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