1.0 ; 1422 { << 1362 for (j=2;j<=nJacobi;j++) 1423 temp = 2.0 + alphaBeta; << 1363 { 1424 nwt1 = (alpha - beta + temp * root) / 2 << 1364 newton3 = newton2 ; 1425 nwt2 = 1.0; << 1365 newton2 = newton1 ; 1426 for(j = 2; j <= nJacobi; ++j) << 1366 temp = 2*j+alphaBeta ; 1427 { << 1367 a = 2*j*(j+alphaBeta)*(temp-2.0) ; 1428 nwt3 = nwt2; << 1368 b = (temp-1.0)*(alpha*alpha-beta*beta+temp*(temp-2.0)*root) ; 1429 nwt2 = nwt1; << 1369 c = 2.0*(j-1+alpha)*(j-1+beta)*temp ; 1430 temp = 2 * j + alphaBeta; << 1370 newton1 = (b*newton2-c*newton3)/a ; 1431 a = 2 * j * (j + alphaBeta) * (tem << 1371 } 1432 b = (temp - 1.0) * << 1372 newton = (nJacobi*(alpha - beta - temp*root)*newton1 + 1433 (alpha * alpha - beta * beta + te << 1373 2.0*(nJacobi + alpha)*(nJacobi + beta)*newton2)/ 1434 c = 2.0 * (j - 1 + alpha) * (j - 1 << 1374 (temp*(1.0 - root*root)) ; 1435 nwt1 = (b * nwt2 - c * nwt3) / a; << 1375 rootTemp = root ; 1436 } << 1376 root = rootTemp - newton1/newton ; 1437 nwt = (nJacobi * (alpha - beta - temp * << 1377 if (fabs(root-rootTemp) <= tolerance) 1438 2.0 * (nJacobi + alpha) * (nJaco << 1378 { 1439 (temp * (1.0 - root * root)); << 1379 break ; 1440 rootTemp = root; << 1380 } 1441 root = rootTemp - nwt1 / nwt; << 1381 } 1442 if(std::fabs(root - rootTemp) <= tolera << 1382 if (k > maxNumber) 1443 { << 1383 { 1444 break; << 1384 G4Exception("Too many iterations (>12) in G4Integrator<T,F>::Jacobi") ; 1445 } << 1385 } 1446 } << 1386 fAbscissa[i-1] = root ; 1447 if(k > maxNumber) << 1387 fWeight[i-1] = exp(GammaLogarithm((G4double)(alpha+nJacobi)) + 1448 { << 1388 GammaLogarithm((G4double)(beta+nJacobi)) - 1449 G4Exception("G4Integrator<T,F>::Jacobi( << 1389 GammaLogarithm((G4double)(nJacobi+1.0)) - 1450 "Too many (>12) iterations. << 1390 GammaLogarithm((G4double)(nJacobi + alphaBeta + 1.0))) 1451 } << 1391 *temp*pow(2.0,alphaBeta)/(newton*newton2) ; 1452 fAbscissa[i - 1] = root; << 1392 } 1453 fWeight[i - 1] = << 1393 // 1454 std::exp(GammaLogarithm((G4double)(alph << 1394 // Calculation of the integral 1455 GammaLogarithm((G4double)(beta << 1395 // 1456 GammaLogarithm((G4double)(nJac << 1396 G4double integral = 0.0 ; 1457 GammaLogarithm((G4double)(nJac << 1397 for(i=0;i<fNumber;i++) 1458 temp * std::pow(2.0, alphaBeta) / (nwt << 1398 { 1459 } << 1399 integral += fWeight[i]*(*f)(fAbscissa[i]) ; >> 1400 } >> 1401 return integral ; >> 1402 } 1460 1403 1461 // << 1462 // Calculation of the integral << 1463 // << 1464 1404 1465 G4double integral = 0.0; << 1466 for(i = 0; i < fNumber; ++i) << 1467 { << 1468 integral += fWeight[i] * (*f)(fAbscissa[i << 1469 } << 1470 delete[] fAbscissa; << 1471 delete[] fWeight; << 1472 return integral; << 1473 } << 1474 1405 1475 // 1406 // 1476 // 1407 // 1477 ///////////////////////////////////////////// 1408 /////////////////////////////////////////////////////////////////// >> 1409 >> 1410 >> 1411 >> 1412 1478 1413