Geant4 Cross Reference |
1 // 1 // 2 // ******************************************* 2 // ******************************************************************** 3 // * License and Disclaimer << 3 // * DISCLAIMER * 4 // * 4 // * * 5 // * The Geant4 software is copyright of th << 5 // * The following disclaimer summarizes all the specific disclaimers * 6 // * the Geant4 Collaboration. It is provided << 6 // * of contributors to this software. The specific disclaimers,which * 7 // * conditions of the Geant4 Software License << 7 // * govern, are listed with their locations in: * 8 // * LICENSE and available at http://cern.ch/ << 8 // * http://cern.ch/geant4/license * 9 // * include a list of copyright holders. << 10 // * 9 // * * 11 // * Neither the authors of this software syst 10 // * Neither the authors of this software system, nor their employing * 12 // * institutes,nor the agencies providing fin 11 // * institutes,nor the agencies providing financial support for this * 13 // * work make any representation or warran 12 // * work make any representation or warranty, express or implied, * 14 // * regarding this software system or assum 13 // * regarding this software system or assume any liability for its * 15 // * use. Please see the license in the file << 14 // * use. * 16 // * for the full disclaimer and the limitatio << 17 // * 15 // * * 18 // * This code implementation is the result << 16 // * This code implementation is the intellectual property of the * 19 // * technical work of the GEANT4 collaboratio << 17 // * GEANT4 collaboration. * 20 // * By using, copying, modifying or distri << 18 // * By copying, distributing or modifying the Program (or any work * 21 // * any work based on the software) you ag << 19 // * based on the Program) you indicate your acceptance of this * 22 // * use in resulting scientific publicati << 20 // * statement, and all its terms. * 23 // * acceptance of all terms of the Geant4 Sof << 24 // ******************************************* 21 // ******************************************************************** 25 // 22 // 26 // G4Integrator inline methods implementation << 27 // 23 // 28 // Author: V.Grichine, 04.09.1999 - First impl << 24 // $Id: G4Integrator.icc,v 1.12 2005/03/15 19:11:35 gcosmo Exp $ 29 // G4SimpleIntegration class with H.P. << 25 // GEANT4 tag $Name: geant4-08-00 $ 30 // E.TCherniaev advises << 26 // 31 // ------------------------------------------- << 27 // Implementation of G4Integrator methods. >> 28 // >> 29 // 32 30 33 ////////////////////////////////////////////// 31 ///////////////////////////////////////////////////////////////////// 34 // 32 // 35 // Sympson integration method 33 // Sympson integration method 36 // 34 // 37 ////////////////////////////////////////////// 35 ///////////////////////////////////////////////////////////////////// 38 // 36 // 39 // Integration of class member functions T::f << 37 // Integration of class member functions T::f by Simpson method. 40 38 41 template <class T, class F> << 39 template <class T, class F> 42 G4double G4Integrator<T, F>::Simpson(T& typeT, << 40 G4double G4Integrator<T,F>::Simpson( T& typeT, 43 G4double << 41 F f, 44 { << 42 G4double xInitial, 45 G4int i; << 43 G4double xFinal, 46 G4double step = (xFinal - xInitial) / itera << 44 G4int iterationNumber ) 47 G4double x = xInitial; << 45 { 48 G4double xPlus = xInitial + 0.5 * step; << 46 G4int i ; 49 G4double mean = ((typeT.*f)(xInitial) + (ty << 47 G4double step = (xFinal - xInitial)/iterationNumber ; 50 G4double sum = (typeT.*f)(xPlus); << 48 G4double x = xInitial ; 51 << 49 G4double xPlus = xInitial + 0.5*step ; 52 for(i = 1; i < iterationNumber; ++i) << 50 G4double mean = ( (typeT.*f)(xInitial) + (typeT.*f)(xFinal) )*0.5 ; 53 { << 51 G4double sum = (typeT.*f)(xPlus) ; 54 x += step; << 52 55 xPlus += step; << 53 for(i=1;i<iterationNumber;i++) 56 mean += (typeT.*f)(x); << 54 { 57 sum += (typeT.*f)(xPlus); << 55 x += step ; 58 } << 56 xPlus += step ; 59 mean += 2.0 * sum; << 57 mean += (typeT.*f)(x) ; >> 58 sum += (typeT.*f)(xPlus) ; >> 59 } >> 60 mean += 2.0*sum ; 60 61 61 return mean * step / 3.0; << 62 return mean*step/3.0 ; 62 } 63 } 63 64 64 ////////////////////////////////////////////// 65 ///////////////////////////////////////////////////////////////////// 65 // 66 // 66 // Integration of class member functions T::f 67 // Integration of class member functions T::f by Simpson method. 67 // Convenient to use with 'this' pointer 68 // Convenient to use with 'this' pointer 68 69 69 template <class T, class F> << 70 template <class T, class F> 70 G4double G4Integrator<T, F>::Simpson(T* ptrT, << 71 G4double G4Integrator<T,F>::Simpson( T* ptrT, 71 G4double << 72 F f, 72 { << 73 G4double xInitial, 73 G4int i; << 74 G4double xFinal, 74 G4double step = (xFinal - xInitial) / itera << 75 G4int iterationNumber ) 75 G4double x = xInitial; << 76 { 76 G4double xPlus = xInitial + 0.5 * step; << 77 G4int i ; 77 G4double mean = ((ptrT->*f)(xInitial) + (pt << 78 G4double step = (xFinal - xInitial)/iterationNumber ; 78 G4double sum = (ptrT->*f)(xPlus); << 79 G4double x = xInitial ; >> 80 G4double xPlus = xInitial + 0.5*step ; >> 81 G4double mean = ( (ptrT->*f)(xInitial) + (ptrT->*f)(xFinal) )*0.5 ; >> 82 G4double sum = (ptrT->*f)(xPlus) ; >> 83 >> 84 for(i=1;i<iterationNumber;i++) >> 85 { >> 86 x += step ; >> 87 xPlus += step ; >> 88 mean += (ptrT->*f)(x) ; >> 89 sum += (ptrT->*f)(xPlus) ; >> 90 } >> 91 mean += 2.0*sum ; 79 92 80 for(i = 1; i < iterationNumber; ++i) << 93 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 } 94 } 91 95 92 ////////////////////////////////////////////// 96 ///////////////////////////////////////////////////////////////////// 93 // 97 // 94 // Integration of class member functions T::f 98 // Integration of class member functions T::f by Simpson method. 95 // Convenient to use, when function f is defin 99 // Convenient to use, when function f is defined in global scope, i.e. in main() 96 // program 100 // program 97 101 98 template <class T, class F> << 102 template <class T, class F> 99 G4double G4Integrator<T, F>::Simpson(G4double << 103 G4double G4Integrator<T,F>::Simpson( G4double (*f)(G4double), 100 G4double << 104 G4double xInitial, 101 { << 105 G4double xFinal, 102 G4int i; << 106 G4int iterationNumber ) 103 G4double step = (xFinal - xInitial) / itera << 107 { 104 G4double x = xInitial; << 108 G4int i ; 105 G4double xPlus = xInitial + 0.5 * step; << 109 G4double step = (xFinal - xInitial)/iterationNumber ; 106 G4double mean = ((*f)(xInitial) + (*f)(xFin << 110 G4double x = xInitial ; 107 G4double sum = (*f)(xPlus); << 111 G4double xPlus = xInitial + 0.5*step ; 108 << 112 G4double mean = ( (*f)(xInitial) + (*f)(xFinal) )*0.5 ; 109 for(i = 1; i < iterationNumber; ++i) << 113 G4double sum = (*f)(xPlus) ; 110 { << 114 111 x += step; << 115 for(i=1;i<iterationNumber;i++) 112 xPlus += step; << 116 { 113 mean += (*f)(x); << 117 x += step ; 114 sum += (*f)(xPlus); << 118 xPlus += step ; 115 } << 119 mean += (*f)(x) ; 116 mean += 2.0 * sum; << 120 sum += (*f)(xPlus) ; >> 121 } >> 122 mean += 2.0*sum ; 117 123 118 return mean * step / 3.0; << 124 return mean*step/3.0 ; 119 } 125 } 120 126 121 ////////////////////////////////////////////// 127 ////////////////////////////////////////////////////////////////////////// 122 // 128 // 123 // Adaptive Gauss method 129 // Adaptive Gauss method 124 // 130 // 125 ////////////////////////////////////////////// 131 ////////////////////////////////////////////////////////////////////////// 126 // 132 // 127 // 133 // 128 134 129 template <class T, class F> << 135 template <class T, class F> 130 G4double G4Integrator<T, F>::Gauss(T& typeT, F << 136 G4double G4Integrator<T,F>::Gauss( T& typeT, F f, 131 G4double xF << 137 G4double xInitial, G4double xFinal ) 132 { << 138 { 133 static const G4double root = 1.0 / std::sqrt << 139 static G4double root = 1.0/std::sqrt(3.0) ; 134 << 140 135 G4double xMean = (xInitial + xFinal) / 2.0; << 141 G4double xMean = (xInitial + xFinal)/2.0 ; 136 G4double Step = (xFinal - xInitial) / 2.0; << 142 G4double Step = (xFinal - xInitial)/2.0 ; 137 G4double delta = Step * root; << 143 G4double delta = Step*root ; 138 G4double sum = ((typeT.*f)(xMean + delta) << 144 G4double sum = ((typeT.*f)(xMean + delta) + 139 << 145 (typeT.*f)(xMean - delta)) ; 140 return sum * Step; << 146 >> 147 return sum*Step ; 141 } 148 } 142 149 143 ////////////////////////////////////////////// 150 ////////////////////////////////////////////////////////////////////// 144 // 151 // 145 // 152 // 146 153 147 template <class T, class F> << 154 template <class T, class F> G4double 148 G4double G4Integrator<T, F>::Gauss(T* ptrT, F << 155 G4Integrator<T,F>::Gauss( T* ptrT, F f, G4double a, G4double b ) 149 { 156 { 150 return Gauss(*ptrT, f, a, b); << 157 return Gauss(*ptrT,f,a,b) ; 151 } 158 } 152 159 153 ////////////////////////////////////////////// 160 /////////////////////////////////////////////////////////////////////// 154 // 161 // 155 // 162 // 156 163 157 template <class T, class F> 164 template <class T, class F> 158 G4double G4Integrator<T, F>::Gauss(G4double (* << 165 G4double G4Integrator<T,F>::Gauss( G4double (*f)(G4double), 159 G4double xF << 166 G4double xInitial, G4double xFinal) 160 { 167 { 161 static const G4double root = 1.0 / std::sqrt << 168 static G4double root = 1.0/std::sqrt(3.0) ; 162 << 169 163 G4double xMean = (xInitial + xFinal) / 2.0; << 170 G4double xMean = (xInitial + xFinal)/2.0 ; 164 G4double Step = (xFinal - xInitial) / 2.0; << 171 G4double Step = (xFinal - xInitial)/2.0 ; 165 G4double delta = Step * root; << 172 G4double delta = Step*root ; 166 G4double sum = ((*f)(xMean + delta) + (*f) << 173 G4double sum = ( (*f)(xMean + delta) + (*f)(xMean - delta) ) ; 167 << 174 168 return sum * Step; << 175 return sum*Step ; 169 } 176 } 170 177 171 ////////////////////////////////////////////// 178 /////////////////////////////////////////////////////////////////////////// 172 // 179 // 173 // 180 // 174 181 175 template <class T, class F> << 182 template <class T, class F> 176 void G4Integrator<T, F>::AdaptGauss(T& typeT, << 183 void G4Integrator<T,F>::AdaptGauss( T& typeT, F f, G4double xInitial, 177 G4double x << 184 G4double xFinal, G4double fTolerance, 178 G4double& << 185 G4double& sum, >> 186 G4int& depth ) >> 187 { >> 188 if(depth > 100) >> 189 { >> 190 G4cout<<"G4Integrator<T,F>::AdaptGauss: WARNING !!!"