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24 // ******************************************* 24 // ********************************************************************
25 // 25 //
26 // G4Integrator inline methods implementation <<
27 // 26 //
28 // Author: V.Grichine, 04.09.1999 - First impl << 27 // $Id: G4Integrator.icc 80312 2014-04-10 12:20:24Z gcosmo $
29 // G4SimpleIntegration class with H.P. << 28 //
30 // E.TCherniaev advises << 29 // Implementation of G4Integrator methods.
31 // ------------------------------------------- << 30 //
>> 31 //
32 32
33 ////////////////////////////////////////////// 33 /////////////////////////////////////////////////////////////////////
34 // 34 //
35 // Sympson integration method 35 // Sympson integration method
36 // 36 //
37 ////////////////////////////////////////////// 37 /////////////////////////////////////////////////////////////////////
38 // 38 //
39 // Integration of class member functions T::f << 39 // Integration of class member functions T::f by Simpson method.
40 40
41 template <class T, class F> << 41 template <class T, class F>
42 G4double G4Integrator<T, F>::Simpson(T& typeT, << 42 G4double G4Integrator<T,F>::Simpson( T& typeT,
43 G4double << 43 F f,
44 { << 44 G4double xInitial,
45 G4int i; << 45 G4double xFinal,
46 G4double step = (xFinal - xInitial) / itera << 46 G4int iterationNumber )
47 G4double x = xInitial; << 47 {
48 G4double xPlus = xInitial + 0.5 * step; << 48 G4int i ;
49 G4double mean = ((typeT.*f)(xInitial) + (ty << 49 G4double step = (xFinal - xInitial)/iterationNumber ;
50 G4double sum = (typeT.*f)(xPlus); << 50 G4double x = xInitial ;
51 << 51 G4double xPlus = xInitial + 0.5*step ;
52 for(i = 1; i < iterationNumber; ++i) << 52 G4double mean = ( (typeT.*f)(xInitial) + (typeT.*f)(xFinal) )*0.5 ;
53 { << 53 G4double sum = (typeT.*f)(xPlus) ;
54 x += step; << 54
55 xPlus += step; << 55 for(i=1;i<iterationNumber;i++)
56 mean += (typeT.*f)(x); << 56 {
57 sum += (typeT.*f)(xPlus); << 57 x += step ;
58 } << 58 xPlus += step ;
59 mean += 2.0 * sum; << 59 mean += (typeT.*f)(x) ;
>> 60 sum += (typeT.*f)(xPlus) ;
>> 61 }
>> 62 mean += 2.0*sum ;
60 63
61 return mean * step / 3.0; << 64 return mean*step/3.0 ;
62 } 65 }
63 66
64 ////////////////////////////////////////////// 67 /////////////////////////////////////////////////////////////////////
65 // 68 //
66 // Integration of class member functions T::f 69 // Integration of class member functions T::f by Simpson method.
67 // Convenient to use with 'this' pointer 70 // Convenient to use with 'this' pointer
68 71
69 template <class T, class F> << 72 template <class T, class F>
70 G4double G4Integrator<T, F>::Simpson(T* ptrT, << 73 G4double G4Integrator<T,F>::Simpson( T* ptrT,
71 G4double << 74 F f,
72 { << 75 G4double xInitial,
73 G4int i; << 76 G4double xFinal,
74 G4double step = (xFinal - xInitial) / itera << 77 G4int iterationNumber )
75 G4double x = xInitial; << 78 {
76 G4double xPlus = xInitial + 0.5 * step; << 79 G4int i ;
77 G4double mean = ((ptrT->*f)(xInitial) + (pt << 80 G4double step = (xFinal - xInitial)/iterationNumber ;
78 G4double sum = (ptrT->*f)(xPlus); << 81 G4double x = xInitial ;
>> 82 G4double xPlus = xInitial + 0.5*step ;
>> 83 G4double mean = ( (ptrT->*f)(xInitial) + (ptrT->*f)(xFinal) )*0.5 ;
>> 84 G4double sum = (ptrT->*f)(xPlus) ;
>> 85
>> 86 for(i=1;i<iterationNumber;i++)
>> 87 {
>> 88 x += step ;
>> 89 xPlus += step ;
>> 90 mean += (ptrT->*f)(x) ;
>> 91 sum += (ptrT->*f)(xPlus) ;
>> 92 }
>> 93 mean += 2.0*sum ;
79 94
80 for(i = 1; i < iterationNumber; ++i) << 95 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 } 96 }
91 97
92 ////////////////////////////////////////////// 98 /////////////////////////////////////////////////////////////////////
93 // 99 //
94 // Integration of class member functions T::f 100 // Integration of class member functions T::f by Simpson method.