<<G4endl ; >> 191 G4cout<<"Function varies too rapidly to get stated accuracy in 100 steps " >> 192 <<G4endl ; >> 193 >> 194 return ; >> 195 } >> 196 G4double xMean = (xInitial + xFinal)/2.0 ; >> 197 G4double leftHalf = Gauss(typeT,f,xInitial,xMean) ; >> 198 G4double rightHalf = Gauss(typeT,f,xMean,xFinal) ; >> 199 G4double full = Gauss(typeT,f,xInitial,xFinal) ; >> 200 if(std::fabs(leftHalf+rightHalf-full) < fTolerance) >> 201 { >> 202 sum += full ; >> 203 } >> 204 else >> 205 { >> 206 depth++ ; >> 207 AdaptGauss(typeT,f,xInitial,xMean,fTolerance,sum,depth) ; >> 208 AdaptGauss(typeT,f,xMean,xFinal,fTolerance,sum,depth) ; >> 209 } >> 210 } >> 211 >> 212 template <class T, class F> >> 213 void G4Integrator<T,F>::AdaptGauss( T* ptrT, F f, G4double xInitial, >> 214 G4double xFinal, G4double fTolerance, >> 215 G4double& sum, >> 216 G4int& depth ) 179 { 217 { 180 if(depth > 100) << 218 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 } 219 } 211 220 212 ////////////////////////////////////////////// 221 ///////////////////////////////////////////////////////////////////////// 213 // 222 // 214 // 223 // 215 template <class T, class F> 224 template <class T, class F> 216 void G4Integrator<T, F>::AdaptGauss(G4double ( << 225 void G4Integrator<T,F>::AdaptGauss( G4double (*f)(G4double), 217 G4double x << 226 G4double xInitial, G4double xFinal, 218 G4double& << 227 G4double fTolerance, G4double& sum, 219 { << 228 G4int& depth ) 220 if(depth > 100) << 229 { 221 { << 230 if(depth > 100) 222 G4cout << "G4SimpleIntegration::AdaptGauss << 231 { 223 G4cout << "Function varies too rapidly to << 232 G4cout<<"G4SimpleIntegration::AdaptGauss: WARNING !!!"<<G4endl ; 224 << G4endl; << 233 G4cout<<"Function varies too rapidly to get stated accuracy in 100 steps " 225 << 234 <<G4endl ; 226 return; << 235 227 } << 236 return ; 228 G4double xMean = (xInitial + xFinal) / 2 << 237 } 229 G4double leftHalf = Gauss(f, xInitial, xMea << 238 G4double xMean = (xInitial + xFinal)/2.0 ; 230 G4double rightHalf = Gauss(f, xMean, xFinal) << 239 G4double leftHalf = Gauss(f,xInitial,xMean) ; 231 G4double full = Gauss(f, xInitial, xFin << 240 G4double rightHalf = Gauss(f,xMean,xFinal) ; 232 if(std::fabs(leftHalf + rightHalf - full) < << 241 G4double full = Gauss(f,xInitial,xFinal) ; 233 { << 242 if(std::fabs(leftHalf+rightHalf-full) < fTolerance) 234 sum += full; << 243 { 235 } << 244 sum += full ; 236 else << 245 } 237 { << 246 else 238 ++depth; << 247 { 239 AdaptGauss(f, xInitial, xMean, fTolerance, << 248 depth++ ; 240 AdaptGauss(f, xMean, xFinal, fTolerance, s << 249 AdaptGauss(f,xInitial,xMean,fTolerance,sum,depth) ; 241 } << 250 AdaptGauss(f,xMean,xFinal,fTolerance,sum,depth) ; >> 251 } 242 } 252 } 243 253 244 ////////////////////////////////////////////// 254 //////////////////////////////////////////////////////////////////////// 245 // 255 // 246 // Adaptive Gauss integration with accuracy 'e 256 // Adaptive Gauss integration with accuracy 'e' 247 // Convenient for using with class object type 257 // Convenient for using with class object typeT 248 << 258 249 template <class T, class F> << 259 template<class T, class F> 250 G4double G4Integrator<T, F>::AdaptiveGauss(T& << 260 G4double G4Integrator<T,F>::AdaptiveGauss( T& typeT, F f, G4double xInitial, 251 G4d << 261 G4double xFinal, G4double e ) 252 { << 262 { 253 G4int depth = 0; << 263 G4int depth = 0 ; 254 G4double sum = 0.0; << 264 G4double sum = 0.0 ; 255 AdaptGauss(typeT, f, xInitial, xFinal, e, su << 265 AdaptGauss(typeT,f,xInitial,xFinal,e,sum,depth) ; 256 return sum; << 266 return sum ; 257 } 267 } 258 268 259 ////////////////////////////////////////////// 269 //////////////////////////////////////////////////////////////////////// 260 // 270 // 261 // Adaptive Gauss integration with accuracy 'e 271 // Adaptive Gauss integration with accuracy 'e' 262 // Convenient for using with 'this' pointer 272 // Convenient for using with 'this' pointer 263 << 273 264 template <class T, class F> << 274 template<class T, class F> 265 G4double G4Integrator<T, F>::AdaptiveGauss(T* << 275 G4double G4Integrator<T,F>::AdaptiveGauss( T* ptrT, F f, G4double xInitial, 266 G4d << 276 G4double xFinal, G4double e ) 267 { 277 { 268 return AdaptiveGauss(*ptrT, f, xInitial, xFi << 278 return AdaptiveGauss(*ptrT,f,xInitial,xFinal,e) ; 269 } 279 } 270 280 271 ////////////////////////////////////////////// 281 //////////////////////////////////////////////////////////////////////// 272 // 282 // 273 // Adaptive Gauss integration with accuracy 'e 283 // Adaptive Gauss integration with accuracy 'e' 274 // Convenient for using with global scope func 284 // Convenient for using with global scope function f 275 << 285 276 template <class T, class F> 286 template <class T, class F> 277 G4double G4Integrator<T, F>::AdaptiveGauss(G4d << 287 G4double G4Integrator<T,F>::AdaptiveGauss( G4double (*f)(G4double), 278 G4d << 288 G4double xInitial, G4double xFinal, G4double e ) 279 G4d << 289 { 280 { << 290 G4int depth = 0 ; 281 G4int depth = 0; << 291 G4double sum = 0.0 ; 282 G4double sum = 0.0; << 292 AdaptGauss(f,xInitial,xFinal,e,sum,depth) ; 283 AdaptGauss(f, xInitial, xFinal, e, sum, dept << 293 return sum ; 284 return sum; << 285 } 294 } 286 295 287 ////////////////////////////////////////////// 296 //////////////////////////////////////////////////////////////////////////// 288 // Gauss integration methods involving ortogon 297 // Gauss integration methods involving ortogonal polynomials 289 ////////////////////////////////////////////// 298 //////////////////////////////////////////////////////////////////////////// 290 // 299 // 291 // Methods involving Legendre polynomials << 300 // Methods involving Legendre polynomials 292 // 301 // 293 ////////////////////////////////////////////// 302 ///////////////////////////////////////////////////////////////////////// 294 // 303 // 295 // The value nLegendre set the accuracy requir 304 // The value nLegendre set the accuracy required, i.e the number of points 296 // where the function pFunction will be evalua 305 // where the function pFunction will be evaluated during integration. 297 // The function creates the arrays for absciss << 306 // The function creates the arrays for abscissas and weights that used 298 // in Gauss-Legendre quadrature method. << 307 // in Gauss-Legendre quadrature method. 299 // The values a and b are the limits of integr 308 // The values a and b are the limits of integration of the function f . 300 // nLegendre MUST BE EVEN !!! 309 // nLegendre MUST BE EVEN !!! 301 // Returns the integral of the function f betw << 310 // Returns the integral of the function f between a and b, by 2*fNumber point 302 // Gauss-Legendre integration: the function is 311 // Gauss-Legendre integration: the function is evaluated exactly 303 // 2*fNumber times at interior points in the r << 312 // 2*fNumber times at interior points in the range of integration. 304 // Since the weights and abscissas are, in thi << 313 // Since the weights and abscissas are, in this case, symmetric around 305 // the midpoint of the range of integration, t << 314 // the midpoint of the range of integration, there are actually only 306 // fNumber distinct values of each. 315 // fNumber distinct values of each. 307 // Convenient for using with some class object 316 // Convenient for using with some class object dataT 308 317 309 template <class T, class F> 318 template <class T, class F> 310 G4double G4Integrator<T, F>::Legendre(T& typeT << 319 G4double G4Integrator<T,F>::Legendre( T& typeT, F f, G4double a, G4double b, 311 G4int nL << 320 G4int nLegendre ) 312 { 321 { 313 G4double nwt, nwt1, temp1, temp2, temp3, tem << 322 G4double newton, newton1, temp1, temp2, temp3, temp ; 314 G4double xDiff, xMean, dx, integral; << 323 G4double xDiff, xMean, dx, integral ; 315 324 316 const G4double tolerance = 1.6e-10; << 325 const G4double tolerance = 1.6e-10 ; 317 G4int i, j, k = nLegendre; << 326 G4int i, j, k = nLegendre ; 318 G4int fNumber = (nLegendre + 1) / 2; << 327 G4int fNumber = (nLegendre + 1)/2 ; 319 << 328 320 if(2 * fNumber != k) << 329 if(2*fNumber != k) 321 { << 330 { 322 G4Exception("G4Integrator<T,F>::Legendre(T << 331 G4Exception("G4Integrator<T,F>::Legendre(T&,F, ...)", "InvalidCall", 323 FatalException, "Invalid (odd) << 332 FatalException, "Invalid (odd) nLegendre in constructor."); 324 } << 333 } 325 << 334 326 G4double* fAbscissa = new G4double[fNumber]; << 335 G4double* fAbscissa = new G4double[fNumber] ; 327 G4double* fWeight = new G4double[fNumber]; << 336 G4double* fWeight = new G4double[fNumber] ; 328 << 337 329 for(i = 1; i <= fNumber; ++i) // Loop over << 338 for(i=1;i<=fNumber;i++) // Loop over the desired roots 330 { << 339 { 331 nwt = std::cos(CLHEP::pi * (i - 0.25) / << 340 newton = std::cos(pi*(i - 0.25)/(k + 0.5)) ; // Initial root approximation 332 (k + 0.5)); // Initial roo << 341 333 << 342 do // loop of Newton's method 334 do // loop of Newton's method << 343 { 335 { << 344 temp1 = 1.0 ; 336 temp1 = 1.0; << 345 temp2 = 0.0 ; 337 temp2 = 0.0; << 346 for(j=1;j<=k;j++) 338 for(j = 1; j <= k; ++j) << 347 { 339 { << 348 temp3 = temp2 ; 340 temp3 = temp2; << 349 temp2 = temp1 ; 341 temp2 = temp1; << 350 temp1 = ((2.0*j - 1.0)*newton*temp2 - (j - 1.0)*temp3)/j ; 342 temp1 = ((2.0 * j - 1.0) * nwt * temp2 << 351 } 343 } << 352 temp = k*(newton*temp1 - temp2)/(newton*newton - 1.0) ; 344 temp = k * (nwt * temp1 - temp2) / (nwt << 353 newton1 = newton ; 345 nwt1 = nwt; << 354 newton = newton1 - temp1/temp ; // Newton's method 346 nwt = nwt1 - temp1 / temp; // Newton's << 355 } 347 } while(std::fabs(nwt - nwt1) > tolerance) << 356 while(std::fabs(newton - newton1) > tolerance) ; 348 << 357 349 fAbscissa[fNumber - i] = nwt; << 358 fAbscissa[fNumber-i] = newton ; 350 fWeight[fNumber - i] = 2.0 / ((1.0 - nwt << 359 fWeight[fNumber-i] = 2.0/((1.0 - newton*newton)*temp*temp) ; 351 } << 360 } 352 << 361 353 // << 362 // 354 // Now we ready to get integral << 363 // Now we ready to get integral 355 // << 364 // 356 << 365 357 xMean = 0.5 * (a + b); << 366 xMean = 0.5*(a + b) ; 358 xDiff = 0.5 * (b - a); << 367 xDiff = 0.5*(b - a) ; 359 integral = 0.0; << 368 integral = 0.0 ; 360 for(i = 0; i < fNumber; ++i) << 369 for(i=0;i<fNumber;i++) 361 { << 370 { 362 dx = xDiff * fAbscissa[i]; << 371 dx = xDiff*fAbscissa[i] ; 363 integral += fWeight[i] * ((typeT.*f)(xMean << 372 integral += fWeight[i]*( (typeT.*f)(xMean + dx) + 364 } << 373 (typeT.*f)(xMean - dx) ) ; 365 delete[] fAbscissa; << 374 } 366 delete[] fWeight; << 375 delete[] fAbscissa; 367 return integral *= xDiff; << 376 delete[] fWeight; 368 } << 377 return integral *= xDiff ; >> 378 } 369 379 370 ////////////////////////////////////////////// 380 /////////////////////////////////////////////////////////////////////// 371 // 381 // 372 // Convenient for using with the pointer 'this 382 // Convenient for using with the pointer 'this' 373 383 374 template <class T, class F> 384 template <class T, class F> 375 G4double G4Integrator<T, F>::Legendre(T* ptrT, << 385 G4double G4Integrator<T,F>::Legendre( T* ptrT, F f, G4double a, 376 G4int nL << 386 G4double b, G4int nLegendre ) 377 { 387 { 378 return Legendre(*ptrT, f, a, b, nLegendre); << 388 return Legendre(*ptrT,f,a,b,nLegendre) ; 379 } 389 } 380 390 381 ////////////////////////////////////////////// 391 /////////////////////////////////////////////////////////////////////// 382 // 392 // 383 // Convenient for using with global scope func 393 // Convenient for using with global scope function f 384 394 385 template <class T, class F> 395 template <class T, class F> 386 G4double G4Integrator<T, F>::Legendre(G4double << 396 G4double G4Integrator<T,F>::Legendre( G4double (*f)(G4double), 387 G4double << 397 G4double a, G4double b, G4int nLegendre) 388 { 398 { 389 G4double nwt, nwt1, temp1, temp2, temp3, tem << 399 G4double newton, newton1, temp1, temp2, temp3, temp ; 390 G4double xDiff, xMean, dx, integral; << 400 G4double xDiff, xMean, dx, integral ; 391 401 392 const G4double tolerance = 1.6e-10; << 402 const G4double tolerance = 1.6e-10 ; 393 G4int i, j, k = nLegendre; << 403 G4int i, j, k = nLegendre ; 394 G4int fNumber = (nLegendre + 1) / 2; << 404 G4int fNumber = (nLegendre + 1)/2 ; 395 << 405 396 if(2 * fNumber != k) << 406 if(2*fNumber != k) 397 { << 407 { 398 G4Exception("G4Integrator<T,F>::Legendre(. << 408 G4Exception("G4Integrator<T,F>::Legendre(...)", "InvalidCall", 399 FatalException, "Invalid (odd) << 409 FatalException, "Invalid (odd) nLegendre in constructor."); 400 } << 410 } 401 << 411 402 G4double* fAbscissa = new G4double[fNumber]; << 412 G4double* fAbscissa = new G4double[fNumber] ; 403 G4double* fWeight = new G4double[fNumber]; << 413 G4double* fWeight = new G4double[fNumber] ; 404 << 414 405 for(i = 1; i <= fNumber; i++) // Loop over << 415 for(i=1;i<=fNumber;i++) // Loop over the desired roots 406 { << 416 { 407 nwt = std::cos(CLHEP::pi * (i - 0.25) / << 417 newton = std::cos(pi*(i - 0.25)/(k + 0.5)) ; // Initial root approximation 408 (k + 0.5)); // Initial roo << 418 409 << 419 do // loop of Newton's method 410 do // loop of Newton's method << 420 { 411 { << 421 temp1 = 1.0 ; 412 temp1 = 1.0; << 422 temp2 = 0.0 ; 413 temp2 = 0.0; << 423 for(j=1;j<=k;j++) 414 for(j = 1; j <= k; ++j) << 424 { 415 { << 425 temp3 = temp2 ; 416 temp3 = temp2; << 426 temp2 = temp1 ; 417 temp2 = temp1; << 427 temp1 = ((2.0*j - 1.0)*newton*temp2 - (j - 1.0)*temp3)/j ; 418 temp1 = ((2.0 * j - 1.0) * nwt * temp2 << 428 } 419 } << 429 temp = k*(newton*temp1 - temp2)/(newton*newton - 1.0) ; 420 temp = k * (nwt * temp1 - temp2) / (nwt << 430 newton1 = newton ; 421 nwt1 = nwt; << 431 newton = newton1 - temp1/temp ; // Newton's method 422 nwt = nwt1 - temp1 / temp; // Newton's << 432 } 423 } while(std::fabs(nwt - nwt1) > tolerance) << 433 while(std::fabs(newton - newton1) > tolerance) ; 424 << 434 425 fAbscissa[fNumber - i] = nwt; << 435 fAbscissa[fNumber-i] = newton ; 426 fWeight[fNumber - i] = 2.0 / ((1.0 - nwt << 436 fWeight[fNumber-i] = 2.0/((1.0 - newton*newton)*temp*temp) ; 427 } << 437 } 428 << 438 429 // << 439 // 430 // Now we ready to get integral << 440 // Now we ready to get integral 431 // << 441 // 432 << 442 433 xMean = 0.5 * (a + b); << 443 xMean = 0.5*(a + b) ; 434 xDiff = 0.5 * (b - a); << 444 xDiff = 0.5*(b - a) ; 435 integral = 0.0; << 445 integral = 0.0 ; 436 for(i = 0; i < fNumber; ++i) << 446 for(i=0;i<fNumber;i++) 437 { << 447 { 438 dx = xDiff * fAbscissa[i]; << 448 dx = xDiff*fAbscissa[i] ; 439 integral += fWeight[i] * ((*f)(xMean + dx) << 449 integral += fWeight[i]*( (*f)(xMean + dx) + (*f)(xMean - dx) ) ; 440 } << 450 } 441 delete[] fAbscissa; << 451 delete[] fAbscissa; 442 delete[] fWeight; << 452 delete[] fWeight; 443 453 444 return integral *= xDiff; << 454 return integral *= xDiff ; 445 } << 455 } 446 456 447 ////////////////////////////////////////////// 457 //////////////////////////////////////////////////////////////////////////// 448 // 458 // 449 // Returns the integral of the function to be 459 // Returns the integral of the function to be pointed by T::f between a and b, 450 // by ten point Gauss-Legendre integration: th 460 // by ten point Gauss-Legendre integration: the function is evaluated exactly 451 // ten times at interior points in the range o 461 // ten times at interior points in the range of integration. Since the weights 452 // and abscissas are, in this case, symmetric << 462 // and abscissas are, in this case, symmetric around the midpoint of the 453 // range of integration, there are actually on 463 // range of integration, there are actually only five distinct values of each 454 // Convenient for using with class object type 464 // Convenient for using with class object typeT 455 465 456 template <class T, class F> << 466 template <class T, class F> 457 G4double G4Integrator<T, F>::Legendre10(T& typ << 467 G4double G4Integrator<T,F>::Legendre10( T& typeT, F f,G4double a, G4double b) 458 { 468 { 459 G4int i; << 469 G4int i ; 460 G4double xDiff, xMean, dx, integral; << 470 G4double xDiff, xMean, dx, integral ; 461 << 471 462 // From Abramowitz M., Stegan I.A. 1964 , Ha << 472 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 916 463 << 473 464 static const G4double abscissa[] = { 0.14887 << 474 static G4double abscissa[] = { 0.148874338981631, 0.433395394129247, 465 0.67940 << 475 0.679409568299024, 0.865063366688985, 466 0.97390 << 476 0.973906528517172 } ; 467 << 477 468 static const G4double weight[] = { 0.2955242 << 478 static G4double weight[] = { 0.295524224714753, 0.269266719309996, 469 0.2190863 << 479 0.219086362515982, 0.149451349150581, 470 0.0666713 << 480 0.066671344308688 } ; 471 xMean = 0.5 * (a + << 481 xMean = 0.5*(a + b) ; 472 xDiff = 0.5 * (b - << 482 xDiff = 0.5*(b - a) ; 473 integral = 0.0; << 483 integral = 0.0 ; 474 for(i = 0; i < 5; ++i) << 484 for(i=0;i<5;i++) 475 { << 485 { 476 dx = xDiff * abscissa[i]; << 486 dx = xDiff*abscissa[i] ; 477 integral += weight[i] * ((typeT.*f)(xMean << 487 integral += weight[i]*( (typeT.*f)(xMean + dx) + (typeT.*f)(xMean - dx)) ; 478 } << 488 } 479 return integral *= xDiff; << 489 return integral *= xDiff ; 480 } 490 } 481 491 482 ////////////////////////////////////////////// 492 /////////////////////////////////////////////////////////////////////////// 483 // 493 // 484 // Convenient for using with the pointer 'this 494 // Convenient for using with the pointer 'this' 485 495 486 template <class T, class F> << 496 template <class T, class F> 487 G4double G4Integrator<T, F>::Legendre10(T* ptr << 497 G4double G4Integrator<T,F>::Legendre10( T* ptrT, F f,G4double a, G4double b) 488 { 498 { 489 return Legendre10(*ptrT, f, a, b); << 499 return Legendre10(*ptrT,f,a,b) ; 490 } << 500 } 491 501 492 ////////////////////////////////////////////// 502 ////////////////////////////////////////////////////////////////////////// 493 // 503 // 494 // Convenient for using with global scope func 504 // Convenient for using with global scope functions 495 505 496 template <class T, class F> 506 template <class T, class F> 497 G4double G4Integrator<T, F>::Legendre10(G4doub << 507 G4double G4Integrator<T,F>::Legendre10( G4double (*f)(G4double), 498 G4doub << 508 G4double a, G4double b ) 499 { 509 { 500 G4int i; << 510 G4int i ; 501 G4double xDiff, xMean, dx, integral; << 511 G4double xDiff, xMean, dx, integral ; 502 << 512 503 // From Abramowitz M., Stegan I.A. 1964 , Ha << 513 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 916 504 << 514 505 static const G4double abscissa[] = { 0.14887 << 515 static G4double abscissa[] = { 0.148874338981631, 0.433395394129247, 506 0.67940 << 516 0.679409568299024, 0.865063366688985, 507 0.97390 << 517 0.973906528517172 } ; 508 << 518 509 static const G4double weight[] = { 0.2955242 << 519 static G4double weight[] = { 0.295524224714753, 0.269266719309996, 510 0.2190863 << 520 0.219086362515982, 0.149451349150581, 511 0.0666713 << 521 0.066671344308688 } ; 512 xMean = 0.5 * (a + << 522 xMean = 0.5*(a + b) ; 513 xDiff = 0.5 * (b - << 523 xDiff = 0.5*(b - a) ; 514 integral = 0.0; << 524 integral = 0.0 ; 515 for(i = 0; i < 5; ++i) << 525 for(i=0;i<5;i++) 516 { << 526 { 517 dx = xDiff * abscissa[i]; << 527 dx = xDiff*abscissa[i] ; 518 integral += weight[i] * ((*f)(xMean + dx) << 528 integral += weight[i]*( (*f)(xMean + dx) + (*f)(xMean - dx)) ; 519 } << 529 } 520 return integral *= xDiff; << 530 return integral *= xDiff ; 521 } 531 } 522 532 523 ////////////////////////////////////////////// 533 /////////////////////////////////////////////////////////////////////// 524 // 534 // 525 // Returns the integral of the function to be 535 // Returns the integral of the function to be pointed by T::f between a and b, 526 // by 96 point Gauss-Legendre integration: the 536 // by 96 point Gauss-Legendre integration: the function is evaluated exactly 527 // ten Times at interior points in the range o 537 // ten Times at interior points in the range of integration. Since the weights 528 // and abscissas are, in this case, symmetric << 538 // and abscissas are, in this case, symmetric around the midpoint of the 529 // range of integration, there are actually on 539 // range of integration, there are actually only five distinct values of each 530 // Convenient for using with some class object 540 // Convenient for using with some class object typeT 531 541 532 template <class T, class F> << 542 template <class T, class F> 533 G4double G4Integrator<T, F>::Legendre96(T& typ << 543 G4double G4Integrator<T,F>::Legendre96( T& typeT, F f,G4double a, G4double b) 534 { 544 { 535 G4int i; << 545 G4int i ; 536 G4double xDiff, xMean, dx, integral; << 546 G4double xDiff, xMean, dx, integral ; 537 << 547 538 // From Abramowitz M., Stegan I.A. 1964 , Ha << 548 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 919 539 << 549 540 static const G4double abscissa[] = { << 550 static G4double 541 0.016276744849602969579, 0.048812985136049 << 551 abscissa[] = { 542 0.081297495464425558994, 0.113695850110665 << 552 