95 // Convenient to use, when function f is defin 101 // Convenient to use, when function f is defined in global scope, i.e. in main()
96 // program 102 // program
97 103
98 template <class T, class F> << 104 template <class T, class F>
99 G4double G4Integrator<T, F>::Simpson(G4double << 105 G4double G4Integrator<T,F>::Simpson( G4double (*f)(G4double),
100 G4double << 106 G4double xInitial,
101 { << 107 G4double xFinal,
102 G4int i; << 108 G4int iterationNumber )
103 G4double step = (xFinal - xInitial) / itera << 109 {
104 G4double x = xInitial; << 110 G4int i ;
105 G4double xPlus = xInitial + 0.5 * step; << 111 G4double step = (xFinal - xInitial)/iterationNumber ;
106 G4double mean = ((*f)(xInitial) + (*f)(xFin << 112 G4double x = xInitial ;
107 G4double sum = (*f)(xPlus); << 113 G4double xPlus = xInitial + 0.5*step ;
108 << 114 G4double mean = ( (*f)(xInitial) + (*f)(xFinal) )*0.5 ;
109 for(i = 1; i < iterationNumber; ++i) << 115 G4double sum = (*f)(xPlus) ;
110 { << 116
111 x += step; << 117 for(i=1;i<iterationNumber;i++)
112 xPlus += step; << 118 {
113 mean += (*f)(x); << 119 x += step ;
114 sum += (*f)(xPlus); << 120 xPlus += step ;
115 } << 121 mean += (*f)(x) ;
116 mean += 2.0 * sum; << 122 sum += (*f)(xPlus) ;
>> 123 }
>> 124 mean += 2.0*sum ;
117 125
118 return mean * step / 3.0; << 126 return mean*step/3.0 ;
119 } 127 }
120 128
121 ////////////////////////////////////////////// 129 //////////////////////////////////////////////////////////////////////////
122 // 130 //
123 // Adaptive Gauss method 131 // Adaptive Gauss method
124 // 132 //
125 ////////////////////////////////////////////// 133 //////////////////////////////////////////////////////////////////////////
126 // 134 //
127 // 135 //
128 136
129 template <class T, class F> << 137 template <class T, class F>
130 G4double G4Integrator<T, F>::Gauss(T& typeT, F << 138 G4double G4Integrator<T,F>::Gauss( T& typeT, F f,
131 G4double xF << 139 G4double xInitial, G4double xFinal )
132 { << 140 {
133 static const G4double root = 1.0 / std::sqrt << 141 static const G4double root = 1.0/std::sqrt(3.0) ;
134 << 142
135 G4double xMean = (xInitial + xFinal) / 2.0; << 143 G4double xMean = (xInitial + xFinal)/2.0 ;
136 G4double Step = (xFinal - xInitial) / 2.0; << 144 G4double Step = (xFinal - xInitial)/2.0 ;
137 G4double delta = Step * root; << 145 G4double delta = Step*root ;
138 G4double sum = ((typeT.*f)(xMean + delta) << 146 G4double sum = ((typeT.*f)(xMean + delta) +
139 << 147 (typeT.*f)(xMean - delta)) ;
140 return sum * Step; << 148
>> 149 return sum*Step ;
141 } 150 }
142 151
143 ////////////////////////////////////////////// 152 //////////////////////////////////////////////////////////////////////
144 // 153 //
145 // 154 //
146 155
147 template <class T, class F> << 156 template <class T, class F> G4double
148 G4double G4Integrator<T, F>::Gauss(T* ptrT, F << 157 G4Integrator<T,F>::Gauss( T* ptrT, F f, G4double a, G4double b )
149 { 158 {
150 return Gauss(*ptrT, f, a, b); << 159 return Gauss(*ptrT,f,a,b) ;
151 } 160 }
152 161
153 ////////////////////////////////////////////// 162 ///////////////////////////////////////////////////////////////////////
154 // 163 //
155 // 164 //
156 165
157 template <class T, class F> 166 template <class T, class F>
158 G4double G4Integrator<T, F>::Gauss(G4double (* << 167 G4double G4Integrator<T,F>::Gauss( G4double (*f)(G4double),
159 G4double xF << 168 G4double xInitial, G4double xFinal)
160 { 169 {
161 static const G4double root = 1.0 / std::sqrt << 170 static const G4double root = 1.0/std::sqrt(3.0) ;
162 171
163 G4double xMean = (xInitial + xFinal) / 2.0; << 172 G4double xMean = (xInitial + xFinal)/2.0 ;
164 G4double Step = (xFinal - xInitial) / 2.0; << 173 G4double Step = (xFinal - xInitial)/2.0 ;