0.016276744849602969579, 0.048812985136049731112, 543 0.145973714654896941989, 0.178096882367618 << 553 0.081297495464425558994, 0.113695850110665920911, 544 << 554 0.145973714654896941989, 0.178096882367618602759, // 6 545 0.210031310460567203603, 0.241743156163840 << 555 546 0.273198812591049141487, 0.304364944354496 << 556 0.210031310460567203603, 0.241743156163840012328, 547 0.335208522892625422616, 0.365696861472313 << 557 0.273198812591049141487, 0.304364944354496353024, 548 << 558 0.335208522892625422616, 0.365696861472313635031, // 12 549 0.395797649828908603285, 0.425478988407300 << 559 550 0.454709422167743008636, 0.483457973920596 << 560 0.395797649828908603285, 0.425478988407300545365, 551 0.511694177154667673586, 0.539388108324357 << 561 0.454709422167743008636, 0.483457973920596359768, 552 << 562 0.511694177154667673586, 0.539388108324357436227, // 18 553 0.566510418561397168404, 0.593032364777572 << 563 554 0.618925840125468570386, 0.644163403784967 << 564 0.566510418561397168404, 0.593032364777572080684, 555 0.668718310043916153953, 0.692564536642171 << 565 0.618925840125468570386, 0.644163403784967106798, 556 << 566 0.668718310043916153953, 0.692564536642171561344, // 24 557 0.715676812348967626225, 0.738030643744400 << 567 558 0.759602341176647498703, 0.780369043867433 << 568 0.715676812348967626225, 0.738030643744400132851, 559 0.800308744139140817229, 0.819400310737931 << 569 0.759602341176647498703, 0.780369043867433217604, 560 << 570 0.800308744139140817229, 0.819400310737931675539, // 30 561 0.837623511228187121494, 0.854959033434601 << 571 562 0.871388505909296502874, 0.886894517402420 << 572 0.837623511228187121494, 0.854959033434601455463, 563 0.901460635315852341319, 0.915071423120898 << 573 0.871388505909296502874, 0.886894517402420416057, 564 << 574 0.901460635315852341319, 0.915071423120898074206, // 36 565 0.927712456722308690965, 0.939370339752755 << 575 566 0.950032717784437635756, 0.959688291448742 << 576 0.927712456722308690965, 0.939370339752755216932, 567 0.968326828463264212174, 0.975939174585136 << 577 0.950032717784437635756, 0.959688291448742539300, 568 << 578 0.968326828463264212174, 0.975939174585136466453, // 42 569 0.982517263563014677447, 0.988054126329623 << 579 570 0.992543900323762624572, 0.995981842987209 << 580 0.982517263563014677447, 0.988054126329623799481, 571 0.998364375863181677724, 0.999689503883230 << 581 0.992543900323762624572, 0.995981842987209290650, 572 }; << 582 0.998364375863181677724, 0.999689503883230766828 // 48 573 << 583 } ; 574 static const G4double weight[] = { << 584 575 0.032550614492363166242, 0.032516118713868 << 585 static G4double 576 0.032447163714064269364, 0.032343822568575 << 586 weight[] = { 577 0.032206204794030250669, 0.032034456231992 << 587 0.032550614492363166242, 0.032516118713868835987, 578 << 588 0.032447163714064269364, 0.032343822568575928429, 579 0.031828758894411006535, 0.031589330770727 << 589 0.032206204794030250669, 0.032034456231992663218, // 6 580 0.031316425596862355813, 0.031010332586313 << 590 581 0.030671376123669149014, 0.030299915420827 << 591 0.031828758894411006535, 0.031589330770727168558, 582 << 592 0.031316425596862355813, 0.031010332586313837423, 583 0.029896344136328385984, 0.029461089958167 << 593 0.030671376123669149014, 0.030299915420827593794, // 12 584 0.028994614150555236543, 0.028497411065085 << 594 585 0.027970007616848334440, 0.027412962726029 << 595 0.029896344136328385984, 0.029461089958167905970, 586 << 596 0.028994614150555236543, 0.028497411065085385646, 587 0.026826866725591762198, 0.026212340735672 << 597 0.027970007616848334440, 0.027412962726029242823, // 18 588 0.025570036005349361499, 0.024900633222483 << 598 589 0.024204841792364691282, 0.023483399085926 << 599 0.026826866725591762198, 0.026212340735672413913, 590 << 600 0.025570036005349361499, 0.024900633222483610288, 591 0.022737069658329374001, 0.021966644438744 << 601 0.024204841792364691282, 0.023483399085926219842, // 24 592 0.021172939892191298988, 0.020356797154333 << 602 593 0.019519081140145022410, 0.018660679627411 << 603 0.022737069658329374001, 0.021966644438744349195, 594 << 604 0.021172939892191298988, 0.020356797154333324595, 595 0.017782502316045260838, 0.016885479864245 << 605 0.019519081140145022410, 0.018660679627411467385, // 30 596 0.015970562902562291381, 0.015038721026994 << 606 597 0.014090941772314860916, 0.013128229566961 << 607 0.017782502316045260838, 0.016885479864245172450, 598 << 608 0.015970562902562291381, 0.015038721026994938006, 599 0.012151604671088319635, 0.011162102099838 << 609 0.014090941772314860916, 0.013128229566961572637, // 36 600 0.010160770535008415758, 0.009148671230783 << 610 601 0.008126876925698759217, 0.007096470791153 << 611 0.012151604671088319635, 0.011162102099838498591, 602 << 612 0.010160770535008415758, 0.009148671230783386633, 603 0.006058545504235961683, 0.005014202742927 << 613 0.008126876925698759217, 0.007096470791153865269, // 42 604 0.003964554338444686674, 0.002910731817934 << 614 605 0.001853960788946921732, 0.000796792065552 << 615 0.006058545504235961683, 0.005014202742927517693, 606 }; << 616 0.003964554338444686674, 0.002910731817934946408, 607 xMean = 0.5 * (a + b); << 617 0.001853960788946921732, 0.000796792065552012429 // 48 608 xDiff = 0.5 * (b - a); << 618 } ; 609 integral = 0.0; << 619 xMean = 0.5*(a + b) ; 610 for(i = 0; i < 48; ++i) << 620 xDiff = 0.5*(b - a) ; 611 { << 621 integral = 0.0 ; 612 dx = xDiff * abscissa[i]; << 622 for(i=0;i<48;i++) 613 integral += weight[i] * ((typeT.*f)(xMean << 623 { 614 } << 624 dx = xDiff*abscissa[i] ; 615 return integral *= xDiff; << 625 integral += weight[i]*((typeT.*f)(xMean + dx) + (typeT.*f)(xMean - dx)) ; >> 626 } >> 627 return integral *= xDiff ; 616 } 628 } 617 629 618 ////////////////////////////////////////////// 630 /////////////////////////////////////////////////////////////////////// 619 // 631 // 620 // Convenient for using with the pointer 'this 632 // Convenient for using with the pointer 'this' 621 633 622 template <class T, class F> << 634 template <class T, class F> 623 G4double G4Integrator<T, F>::Legendre96(T* ptr << 635 G4double G4Integrator<T,F>::Legendre96( T* ptrT, F f,G4double a, G4double b) 624 { 636 { 625 return Legendre96(*ptrT, f, a, b); << 637 return Legendre96(*ptrT,f,a,b) ; 626 } << 638 } 627 639 628 ////////////////////////////////////////////// 640 /////////////////////////////////////////////////////////////////////// 629 // 641 // 630 // Convenient for using with global scope func << 642 // Convenient for using with global scope function f 631 643 632 template <class T, class F> 644 template <class T, class F> 633 G4double G4Integrator<T, F>::Legendre96(G4doub << 645 G4double G4Integrator<T,F>::Legendre96( G4double (*f)(G4double), 634 G4doub << 646 G4double a, G4double b ) 635 { 647 { 636 G4int i; << 648 G4int i ; 637 G4double xDiff, xMean, dx, integral; << 649 G4double xDiff, xMean, dx, integral ; 638 << 650 639 // From Abramowitz M., Stegan I.A. 1964 , Ha << 651 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 919 640 << 652 641 static const G4double abscissa[] = { << 653 static G4double 642 0.016276744849602969579, 0.048812985136049 << 654 abscissa[] = { 643 0.081297495464425558994, 0.113695850110665 << 655 0.016276744849602969579, 0.048812985136049731112, 644 0.145973714654896941989, 0.178096882367618 << 656 0.081297495464425558994, 0.113695850110665920911, 645 << 657 0.145973714654896941989, 0.178096882367618602759, // 6 646 0.210031310460567203603, 0.241743156163840 << 658 647 0.273198812591049141487, 0.304364944354496 << 659 0.210031310460567203603, 0.241743156163840012328, 648 0.335208522892625422616, 0.365696861472313 << 660 0.273198812591049141487, 0.304364944354496353024, 649 << 661 0.335208522892625422616, 0.365696861472313635031, // 12 650 0.395797649828908603285, 0.425478988407300 << 662 651 0.454709422167743008636, 0.483457973920596 << 663 0.395797649828908603285, 0.425478988407300545365, 652 0.511694177154667673586, 0.539388108324357 << 664 0.454709422167743008636, 0.483457973920596359768, 653 << 665 0.511694177154667673586, 0.539388108324357436227, // 18 654 0.566510418561397168404, 0.593032364777572 << 666 655 0.618925840125468570386, 0.644163403784967 << 667 0.566510418561397168404, 0.593032364777572080684, 656 0.668718310043916153953, 0.692564536642171 << 668 0.618925840125468570386, 0.644163403784967106798, 657 << 669 0.668718310043916153953, 0.692564536642171561344, // 24 658 0.715676812348967626225, 0.738030643744400 << 670 659 0.759602341176647498703, 0.780369043867433 << 671 0.715676812348967626225, 0.738030643744400132851, 660 0.800308744139140817229, 0.819400310737931 << 672 0.759602341176647498703, 0.780369043867433217604, 661 << 673 0.800308744139140817229, 0.819400310737931675539, // 30 662 0.837623511228187121494, 0.854959033434601 << 674 663 0.871388505909296502874, 0.886894517402420 << 675 0.837623511228187121494, 0.854959033434601455463, 664 0.901460635315852341319, 0.915071423120898 << 676 0.871388505909296502874, 0.886894517402420416057, 665 << 677 0.901460635315852341319, 0.915071423120898074206, // 36 666 0.927712456722308690965, 0.939370339752755 << 678 667 0.950032717784437635756, 0.959688291448742 << 679 0.927712456722308690965, 0.939370339752755216932, 668 0.968326828463264212174, 0.975939174585136 << 680 0.950032717784437635756, 0.959688291448742539300, 669 << 681 0.968326828463264212174, 0.975939174585136466453, // 42 670 0.982517263563014677447, 0.988054126329623 << 682 671 0.992543900323762624572, 0.995981842987209 << 683 0.982517263563014677447, 0.988054126329623799481, 672 0.998364375863181677724, 0.999689503883230 << 684 0.992543900323762624572, 0.995981842987209290650, 673 }; << 685 0.998364375863181677724, 0.999689503883230766828 // 48 674 << 686 } ; 675 static const G4double weight[] = { << 687 676 0.032550614492363166242, 0.032516118713868 << 688 static G4double 677 0.032447163714064269364, 0.032343822568575 << 689 weight[] = { 678 0.032206204794030250669, 0.032034456231992 << 690 0.032550614492363166242, 0.032516118713868835987, 679 << 691 0.032447163714064269364, 0.032343822568575928429, 680 0.031828758894411006535, 0.031589330770727 << 692 0.032206204794030250669, 0.032034456231992663218, // 6 681 0.031316425596862355813, 0.031010332586313 << 693 682 0.030671376123669149014, 0.030299915420827 << 694 0.031828758894411006535, 0.031589330770727168558, 683 << 695 0.031316425596862355813, 0.031010332586313837423, 684 