165 G4double delta = Step * root; << 174 G4double delta = Step*root ;
166 G4double sum = ((*f)(xMean + delta) + (*f) << 175 G4double sum = ( (*f)(xMean + delta) + (*f)(xMean - delta) ) ;
167 << 176
168 return sum * Step; << 177 return sum*Step ;
169 } 178 }
170 179
171 ////////////////////////////////////////////// 180 ///////////////////////////////////////////////////////////////////////////
172 // 181 //
173 // 182 //
174 183
175 template <class T, class F> << 184 template <class T, class F>
176 void G4Integrator<T, F>::AdaptGauss(T& typeT, << 185 void G4Integrator<T,F>::AdaptGauss( T& typeT, F f, G4double xInitial,
177 G4double x << 186 G4double xFinal, G4double fTolerance,
178 G4double& << 187 G4double& sum,
>> 188 G4int& depth )
>> 189 {
>> 190 if(depth > 100)
>> 191 {
>> 192 G4cout<<"G4Integrator<T,F>::AdaptGauss: WARNING !!!"<<G4endl ;
>> 193 G4cout<<"Function varies too rapidly to get stated accuracy in 100 steps "
>> 194 <<G4endl ;
>> 195
>> 196 return ;
>> 197 }
>> 198 G4double xMean = (xInitial + xFinal)/2.0 ;
>> 199 G4double leftHalf = Gauss(typeT,f,xInitial,xMean) ;
>> 200 G4double rightHalf = Gauss(typeT,f,xMean,xFinal) ;
>> 201 G4double full = Gauss(typeT,f,xInitial,xFinal) ;
>> 202 if(std::fabs(leftHalf+rightHalf-full) < fTolerance)
>> 203 {
>> 204 sum += full ;
>> 205 }
>> 206 else
>> 207 {
>> 208 depth++ ;
>> 209 AdaptGauss(typeT,f,xInitial,xMean,fTolerance,sum,depth) ;
>> 210 AdaptGauss(typeT,f,xMean,xFinal,fTolerance,sum,depth) ;
>> 211 }
>> 212 }
>> 213
>> 214 template <class T, class F>
>> 215 void G4Integrator<T,F>::AdaptGauss( T* ptrT, F f, G4double xInitial,
>> 216 G4double xFinal, G4double fTolerance,
>> 217 G4double& sum,
>> 218 G4int& depth )
179 { 219 {
180 if(depth > 100) << 220 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 } 221 }
211 222
212 ////////////////////////////////////////////// 223 /////////////////////////////////////////////////////////////////////////
213 // 224 //
214 // 225 //
215 template <class T, class F> 226 template <class T, class F>
216 void G4Integrator<T, F>::AdaptGauss(G4double ( << 227 void G4Integrator<T,F>::AdaptGauss( G4double (*f)(G4double),
217 G4double x << 228 G4double xInitial, G4double xFinal,
218 G4double& << 229 G4double fTolerance, G4double& sum,
219 { << 230 G4int& depth )
220 if(depth > 100) << 231 {
221 { << 232 if(depth > 100)
222 G4cout << "G4SimpleIntegration::AdaptGauss << 233 {
223 G4cout << "Function varies too rapidly to << 234 G4cout<<"G4SimpleIntegration::AdaptGauss: WARNING !!!"<<G4endl ;
224 << G4endl; << 235 G4cout<<"Function varies too rapidly to get stated accuracy in 100 steps "
225 << 236 <<G4endl ;
226 return; << 237
227 } << 238 return ;
228 G4double xMean = (xInitial + xFinal) / 2 << 239 }
229 G4double leftHalf = Gauss(f, xInitial, xMea << 240 G4double xMean = (xInitial + xFinal)/2.0 ;
230 G4double rightHalf = Gauss(f, xMean, xFinal) << 241 G4double leftHalf = Gauss(f,xInitial,xMean) ;
231 G4double full = Gauss(f, xInitial, xFin << 242 G4double rightHalf = Gauss(f,xMean,xFinal) ;
232 if(std::fabs(leftHalf + rightHalf - full) < << 243 G4double full = Gauss(f,xInitial,xFinal) ;
233 { << 244 if(std::fabs(leftHalf+rightHalf-full) < fTolerance)
234 sum += full; << 245 {
235 } << 246 sum += full ;
236 else << 247 }
237 { << 248 else
238 ++depth; << 249 {
239 AdaptGauss(f, xInitial, xMean, fTolerance, << 250 depth++ ;
240 AdaptGauss(f, xMean, xFinal, fTolerance, s << 251 AdaptGauss(f,xInitial,xMean,fTolerance,sum,depth) ;
241 } << 252 AdaptGauss(f,xMean,xFinal,fTolerance,sum,depth) ;
>> 253 }
242 } 254 }
243 255
244 ////////////////////////////////////////////// 256 ////////////////////////////////////////////////////////////////////////