0.029896344136328385984, 0.029461089958167 << 696 0.030671376123669149014, 0.030299915420827593794, // 12 685 0.028994614150555236543, 0.028497411065085 << 697 686 0.027970007616848334440, 0.027412962726029 << 698 0.029896344136328385984, 0.029461089958167905970, 687 << 699 0.028994614150555236543, 0.028497411065085385646, 688 0.026826866725591762198, 0.026212340735672 << 700 0.027970007616848334440, 0.027412962726029242823, // 18 689 0.025570036005349361499, 0.024900633222483 << 701 690 0.024204841792364691282, 0.023483399085926 << 702 0.026826866725591762198, 0.026212340735672413913, 691 << 703 0.025570036005349361499, 0.024900633222483610288, 692 0.022737069658329374001, 0.021966644438744 << 704 0.024204841792364691282, 0.023483399085926219842, // 24 693 0.021172939892191298988, 0.020356797154333 << 705 694 0.019519081140145022410, 0.018660679627411 << 706 0.022737069658329374001, 0.021966644438744349195, 695 << 707 0.021172939892191298988, 0.020356797154333324595, 696 0.017782502316045260838, 0.016885479864245 << 708 0.019519081140145022410, 0.018660679627411467385, // 30 697 0.015970562902562291381, 0.015038721026994 << 709 698 0.014090941772314860916, 0.013128229566961 << 710 0.017782502316045260838, 0.016885479864245172450, 699 << 711 0.015970562902562291381, 0.015038721026994938006, 700 0.012151604671088319635, 0.011162102099838 << 712 0.014090941772314860916, 0.013128229566961572637, // 36 701 0.010160770535008415758, 0.009148671230783 << 713 702 0.008126876925698759217, 0.007096470791153 << 714 0.012151604671088319635, 0.011162102099838498591, 703 << 715 0.010160770535008415758, 0.009148671230783386633, 704 0.006058545504235961683, 0.005014202742927 << 716 0.008126876925698759217, 0.007096470791153865269, // 42 705 0.003964554338444686674, 0.002910731817934 << 717 706 0.001853960788946921732, 0.000796792065552 << 718 0.006058545504235961683, 0.005014202742927517693, 707 }; << 719 0.003964554338444686674, 0.002910731817934946408, 708 xMean = 0.5 * (a + b); << 720 0.001853960788946921732, 0.000796792065552012429 // 48 709 xDiff = 0.5 * (b - a); << 721 } ; 710 integral = 0.0; << 722 xMean = 0.5*(a + b) ; 711 for(i = 0; i < 48; ++i) << 723 xDiff = 0.5*(b - a) ; 712 { << 724 integral = 0.0 ; 713 dx = xDiff * abscissa[i]; << 725 for(i=0;i<48;i++) 714 integral += weight[i] * ((*f)(xMean + dx) << 726 { 715 } << 727 dx = xDiff*abscissa[i] ; 716 return integral *= xDiff; << 728 integral += weight[i]*((*f)(xMean + dx) + (*f)(xMean - dx)) ; >> 729 } >> 730 return integral *= xDiff ; 717 } 731 } 718 732 719 ////////////////////////////////////////////// 733 ////////////////////////////////////////////////////////////////////////////// 720 // 734 // 721 // Methods involving Chebyshev polynomials << 735 // Methods involving Chebyshev polynomials 722 // 736 // 723 ////////////////////////////////////////////// 737 /////////////////////////////////////////////////////////////////////////// 724 // 738 // 725 // Integrates function pointed by T::f from a << 739 // Integrates function pointed by T::f from a to b by Gauss-Chebyshev 726 // quadrature method. 740 // quadrature method. 727 // Convenient for using with class object type 741 // Convenient for using with class object typeT 728 742 729 template <class T, class F> 743 template <class T, class F> 730 G4double G4Integrator<T, F>::Chebyshev(T& type << 744 G4double G4Integrator<T,F>::Chebyshev( T& typeT, F f, G4double a, 731 G4int n << 745 G4double b, G4int nChebyshev ) 732 { 746 { 733 G4int i; << 747 G4int i ; 734 G4double xDiff, xMean, dx, integral = 0.0; << 748 G4double xDiff, xMean, dx, integral = 0.0 ; 735 << 749 736 G4int fNumber = nChebyshev; // Try to << 750 G4int fNumber = nChebyshev ; // Try to reduce fNumber twice ?? 737 G4double cof = CLHEP::pi / fNumber; << 751 G4double cof = pi/fNumber ; 738 G4double* fAbscissa = new G4double[fNumber]; << 752 G4double* fAbscissa = new G4double[fNumber] ; 739 G4double* fWeight = new G4double[fNumber]; << 753 G4double* fWeight = new G4double[fNumber] ; 740 for(i = 0; i < fNumber; ++i) << 754 for(i=0;i<fNumber;i++) 741 { << 755 { 742 fAbscissa[i] = std::cos(cof * (i + 0.5)); << 756 fAbscissa[i] = std::cos(cof*(i + 0.5)) ; 743 fWeight[i] = cof * std::sqrt(1 - fAbscis << 757 fWeight[i] = cof*std::sqrt(1 - fAbscissa[i]*fAbscissa[i]) ; 744 } << 758 } 745 << 759 746 // << 760 // 747 // Now we ready to estimate the integral << 761 // Now we ready to estimate the integral 748 // << 762 // 749 << 763 750 xMean = 0.5 * (a + b); << 764 xMean = 0.5*(a + b) ; 751 xDiff = 0.5 * (b - a); << 765 xDiff = 0.5*(b - a) ; 752 for(i = 0; i < fNumber; ++i) << 766 for(i=0;i<fNumber;i++) 753 { << 767 { 754 dx = xDiff * fAbscissa[i]; << 768 dx = xDiff*fAbscissa[i] ; 755 integral += fWeight[i] * (typeT.*f)(xMean << 769 integral += fWeight[i]*(typeT.*f)(xMean + dx) ; 756 } << 770 } 757 delete[] fAbscissa; << 771 delete[] fAbscissa; 758 delete[] fWeight; << 772 delete[] fWeight; 759 return integral *= xDiff; << 773 return integral *= xDiff ; 760 } 774 } 761 775 762 ////////////////////////////////////////////// 776 /////////////////////////////////////////////////////////////////////// 763 // 777 // 764 // Convenient for using with 'this' pointer 778 // Convenient for using with 'this' pointer 765 779 766 template <class T, class F> 780 template <class T, class F> 767 G4double G4Integrator<T, F>::Chebyshev(T* ptrT << 781 G4double G4Integrator<T,F>::Chebyshev( T* ptrT, F f, G4double a, 768 G4int n << 782 G4double b, G4int n ) 769 { 783 { 770 return Chebyshev(*ptrT, f, a, b, n); << 784 return Chebyshev(*ptrT,f,a,b,n) ; 771 } << 785 } 772 786 773 ////////////////////////////////////////////// 787 //////////////////////////////////////////////////////////////////////// 774 // 788 // 775 // For use with global scope functions f << 789 // For use with global scope functions f 776 790 777 template <class T, class F> 791 template <class T, class F> 778 G4double G4Integrator<T, F>::Chebyshev(G4doubl << 792 G4double G4Integrator<T,F>::Chebyshev( G4double (*f)(G4double), 779 G4doubl << 793 G4double a, G4double b, G4int nChebyshev ) 780 { 794 { 781 G4int i; << 795 G4int i ; 782 G4double xDiff, xMean, dx, integral = 0.0; << 796 G4double xDiff, xMean, dx, integral = 0.0 ; 783 << 797 784 G4int fNumber = nChebyshev; // Try to << 798 G4int fNumber = nChebyshev ; // Try to reduce fNumber twice ?? 785 G4double cof = CLHEP::pi / fNumber; << 799 G4double cof = pi/fNumber ; 786 G4double* fAbscissa = new G4double[fNumber]; << 800 G4double* fAbscissa = new G4double[fNumber] ; 787 G4double* fWeight = new G4double[fNumber]; << 801 G4double* fWeight = new G4double[fNumber] ; 788 for(i = 0; i < fNumber; ++i) << 802 for(i=0;i<fNumber;i++) 789 { << 803 { 790 fAbscissa[i] = std::cos(cof * (i + 0.5)); << 804 fAbscissa[i] = std::cos(cof*(i + 0.5)) ; 791 fWeight[i] = cof * std::sqrt(1 - fAbscis << 805 fWeight[i] = cof*std::sqrt(1 - fAbscissa[i]*fAbscissa[i]) ; 792 } << 806 } 793 << 807 794 // << 808 // 795 // Now we ready to estimate the integral << 809 // Now we ready to estimate the integral 796 // << 810 // 797 << 811 798 xMean = 0.5 * (a + b); << 812 xMean = 0.5*(a + b) ; 799 xDiff = 0.5 * (b - a); << 813 xDiff = 0.5*(b - a) ; 800 for(i = 0; i < fNumber; ++i) << 814 for(i=0;i<fNumber;i++) 801 { << 815 { 802 dx = xDiff * fAbscissa[i]; << 816 dx = xDiff*fAbscissa[i] ; 803 integral += fWeight[i] * (*f)(xMean + dx); << 817 integral += fWeight[i]*(*f)(xMean + dx) ; 804 } << 818 } 805 delete[] fAbscissa; << 819 delete[] fAbscissa; 806 delete[] fWeight; << 820 delete[] fWeight; 807 return integral *= xDiff; << 821 return integral *= xDiff ; 808 } 822 } 809 823 810 ////////////////////////////////////////////// 824 ////////////////////////////////////////////////////////////////////// 811 // 825 // 812 // Method involving Laguerre polynomials 826 // Method involving Laguerre polynomials 813 // 827 // 814 ////////////////////////////////////////////// 828 ////////////////////////////////////////////////////////////////////// 815 // 829 // 816 // Integral from zero to infinity of std::pow( << 830 // Integral from zero to infinity of std::pow(x,alpha)*std::exp(-x)*f(x). 817 // The value of nLaguerre sets the accuracy. 831 // The value of nLaguerre sets the accuracy. 818 // The function creates arrays fAbscissa[0,.., << 832 // The function creates arrays fAbscissa[0,..,nLaguerre-1] and 819 // fWeight[0,..,nLaguerre-1] . << 833 // fWeight[0,..,nLaguerre-1] . 820 // Convenient for using with class object 'typ 834 // Convenient for using with class object 'typeT' and (typeT.*f) function 821 // (T::f) 835 // (T::f) 822 836 823 template <class T, class F> 837 template <class T, class F> 824 G4double G4Integrator<T, F>::Laguerre(T& typeT << 838 G4double G4Integrator<T,F>::Laguerre( T& typeT, F f, G4double alpha, 825 G4int nL << 839 G4int nLaguerre ) 826 { 840 { 827 const G4double tolerance = 1.0e-10; << 841 const G4double tolerance = 1.0e-10 ; 828 const G4int maxNumber = 12; << 842 const G4int maxNumber = 12 ; 829 G4int i, j, k; << 843 G4int i, j, k ; 830 G4double nwt = 0., nwt1, temp1, temp2, << 844 G4double newton=0., newton1, temp1, temp2, temp3, temp, cofi ; 831 G4double integral = 0.0; << 845 G4double integral = 0.0 ; 832 << 846 833 G4int fNumber = nLaguerre; << 847 G4int fNumber = nLaguerre ; 834 G4double* fAbscissa = new G4double[fNumber]; << 848 G4double* fAbscissa = new G4double[fNumber] ; 835 G4double* fWeight = new G4double[fNumber]; << 849 G4double* fWeight = new G4double[fNumber] ; >> 850 >> 851 for(i=1;i<=fNumber;i++) // Loop over the desired roots >> 852 { >> 853 if(i == 1) >> 854 { >> 855 newton = (1.0 + alpha)*(3.0 + 0.92*alpha) >> 856 / (1.0 + 2.4*fNumber + 1.8*alpha) ; >> 857 } >> 858 else if(i == 2) >> 859 { >> 860 newton += (15.0 + 6.25*alpha)/(1.0 + 0.9*alpha + 2.5*fNumber) ; >> 861 } >> 862 else >> 863 { >> 864 cofi = i - 2 ; >> 865 newton += ((1.0+2.55*cofi)/(1.9*cofi) >> 866 + 1.26*cofi*alpha/(1.0+3.5*cofi)) >> 867 * (newton - fAbscissa[i-3])/(1.0 + 0.3*alpha) ; >> 868 } >> 869 for(k=1;k<=maxNumber;k++) >> 870 { >> 871 temp1 = 1.0 ; >> 872 temp2 = 0.0 ; 836 873 837 for(i = 1; i <= fNumber; ++i) // Loop over << 874 for(j=1;j<=fNumber;j++) 838 { << 875 { 839 if(i == 1) << 876 temp3 = temp2 ; 840 { << 877 temp2 = temp1 ; 841 nwt = (1.0 + alpha) * (3.0 + 0.92 * alph << 878 temp1 = ((2*j - 1 + alpha - newton)*temp2 - (j - 1 + alpha)*temp3)/j ; 842 (1.0 + 2.4 * fNumber + 1.8 * alpha << 879 } 843 } << 880 temp = (fNumber*temp1 - (fNumber +alpha)*temp2)/newton ; 844 else if(i == 2) << 881 newton1 = newton ; 845 { << 882 newton = newton1 - temp1/temp ; 846 nwt += (15.0 + 6.25 * alpha) / (1.0 + 0. << 883 847 } << 884 if(std::fabs(newton - newton1) <= tolerance) 848 else << 885 { 849 { << 886 break ; 850 cofi = i - 2; << 887 } 851 nwt += ((1.0 + 2.55 * cofi) / (1.9 * cof << 888 } 852 1.26 * cofi * alpha / (1.0 + 3.5 << 889 if(k > maxNumber) 853 (nwt - fAbscissa[i - 3]) / (1.0 + << 890 { 854 } << 891 G4Exception("G4Integrator<T,F>::Laguerre(T,F, ...)", "Error", 855 for(k = 1; k <= maxNumber; ++k) << 892 FatalException, "Too many (>12) iterations."); 856 { << 893 } 857 temp1 = 1.0; << 894 858 temp2 = 0.0; << 895 fAbscissa[i-1] = newton ; 859 << 896 fWeight[i-1] = -std::exp(GammaLogarithm(alpha + fNumber) - 860 for(j = 1; j <= fNumber; ++j) << 897 GammaLogarithm((G4double)fNumber))/(temp*fNumber*temp2) ; 861 { << 898 } 862 temp3 = temp2; << 899 863 temp2 = temp1; << 900 // 864 temp1 = << 901 // Integral evaluation 865 ((2 * j - 1 + alpha - nwt) * temp2 - << 902 // 866 } << 903 867 temp = (fNumber * temp1 - (fNumber + alp << 904 for(i=0;i<fNumber;i++) 868 nwt1 = nwt; << 905 { 869 nwt = nwt1 - temp1 / temp; << 906 integral += fWeight[i]*(typeT.*f)(fAbscissa[i]) ; 870 << 907 } 871 if(std::fabs(nwt - nwt1) <= tolerance) << 908 delete[] fAbscissa; 872 { << 909 delete[] fWeight; 873 break; << 910 return integral ; 874 } << 911 } 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 } << 887 912 888 // << 889 // Integral evaluation << 890 // << 891 913 892 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 } << 900 914 901 ////////////////////////////////////////////// 915 ////////////////////////////////////////////////////////////////////// 902 // 916 // 903 // 917 // 904 918 905 template <class T, class F> << 919 template <class T, class F> G4double 906 G4double G4Integrator<T, F>::Laguerre(T* ptrT, << 920 G4Integrator<T,F>::Laguerre( T* ptrT, F f, G4double alpha, G4int nLaguerre ) 907 G4int nL << 908 { 921 { 909 return Laguerre(*ptrT, f, alpha, nLaguerre); << 922 return Laguerre(*ptrT,f,alpha,nLaguerre) ; 910 } 923 } 911 924 912 ////////////////////////////////////////////// 925 //////////////////////////////////////////////////////////////////////// 913 // 926 // 914 // For use with global scope functions f << 927 // For use with global scope functions f 915 928 916 template <class T, class F> << 929 template <class T, class F> G4double 917 G4double G4Integrator<T, F>::Laguerre(G4double << 930 G4Integrator<T,F>::Laguerre( G4double (*f)(G4double), 918 G4int nL << 931 G4double alpha, G4int nLaguerre ) 919 { << 932 { 920 const G4double tolerance = 1.0e-10; << 933 const G4double tolerance = 1.0e-10 ; 921 const G4int maxNumber = 12; << 934 const G4int maxNumber = 12 ; 922 G4int i, j, k; << 935 G4int i, j, k ; 923 G4double nwt = 0., nwt1, temp1, temp2, << 936 G4double newton=0., newton1, temp1, temp2, temp3, temp, cofi ; 924 G4double integral = 0.0; << 937 G4double integral = 0.0 ; 925 << 938 926 G4int fNumber = nLaguerre; << 939 G4int fNumber = nLaguerre ; 927 G4double* fAbscissa = new G4double[fNumber]; << 940 G4double* fAbscissa = new G4double[fNumber] ; 928 G4double* fWeight = new G4double[fNumber]; << 941 G4double* fWeight = new G4double[fNumber] ; 929 << 942 930 for(i = 1; i <= fNumber; ++i) // Loop over << 943 for(i=1;i<=fNumber;i++) // Loop over the desired roots 931 { << 944 { 932 if(i == 1) << 945 if(i == 1) 933 { << 946 { 934 nwt = (1.0 + alpha) * (3.0 + 0.92 * alph << 947 newton = (1.0 + alpha)*(3.0 + 0.92*alpha) 935 (1.0 + 2.4 * fNumber + 1.8 * alpha << 948 / (1.0 + 2.4*fNumber + 1.8*alpha) ; 936 } << 949 } 937 else if(i == 2) << 950 else if(i == 2) 938 { << 951 { 939 nwt += (15.0 + 6.25 * alpha) / (1.0 + 0. << 952 newton += (15.0 + 6.25*alpha)/(1.0 + 0.9*alpha + 2.5*fNumber) ; 940 } << 953 } 941 else << 954 else 942 { << 955 { 943 cofi = i - 2; << 956 cofi = i - 2 ; 944 nwt += ((1.0 + 2.55 * cofi) / (1.9 * cof << 957 newton += ((1.0+2.55*cofi)/(1.9*cofi) 945 1.26 * cofi * alpha / (1.0 + 3.5 << 958 + 1.26*cofi*alpha/(1.0+3.5*cofi)) 946 (nwt - fAbscissa[i - 3]) / (1.0 + << 959 * (newton - fAbscissa[i-3])/(1.0 + 0.3*alpha) ; 947 } << 960 } 948 for(k = 1; k <= maxNumber; ++k) << 961 for(k=1;k<=maxNumber;k++) 949 { << 962 { 950 temp1 = 1.0; << 963 temp1 = 1.0 ; 951 temp2 = 0.0; << 964 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 // << 984 965 985 for(i = 0; i < fNumber; i++) << 966 for(j=1;j<=fNumber;j++) 986 { << 967 { 987 integral += fWeight[i] * (*f)(fAbscissa[i] << 968 temp3 = temp2 ; 988 } << 969 temp2 = temp1 ; 989 delete[] fAbscissa; << 970 temp1 = ((2*j - 1 + alpha - newton)*temp2 - (j - 1 + alpha)*temp3)/j ; 990 delete[] fWeight; << 971 } 991 return integral; << 972 temp = (fNumber*temp1 - (fNumber +alpha)*temp2)/newton ; >> 973 newton1 = newton ; >> 974 newton = newton1 - temp1/temp ; >> 975 >> 976 if(std::fabs(newton - newton1) <= tolerance) >> 977 { >> 978 break ; >> 979 } >> 980 } >> 981 if(k > maxNumber) >> 982 { >> 983 G4Exception("G4Integrator<T,F>::Laguerre( ...)", "Error", >> 984 FatalException, "Too many (>12) iterations."); >> 985 } >> 986 >> 987 fAbscissa[i-1] = newton ; >> 988 fWeight[i-1] = -std::exp(GammaLogarithm(alpha + fNumber) - >> 989 GammaLogarithm((G4double)fNumber))/(temp*fNumber*temp2) ; >> 990 } >> 991 >> 992 // >> 993 // Integral evaluation >> 994 // >> 995 >> 996 for(i=0;i<fNumber;i++) >> 997 { >> 998 integral += fWeight[i]*(*f)(fAbscissa[i]) ; >> 999 } >> 1000 delete[] fAbscissa; >> 1001 delete[] fWeight; >> 1002 return integral ; 992 } 1003 } 993 1004 994 ////////////////////////////////////////////// 1005 /////////////////////////////////////////////////////////////////////// 995 // 1006 // 996 // Auxiliary function which returns the value 1007 // Auxiliary function which returns the value of std::log(gamma-function(x)) 997 // Returns the value ln(Gamma(xx) for xx > 0. << 1008 // Returns the value ln(Gamma(xx) for xx > 0. Full accuracy is obtained for 998 // xx > 1. For 0 < xx < 1. the reflection form 1009 // xx > 1. For 0 < xx < 1. the reflection formula (6.1.4) can be used first. 999 // (Adapted from Numerical Recipes in C) 1010 // (Adapted from Numerical Recipes in C) 1000 // 1011 // 1001 1012 1002 template <class T, class F> 1013 template <class T, class F> 1003 G4double G4Integrator<T, F>::GammaLogarithm(G << 1014 G4double G4Integrator<T,F>::GammaLogarithm(G4double xx) 1004 { 1015 { 1005 static const G4double cof[6] = { 76.1800917 << 1016 static G4double cof[6] = { 76.18009172947146, -86.50532032941677, 1006 24.0140982 << 1017 24.01409824083091, -1.231739572450155, 1007 0.12086509 << 1018 0.1208650973866179e-2, -0.5395239384953e-5 } ; 1008 G4int j; << 1019 register G4int j; 1009 G4double x = xx - 1.0; << 1020 G4double x = xx - 1.0 ; 1010 G4double tmp = x + 5.5; << 1021 G4double tmp = x + 5.5 ; 1011 tmp -= (x + 0.5) * std::log(tmp); << 1022 tmp -= (x + 0.5) * std::log(tmp) ; 1012 G4double ser = 1.000000000190015; << 1023 G4double ser = 1.000000000190015 ; 1013 1024 1014 for(j = 0; j <= 5; ++j) << 1025 for ( j = 0; j <= 5; j++ ) 1015 { 1026 { 1016 x += 1.0; << 1027 x += 1.0 ; 1017 ser += cof[j] / x; << 1028 ser += cof[j]/x ; 1018 } 1029 } 1019 return -tmp + std::log(2.5066282746310005 * << 1030 return -tmp + std::log(2.5066282746310005*ser) ; 1020 } 1031 } 1021 1032 1022 ///////////////////////////////////////////// 1033 /////////////////////////////////////////////////////////////////////// 1023 // 1034 // 1024 // Method involving Hermite polynomials 1035 // Method involving Hermite polynomials 1025 // 1036 // 1026 ///////////////////////////////////////////// 1037 /////////////////////////////////////////////////////////////////////// 1027 // 1038 // 1028 // 1039 // 1029 // Gauss-Hermite method for integration of st << 1040 // Gauss-Hermite method for integration of std::exp(-x*x)*f(x) 1030 // from minus infinity to plus infinity . << 1041 // from minus infinity to plus infinity . 1031 // 1042 // 1032 1043 1033 template <class T, class F> << 1044 template <class T, class F> 1034 G4double G4Integrator<T, F>::Hermite(T& typeT << 1045 G4double G4Integrator<T,F>::Hermite( T& typeT, F f, G4int nHermite ) 1035 { 1046 { 1036 const G4double tolerance = 1.0e-12; << 1047 const G4double tolerance = 1.0e-12 ; 1037 const G4int maxNumber = 12; << 1048 const G4int maxNumber = 12 ; 1038 << 1049 1039 G4int i, j, k; << 1050 G4int i, j, k ; 1040 G4double integral = 0.0; << 1051 G4double integral = 0.0 ; 1041 G4double nwt = 0., nwt1, temp1, temp2, << 1052 G4double newton=0., newton1, temp1, temp2, temp3, temp ; 1042 1053 1043 G4double piInMinusQ = << 1054 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 1055 1046 G4int fNumber = (nHermite + 1) / 2; << 1056 G4int fNumber = (nHermite +1)/2 ; 1047 G4double* fAbscissa = new G4double[fNumber] << 1057 G4double* fAbscissa = new G4double[fNumber] ; 1048 G4double* fWeight = new G4double[fNumber] << 1058 G4double* fWeight = new G4double[fNumber] ; 1049 1059 1050 for(i = 1; i <= fNumber; ++i) << 1060 for(i=1;i<=fNumber;i++) 1051 { << 1061 { 1052 if(i == 1) << 1062 if(i == 1) 1053 { << 1063 { 1054 nwt = std::sqrt((G4double)(2 * nHermite << 1064 newton = std::sqrt((G4double)(2*nHermite + 1)) - 1055 1.85575001 * std::pow((G4double)( << 1065 1.85575001*std::pow((G4double)(2*nHermite + 1),-0.16666999) ; 1056 } << 1066 } 1057 else if(i == 2) << 1067 else if(i == 2) 1058 { << 1068 { 1059 nwt -= 1.14001 * std::pow((G4double) nH << 1069 newton -= 1.14001*std::pow((G4double)nHermite,0.425999)/newton ; 1060 } << 1070 } 1061 else if(i == 3) << 1071 else if(i == 3) 1062 { << 1072 { 1063 nwt = 1.86002 * nwt - 0.86002 * fAbscis << 1073 newton = 1.86002*newton - 0.86002*fAbscissa[0] ; 1064 } << 1074 } 1065 else if(i == 4) << 1075 else if(i == 4) 1066 { << 1076 { 1067 nwt = 1.91001 * nwt - 0.91001 * fAbscis << 1077 newton = 1.91001*newton - 0.91001*fAbscissa[1] ; 1068 } << 1078 } 1069 else << 1079 else 1070 { << 1080 { 1071 nwt = 2.0 * nwt - fAbscissa[i - 3]; << 1081 newton = 2.0*newton - fAbscissa[i - 3] ; 1072 } << 1082 } 1073 for(k = 1; k <= maxNumber; ++k) << 1083 for(k=1;k<=maxNumber;k++) 1074 { << 1084 { 1075 temp1 = piInMinusQ; << 1085 temp1 = piInMinusQ ; 1076 temp2 = 0.0; << 1086 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 1087 1107 for(i = 0; i < fNumber; ++i) << 1088 for(j=1;j<=nHermite;j++) 1108 { << 1089 { 1109 integral += << 1090 temp3 = temp2 ; 1110 fWeight[i] * ((typeT.*f)(fAbscissa[i]) << 1091 temp2 = temp1 ; 1111 } << 1092 temp1 = newton*std::sqrt(2.0/j)*temp2 - 1112 delete[] fAbscissa; << 1093 std::sqrt(((G4double)(j - 1))/j)*temp3 ; 1113 delete[] fWeight; << 1094 } 1114 return integral; << 1095 temp = std::sqrt((G4double)2*nHermite)*temp2 ; >> 1096 newton1 = newton ; >> 1097 newton = newton1 - temp1/temp ; >> 1098 >> 1099 if(std::fabs(newton - newton1) <= tolerance) >> 1100 { >> 1101 break ; >> 1102 } >> 1103 } >> 1104 if(k > maxNumber) >> 1105 { >> 1106 G4Exception("G4Integrator<T,F>::Hermite(T,F, ...)", "Error", >> 1107 FatalException, "Too many (>12) iterations."); >> 1108 } >> 1109 fAbscissa[i-1] = newton ; >> 1110 fWeight[i-1] = 2.0/(temp*temp) ; >> 1111 } >> 1112 >> 1113 // >> 1114 // Integral calculation >> 1115 // >> 1116 >> 1117 for(i=0;i<fNumber;i++) >> 1118 { >> 1119 integral += fWeight[i]*( (typeT.*f)(fAbscissa[i]) + >> 1120 (typeT.*f)(-fAbscissa[i]) ) ; >> 1121 } >> 1122 delete[] fAbscissa; >> 1123 delete[] fWeight; >> 1124 return integral ; 1115 } 1125 } 1116 1126 >> 1127 1117 ///////////////////////////////////////////// 1128 //////////////////////////////////////////////////////////////////////// 1118 // 1129 // 1119 // For use with 'this' pointer 1130 // For use with 'this' pointer 1120 1131 1121 template <class T, class F> << 1132 template <class T, class F> 1122 G4double G4Integrator<T, F>::Hermite(T* ptrT, << 1133 G4double G4Integrator<T,F>::Hermite( T* ptrT, F f, G4int n ) 1123 { 1134 { 1124 return Hermite(*ptrT, f, n); << 1135 return Hermite(*ptrT,f,n) ; 1125 } << 1136 } 1126 1137 1127 ///////////////////////////////////////////// 1138 //////////////////////////////////////////////////////////////////////// 1128 // 1139 // 1129 // For use with global scope f 1140 // For use with global scope f 1130 1141 1131 template <class T, class F> 1142 template <class T, class F> 1132 G4double G4Integrator<T, F>::Hermite(G4double << 1143 G4double G4Integrator<T,F>::Hermite( G4double (*f)(G4double), G4int nHermite) 1133 { 1144 { 1134 const G4double tolerance = 1.0e-12; << 1145 const G4double tolerance = 1.0e-12 ; 1135 const G4int maxNumber = 12; << 1146 const G4int maxNumber = 12 ; 1136 << 1147 1137 G4int i, j, k; << 1148 G4int i, j, k ; 1138 G4double integral = 0.0; << 1149 G4double integral = 0.0 ; 1139 G4double nwt = 0., nwt1, temp1, temp2, << 1150 G4double newton=0., newton1, temp1, temp2, temp3, temp ; 1140 << 1151 1141 G4double piInMinusQ = << 1152 G4double piInMinusQ = std::pow(pi,-0.25) ; // 1.0/std::sqrt(std::sqrt(pi)) ?? 1142 std::pow(CLHEP::pi, -0.25); // 1.0/std:: << 1153 1143 << 1154 G4int fNumber = (nHermite +1)/2 ; 1144 G4int fNumber = (nHermite + 1) / 2; << 1155 G4double* fAbscissa = new G4double[fNumber] ; 1145 G4double* fAbscissa = new G4double[fNumber] << 1156 G4double* fWeight = new G4double[fNumber] ; 1146 G4double* fWeight = new G4double[fNumber] << 1157 1147 << 1158 for(i=1;i<=fNumber;i++) 1148 for(i = 1; i <= fNumber; ++i) << 1159 { 1149 { << 1160 if(i == 1) 1150 if(i == 1) << 1161 { 1151 { << 1162 newton = std::sqrt((G4double)(2*nHermite + 1)) - 1152 nwt = std::sqrt((G4double)(2 * nHermite << 1163 1.85575001*std::pow((G4double)(2*nHermite + 1),-0.16666999) ; 1153 1.85575001 * std::pow((G4double)( << 1164 } 1154 } << 1165 else if(i == 2) 1155 else if(i == 2) << 1166 { 1156 { << 1167 newton -= 1.14001*std::pow((G4double)nHermite,0.425999)/newton ; 1157 nwt -= 1.14001 * std::pow((G4double) nH << 1168 } 1158 } << 1169 else if(i == 3) 1159 else if(i == 3) << 1170 { 1160 { << 1171 newton = 1.86002*newton - 0.86002*fAbscissa[0] ; 1161 nwt = 1.86002 * nwt - 0.86002 * fAbscis << 1172 } 1162 } << 1173 else if(i == 4) 1163 else if(i == 4) << 1174 { 1164 { << 1175 newton = 1.91001*newton - 0.91001*fAbscissa[1] ; 1165 nwt = 1.91001 * nwt - 0.91001 * fAbscis << 1176 } 1166 } << 1177 else 1167 else << 1178 { 1168 { << 1179 newton = 2.0*newton - fAbscissa[i - 3] ; 1169 nwt = 2.0 * nwt - fAbscissa[i - 3]; << 1180 } 1170 } << 1181 for(k=1;k<=maxNumber;k++) 1171 for(k = 1; k <= maxNumber; ++k) << 1182 { 1172 { << 1183 temp1 = piInMinusQ ; 1173 temp1 = piInMinusQ; << 1184 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 1185 1205 for(i = 0; i < fNumber; ++i) << 1186 for(j=1;j<=nHermite;j++) 1206 { << 1187 { 1207 integral += fWeight[i] * ((*f)(fAbscissa[ << 1188 temp3 = temp2 ; 1208 } << 1189 temp2 = temp1 ; 1209 delete[] fAbscissa; << 1190 temp1 = newton*std::sqrt(2.0/j)*temp2 - 1210 delete[] fWeight; << 1191 std::sqrt(((G4double)(j - 1))/j)*temp3 ; 1211 return integral; << 1192 } >> 1193 temp = std::sqrt((G4double)2*nHermite)*temp2 ; >> 1194 newton1 = newton ; >> 1195 newton = newton1 - temp1/temp ; >> 1196 >> 1197 if(std::fabs(newton - newton1) <= tolerance) >> 1198 { >> 1199 break ; >> 1200 } >> 1201 } >> 1202 if(k > maxNumber) >> 1203 { >> 1204 G4Exception("G4Integrator<T,F>::Hermite(...)", "Error", >> 1205 FatalException, "Too many (>12) iterations."); >> 1206 } >> 1207 fAbscissa[i-1] = newton ; >> 1208 fWeight[i-1] = 2.0/(temp*temp) ; >> 1209 } >> 1210 >> 1211 // >> 1212 // Integral calculation >> 1213 // >> 1214 >> 1215 for(i=0;i<fNumber;i++) >> 1216 { >> 1217 integral += fWeight[i]*( (*f)(fAbscissa[i]) + (*f)(-fAbscissa[i]) ) ; >> 1218 } >> 1219 delete[] fAbscissa; >> 1220 delete[] fWeight; >> 1221 return integral ; 1212 } 1222 } 1213 1223 1214 ///////////////////////////////////////////// 1224 //////////////////////////////////////////////////////////////////////////// 1215 // 1225 // 1216 // Method involving Jacobi polynomials 1226 // Method involving Jacobi polynomials 1217 // 1227 // 1218 ///////////////////////////////////////////// 1228 //////////////////////////////////////////////////////////////////////////// 1219 // 1229 // 1220 // Gauss-Jacobi method for integration of ((1 1230 // Gauss-Jacobi method for integration of ((1-x)^alpha)*((1+x)^beta)*f(x) 1221 // from minus unit to plus unit . 1231 // from minus unit to plus unit . 1222 // 1232 // 1223 1233 1224 template <class T, class F> << 1234 template <class T, class F> 1225 G4double G4Integrator<T, F>::Jacobi(T& typeT, << 1235 G4double G4Integrator<T,F>::Jacobi( T& typeT, F f, G4double alpha, 1226 G4double << 1236 G4double beta, G4int nJacobi) 1227 { << 1237 { 1228 const G4double tolerance = 1.0e-12; << 1238 const G4double tolerance = 1.0e-12 ; 1229 const G4double maxNumber = 12; << 1239 const G4double maxNumber = 12 ; 1230 G4int i, k, j; << 1240 G4int i, k, j ; 1231 G4double alphaBeta, alphaReduced, betaReduc << 1241 G4double alphaBeta, alphaReduced, betaReduced, root1=0., root2=0., root3=0. ; 1232 << 1242 G4double a, b, c, newton1, newton2, newton3, newton, temp, root=0., rootTemp ; 1233 G4double a, b, c, nwt1, nwt2, nwt3, nwt, te << 1243 1234 << 1244 G4int fNumber = nJacobi ; 1235 G4int fNumber = nJacobi; << 1245 G4double* fAbscissa = new G4double[fNumber] ; 1236 G4double* fAbscissa = new G4double[fNumber] << 1246 G4double* fWeight = new G4double[fNumber] ; 1237 G4double* fWeight = new G4double[fNumber] << 1247 1238 << 1248 for (i=1;i<=nJacobi;i++) 1239 for(i = 1; i <= nJacobi; ++i) << 1249 { 1240 { << 1250 if (i == 1) 1241 if(i == 1) << 1251 { 1242 { << 1252 alphaReduced = alpha/nJacobi ; 1243 alphaReduced = alpha / nJacobi; << 1253 betaReduced = beta/nJacobi ; 1244 betaReduced = beta / nJacobi; << 1254 root1 = (1.0+alpha)*(2.78002/(4.0+nJacobi*nJacobi)+ 1245 root1 = (1.0 + alpha) * (2.78002 << 1255 0.767999*alphaReduced/nJacobi) ; 1246 0.767999 * alp << 1256 root2 = 1.0+1.48*alphaReduced+0.96002*betaReduced + 1247 root2 = 1.0 + 1.48 * alphaReduce << 1257 0.451998*alphaReduced*alphaReduced + 1248 0.451998 * alphaReduced * alpha << 1258 0.83001*alphaReduced*betaReduced ; 1249 0.83001 * alphaReduced * betaRe << 1259 root = 1.0-root1/root2 ; 1250 root = 1.0 - root1 / root2; << 1260 } 1251 } << 1261 else if (i == 2) 1252 else if(i == 2) << 1262 { 1253 { << 1263 root1=(4.1002+alpha)/((1.0+alpha)*(1.0+0.155998*alpha)) ; 1254 root1 = (4.1002 + alpha) / ((1.0 + alph << 1264 root2=1.0+0.06*(nJacobi-8.0)*(1.0+0.12*alpha)/nJacobi ; 1255 root2 = 1.0 + 0.06 * (nJacobi - 8.0) * << 1265 root3=1.0+0.012002*beta*(1.0+0.24997*std::fabs(alpha))/nJacobi ; 1256 root3 = << 1266 root -= (1.0-root)*root1*root2*root3 ; 1257 1.0 + 0.012002 * beta * (1.0 + 0.2499 << 1267 } 1258 root -= (1.0 - root) * root1 * root2 * << 1268 else if (i == 3) 1259 } << 1269 { 1260 else if(i == 3) << 1270 root1=(1.67001+0.27998*alpha)/(1.0+0.37002*alpha) ; 1261 { << 1271 root2=1.0+0.22*(nJacobi-8.0)/nJacobi ; 1262 root1 = (1.67001 + 0.27998 * alpha) / ( << 1272 root3=1.0+8.0*beta/((6.28001+beta)*nJacobi*nJacobi) ; 1263 root2 = 1.0 + 0.22 * (nJacobi - 8.0) / << 1273 root -= (fAbscissa[0]-root)*root1*root2*root3 ; 1264 root3 = 1.0 + 8.0 * beta / ((6.28001 + << 1274 } 1265 root -= (fAbscissa[0] - root) * root1 * << 1275 else if (i == nJacobi-1) 1266 } << 1276 { 1267 else if(i == nJacobi - 1) << 1277 root1=(1.0+0.235002*beta)/(0.766001+0.118998*beta) ; 1268 { << 1278 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 << 1279 root3=1.0/(1.0+20.0*alpha/((7.5+alpha)*nJacobi*nJacobi)) ; 1270 root2 = 1.0 / (1.0 + 0.639002 * (nJacob << 1280 root += (root-fAbscissa[nJacobi-4])*root1*root2*root3 ; 1271 (1.0 + 0.71001 * << 1281 } 1272 root3 = 1.0 / (1.0 + 20.0 * alpha / ((7 << 1282 else if (i == nJacobi) 1273 root += (root - fAbscissa[nJacobi - 4]) << 1283 { 1274 } << 1284 root1 = (1.0+0.37002*beta)/(1.67001+0.27998*beta) ; 1275 else if(i == nJacobi) << 1285 root2 = 1.0/(1.0+0.22*(nJacobi-8.0)/nJacobi) ; 1276 { << 1286 root3 = 1.0/(1.0+8.0*alpha/((6.28002+alpha)*nJacobi*nJacobi)) ; 1277 root1 = (1.0 + 0.37002 * beta) / (1.670 << 1287 root += (root-fAbscissa[nJacobi-3])*root1*root2*root3 ; 1278 root2 = 1.0 / (1.0 + 0.22 * (nJacobi - << 1288 } 1279 root3 = << 1289 else 1280 1.0 / (1.0 + 8.0 * alpha / ((6.28002 << 1290 { 1281 root += (root - fAbscissa[nJacobi - 3]) << 1291 root = 3.0*fAbscissa[i-2]-3.0*fAbscissa[i-3]+fAbscissa[i-4] ; 1282 } << 1292 } 1283 else << 1293 alphaBeta = alpha + beta ; 1284 { << 1294 for (k=1;k<=maxNumber;k++) 1285 root = 3.0 * fAbscissa[i - 2] - 3.0 * f << 1295 { 1286 } << 1296 temp = 2.0 + alphaBeta ; 1287 alphaBeta = alpha + beta; << 1297 newton1 = (alpha-beta+temp*root)/2.0 ; 1288 for(k = 1; k <= maxNumber; ++k) << 1298 newton2 = 1.0 ; 1289 { << 1299 for (j=2;j<=nJacobi;j++) 1290 temp = 2.0 + alphaBeta; << 1300 { 1291 nwt1 = (alpha - beta + temp * root) / 2 << 1301 newton3 = newton2 ; 1292 nwt2 = 1.0; << 1302 newton2 = newton1 ; 1293 for(j = 2; j <= nJacobi; ++j) << 1303 temp = 2*j+alphaBeta ; 1294 { << 1304 a = 2*j*(j+alphaBeta)*(temp-2.0) ; 1295 nwt3 = nwt2; << 1305 b = (temp-1.0)*(alpha*alpha-beta*beta+temp*(temp-2.0)*root) ; 1296 nwt2 = nwt1; << 1306 c = 2.0*(j-1+alpha)*(j-1+beta)*temp ; 1297 temp = 2 * j + alphaBeta; << 1307 newton1 = (b*newton2-c*newton3)/a ; 1298 a = 2 * j * (j + alphaBeta) * (tem << 1308 } 1299 b = (temp - 1.0) * << 1309 newton = (nJacobi*(alpha - beta - temp*root)*newton1 + 1300 (alpha * alpha - beta * beta + te << 1310 2.0*(nJacobi + alpha)*(nJacobi + beta)*newton2)/ 1301 c = 2.0 * (j - 1 + alpha) * (j - 1 << 1311 (temp*(1.0 - root*root)) ; 1302 nwt1 = (b * nwt2 - c * nwt3) / a; << 1312 rootTemp = root ; 1303 } << 1313 root = rootTemp - newton1/newton ; 1304 nwt = (nJacobi * (alpha - beta - temp * << 1314 if (std::fabs(root-rootTemp) <= tolerance) 1305 2.0 * (nJacobi + alpha) * (nJaco << 1315 { 1306 (temp * (1.0 - root * root)); << 1316 break ; 1307 rootTemp = root; << 1317 } 1308 root = rootTemp - nwt1 / nwt; << 1318 } 1309 if(std::fabs(root - rootTemp) <= tolera << 1319 if (k > maxNumber) 1310 { << 1320 { 1311 break; << 1321 G4Exception("G4Integrator<T,F>::Jacobi(T,F, ...)", "Error", 1312 } << 1322 FatalException, "Too many (>12) iterations."); 1313 } << 1323 } 1314 if(k > maxNumber) << 1324 fAbscissa[i-1] = root ; 1315 { << 1325 fWeight[i-1] = std::exp(GammaLogarithm((G4double)(alpha+nJacobi)) + 1316 G4Exception("G4Integrator<T,F>::Jacobi( << 1326 GammaLogarithm((G4double)(beta+nJacobi)) - 1317 FatalException, "Too many ( << 1327 GammaLogarithm((G4double)(nJacobi+1.0)) - 1318 } << 1328 GammaLogarithm((G4double)(nJacobi + alphaBeta + 1.0))) 1319 fAbscissa[i - 1] = root; << 1329 *temp*std::pow(2.0,alphaBeta)/(newton*newton2) ; 1320 fWeight[i - 1] = << 1330 } 1321 std::exp(GammaLogarithm((G4double)(alph << 1331 1322 GammaLogarithm((G4double)(beta << 1332 // 1323 GammaLogarithm((G4double)(nJac << 1333 // Calculation of the integral 1324 GammaLogarithm((G4double)(nJac << 1334 // 1325 temp * std::pow(2.0, alphaBeta) / (nwt << 1335 1326 } << 1336 G4double integral = 0.0 ; 1327 << 1337 for(i=0;i<fNumber;i++) 1328 // << 1338 { 1329 // Calculation of the integral << 1339 integral += fWeight[i]*(typeT.*f)(fAbscissa[i]) ; 1330 // << 1340 } 1331 << 1341 delete[] fAbscissa; 1332 G4double integral = 0.0; << 1342 delete[] fWeight; 1333 for(i = 0; i < fNumber; ++i) << 1343 return integral ; 1334 { << 1335 integral += fWeight[i] * (typeT.*f)(fAbsc << 1336 } << 1337 delete[] fAbscissa; << 1338 delete[] fWeight; << 1339 return integral; << 1340 } 1344 } 1341 1345 >> 1346 1342 ///////////////////////////////////////////// 1347 ///////////////////////////////////////////////////////////////////////// 1343 // 1348 // 1344 // For use with 'this' pointer 1349 // For use with 'this' pointer 1345 1350 1346 template <class T, class F> << 1351 template <class T, class F> 1347 G4double G4Integrator<T, F>::Jacobi(T* ptrT, << 1352 G4double G4Integrator<T,F>::Jacobi( T* ptrT, F f, G4double alpha, 1348 G4int n) << 1353 G4double beta, G4int n) 1349 { 1354 { 1350 return Jacobi(*ptrT, f, alpha, beta, n); << 1355 return Jacobi(*ptrT,f,alpha,beta,n) ; 1351 } << 1356 } 1352 1357 1353 ///////////////////////////////////////////// 1358 ///////////////////////////////////////////////////////////////////////// 1354 // 1359 // 1355 // For use with global scope f << 1360 // For use with global scope f 1356 1361 1357 template <class T, class F> 1362 template <class T, class F> 1358 G4double G4Integrator<T, F>::Jacobi(G4double << 1363 G4double G4Integrator<T,F>::Jacobi( G4double (*f)(G4double), G4double alpha, 1359 G4double << 1364 G4double beta, G4int nJacobi) 1360 { 1365 { 1361 const G4double tolerance = 1.0e-12; << 1366 const G4double tolerance = 1.0e-12 ; 1362 const G4double maxNumber = 12; << 1367 const G4double maxNumber = 12 ; 1363 G4int i, k, j; << 1368 G4int i, k, j ; 1364 G4double alphaBeta, alphaReduced, betaReduc << 1369 G4double alphaBeta, alphaReduced, betaReduced, root1=0., root2=0., root3=0. ; 1365 << 1370 G4double a, b, c, newton1, newton2, newton3, newton, temp, root=0., rootTemp ; 1366 G4double a, b, c, nwt1, nwt2, nwt3, nwt, te << 1371 1367 << 1372 G4int fNumber = nJacobi ; 1368 G4int fNumber = nJacobi; << 1373 G4double* fAbscissa = new G4double[fNumber] ; 1369 G4double* fAbscissa = new G4double[fNumber] << 1374 G4double* fWeight = new G4double[fNumber] ; 1370 G4double* fWeight = new G4double[fNumber] << 1375 1371 << 1376 for (i=1;i<=nJacobi;i++) 1372 for(i = 1; i <= nJacobi; ++i) << 1377 { 1373 { << 1378 if (i == 1) 1374 if(i == 1) << 1379 { 1375 { << 1380 alphaReduced = alpha/nJacobi ; 1376 alphaReduced = alpha / nJacobi; << 1381 betaReduced = beta/nJacobi ; 1377 betaReduced = beta / nJacobi; << 1382 root1 = (1.0+alpha)*(2.78002/(4.0+nJacobi*nJacobi)+ 1378 root1 = (1.0 + alpha) * (2.78002 << 1383 0.767999*alphaReduced/nJacobi) ; 1379 0.767999 * alp << 1384 root2 = 1.0+1.48*alphaReduced+0.96002*betaReduced + 1380 root2 = 1.0 + 1.48 * alphaReduce << 1385 0.451998*alphaReduced*alphaReduced + 1381 0.451998 * alphaReduced * alpha << 1386 0.83001*alphaReduced*betaReduced ; 1382 0.83001 * alphaReduced * betaRe << 1387 root = 1.0-root1/root2 ; 1383 root = 1.0 - root1 / root2; << 1388 } 1384 } << 1389 else if (i == 2) 1385 else if(i == 2) << 1390 { 1386 { << 1391 root1=(4.1002+alpha)/((1.0+alpha)*(1.0+0.155998*alpha)) ; 1387 root1 = (4.1002 + alpha) / ((1.0 + alph << 1392 root2=1.0+0.06*(nJacobi-8.0)*(1.0+0.12*alpha)/nJacobi ; 1388 root2 = 1.0 + 0.06 * (nJacobi - 8.0) * << 1393 root3=1.0+0.012002*beta*(1.0+0.24997*std::fabs(alpha))/nJacobi ; 1389 root3 = << 1394 root -= (1.0-root)*root1*root2*root3 ; 1390 1.0 + 0.012002 * beta * (1.0 + 0.2499 << 1395 } 1391 root -= (1.0 - root) * root1 * root2 * << 1396 else if (i == 3) 1392 } << 1397 { 1393 else if(i == 3) << 1398 root1=(1.67001+0.27998*alpha)/(1.0+0.37002*alpha) ; 1394 { << 1399 root2=1.0+0.22*(nJacobi-8.0)/nJacobi ; 1395 root1 = (1.67001 + 0.27998 * alpha) / ( << 1400 root3=1.0+8.0*beta/((6.28001+beta)*nJacobi*nJacobi) ; 1396 root2 = 1.0 + 0.22 * (nJacobi - 8.0) / << 1401 root -= (fAbscissa[0]-root)*root1*root2*root3 ; 1397 root3 = 1.0 + 8.0 * beta / ((6.28001 + << 1402 } 1398 root -= (fAbscissa[0] - root) * root1 * << 1403 else if (i == nJacobi-1) 1399 } << 1404 { 1400 else if(i == nJacobi - 1) << 1405 root1=(1.0+0.235002*beta)/(0.766001+0.118998*beta) ; 1401 { << 1406 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 << 1407 root3=1.0/(1.0+20.0*alpha/((7.5+alpha)*nJacobi*nJacobi)) ; 1403 root2 = 1.0 / (1.0 + 0.639002 * (nJacob << 1408 root += (root-fAbscissa[nJacobi-4])*root1*root2*root3 ; 1404 (1.0 + 0.71001 * << 1409 } 1405 root3 = 1.0 / (1.0 + 20.0 * alpha / ((7 << 1410 else if (i == nJacobi) 1406 root += (root - fAbscissa[nJacobi - 4]) << 1411 { 1407 } << 1412 root1 = (1.0+0.37002*beta)/(1.67001+0.27998*beta) ; 1408 else if(i == nJacobi) << 1413 root2 = 1.0/(1.0+0.22*(nJacobi-8.0)/nJacobi) ; 1409 { << 1414 root3 = 1.0/(1.0+8.0*alpha/((6.28002+alpha)*nJacobi*nJacobi)) ; 1410 root1 = (1.0 + 0.37002 * beta) / (1.670 << 1415 root += (root-fAbscissa[nJacobi-3])*root1*root2*root3 ; 1411 root2 = 1.0 / (1.0 + 0.22 * (nJacobi - << 1416 } 1412 root3 = << 1417 else 1413 1.0 / (1.0 + 8.0 * alpha / ((6.28002 << 1418 { 1414 root += (root - fAbscissa[nJacobi - 3]) << 1419 root = 3.0*fAbscissa[i-2]-3.0*fAbscissa[i-3]+fAbscissa[i-4] ; 1415 } << 1420 } 1416 else << 1421 alphaBeta = alpha + beta ; 1417 { << 1422 for (k=1;k<=maxNumber;k++) 1418 root = 3.0 * fAbscissa[i - 2] - 3.0 * f << 1423 { 1419 } << 1424 temp = 2.0 + alphaBeta ; 1420 alphaBeta = alpha + beta; << 1425 newton1 = (alpha-beta+temp*root)/2.0 ; 1421 for(k = 1; k <= maxNumber; ++k) << 1426 newton2 = 1.0 ; 1422 { << 1427 for (j=2;j<=nJacobi;j++) 1423 temp = 2.0 + alphaBeta; << 1428 { 1424 nwt1 = (alpha - beta + temp * root) / 2 << 1429 newton3 = newton2 ; 1425 nwt2 = 1.0; << 1430 newton2 = newton1 ; 1426 for(j = 2; j <= nJacobi; ++j) << 1431 temp = 2*j+alphaBeta ; 1427 { << 1432 a = 2*j*(j+alphaBeta)*(temp-2.0) ; 1428 nwt3 = nwt2; << 1433 b = (temp-1.0)*(alpha*alpha-beta*beta+temp*(temp-2.0)*root) ; 1429 nwt2 = nwt1; << 1434 c = 2.0*(j-1+alpha)*(j-1+beta)*temp ; 1430 temp = 2 * j + alphaBeta; << 1435 newton1 = (b*newton2-c*newton3)/a ; 1431 a = 2 * j * (j + alphaBeta) * (tem << 1436 } 1432 b = (temp - 1.0) * << 1437 newton = (nJacobi*(alpha - beta - temp*root)*newton1 + 1433 (alpha * alpha - beta * beta + te << 1438 2.0*(nJacobi + alpha)*(nJacobi + beta)*newton2) / 1434 c = 2.0 * (j - 1 + alpha) * (j - 1 << 1439 (temp*(1.0 - root*root)) ; 1435 nwt1 = (b * nwt2 - c * nwt3) / a; << 1440 rootTemp = root ; 1436 } << 1441 root = rootTemp - newton1/newton ; 1437 nwt = (nJacobi * (alpha - beta - temp * << 1442 if (std::fabs(root-rootTemp) <= tolerance) 1438 2.0 * (nJacobi + alpha) * (nJaco << 1443 { 1439 (temp * (1.0 - root * root)); << 1444 break ; 1440 rootTemp = root; << 1445 } 1441 root = rootTemp - nwt1 / nwt; << 1446 } 1442 if(std::fabs(root - rootTemp) <= tolera << 1447 if (k > maxNumber) 1443 { << 1448 { 1444 break; << 1449 G4Exception("G4Integrator<T,F>::Jacobi(...)", "Error", 1445 } << 1450 FatalException, "Too many (>12) iterations."); 1446 } << 1451 } 1447 if(k > maxNumber) << 1452 fAbscissa[i-1] = root ; 1448 { << 1453 fWeight[i-1] = 1449 G4Exception("G4Integrator<T,F>::Jacobi( << 1454 std::exp(GammaLogarithm((G4double)(alpha+nJacobi)) + 1450 "Too many (>12) iterations. << 1455 GammaLogarithm((G4double)(beta+nJacobi)) - 1451 } << 1456 GammaLogarithm((G4double)(nJacobi+1.0)) - 1452 fAbscissa[i - 1] = root; << 1457 GammaLogarithm((G4double)(nJacobi + alphaBeta + 1.0))) 1453 fWeight[i - 1] = << 1458 *temp*std::pow(2.0,alphaBeta)/(newton*newton2); 1454 std::exp(GammaLogarithm((G4double)(alph << 1459 } 1455 GammaLogarithm((G4double)(beta << 1460 1456 GammaLogarithm((G4double)(nJac << 1461 // 1457 GammaLogarithm((G4double)(nJac << 1462 // Calculation of the integral 1458 temp * std::pow(2.0, alphaBeta) / (nwt << 1463 // 1459 } << 1464 1460 << 1465 G4double integral = 0.0 ; 1461 // << 1466 for(i=0;i<fNumber;i++) 1462 // Calculation of the integral << 1467 { 1463 // << 1468 integral += fWeight[i]*(*f)(fAbscissa[i]) ; 1464 << 1469 } 1465 G4double integral = 0.0; << 1470 delete[] fAbscissa; 1466 for(i = 0; i < fNumber; ++i) << 1471 delete[] fWeight; 1467 { << 1472 return integral ; 1468 integral += fWeight[i] * (*f)(fAbscissa[i << 1469 } << 1470 delete[] fAbscissa; << 1471 delete[] fWeight; << 1472 return integral; << 1473 } 1473 } 1474 1474 1475 // 1475 // 1476 // 1476 // 1477 ///////////////////////////////////////////// 1477 /////////////////////////////////////////////////////////////////// 1478 1478