245 // 257 //
246 // Adaptive Gauss integration with accuracy 'e 258 // Adaptive Gauss integration with accuracy 'e'
247 // Convenient for using with class object type 259 // Convenient for using with class object typeT
248 << 260
249 template <class T, class F> << 261 template<class T, class F>
250 G4double G4Integrator<T, F>::AdaptiveGauss(T& << 262 G4double G4Integrator<T,F>::AdaptiveGauss( T& typeT, F f, G4double xInitial,
251 G4d << 263 G4double xFinal, G4double e )
252 { << 264 {
253 G4int depth = 0; << 265 G4int depth = 0 ;
254 G4double sum = 0.0; << 266 G4double sum = 0.0 ;
255 AdaptGauss(typeT, f, xInitial, xFinal, e, su << 267 AdaptGauss(typeT,f,xInitial,xFinal,e,sum,depth) ;
256 return sum; << 268 return sum ;
257 } 269 }
258 270
259 ////////////////////////////////////////////// 271 ////////////////////////////////////////////////////////////////////////
260 // 272 //
261 // Adaptive Gauss integration with accuracy 'e 273 // Adaptive Gauss integration with accuracy 'e'
262 // Convenient for using with 'this' pointer 274 // Convenient for using with 'this' pointer
263 << 275
264 template <class T, class F> << 276 template<class T, class F>
265 G4double G4Integrator<T, F>::AdaptiveGauss(T* << 277 G4double G4Integrator<T,F>::AdaptiveGauss( T* ptrT, F f, G4double xInitial,
266 G4d << 278 G4double xFinal, G4double e )
267 { 279 {
268 return AdaptiveGauss(*ptrT, f, xInitial, xFi << 280 return AdaptiveGauss(*ptrT,f,xInitial,xFinal,e) ;
269 } 281 }
270 282
271 ////////////////////////////////////////////// 283 ////////////////////////////////////////////////////////////////////////
272 // 284 //
273 // Adaptive Gauss integration with accuracy 'e 285 // Adaptive Gauss integration with accuracy 'e'
274 // Convenient for using with global scope func 286 // Convenient for using with global scope function f
275 << 287
276 template <class T, class F> 288 template <class T, class F>
277 G4double G4Integrator<T, F>::AdaptiveGauss(G4d << 289 G4double G4Integrator<T,F>::AdaptiveGauss( G4double (*f)(G4double),
278 G4d << 290 G4double xInitial, G4double xFinal, G4double e )
279 G4d << 291 {
280 { << 292 G4int depth = 0 ;
281 G4int depth = 0; << 293 G4double sum = 0.0 ;
282 G4double sum = 0.0; << 294 AdaptGauss(f,xInitial,xFinal,e,sum,depth) ;
283 AdaptGauss(f, xInitial, xFinal, e, sum, dept << 295 return sum ;
284 return sum; <<
285 } 296 }
286 297
287 ////////////////////////////////////////////// 298 ////////////////////////////////////////////////////////////////////////////
288 // Gauss integration methods involving ortogon 299 // Gauss integration methods involving ortogonal polynomials
289 ////////////////////////////////////////////// 300 ////////////////////////////////////////////////////////////////////////////
290 // 301 //
291 // Methods involving Legendre polynomials << 302 // Methods involving Legendre polynomials
292 // 303 //
293 ////////////////////////////////////////////// 304 /////////////////////////////////////////////////////////////////////////
294 // 305 //
295 // The value nLegendre set the accuracy requir 306 // The value nLegendre set the accuracy required, i.e the number of points
296 // where the function pFunction will be evalua 307 // where the function pFunction will be evaluated during integration.
297 // The function creates the arrays for absciss << 308 // The function creates the arrays for abscissas and weights that used
298 // in Gauss-Legendre quadrature method. << 309 // in Gauss-Legendre quadrature method.
299 // The values a and b are the limits of integr 310 // The values a and b are the limits of integration of the function f .
300 // nLegendre MUST BE EVEN !!! 311 // nLegendre MUST BE EVEN !!!
301 // Returns the integral of the function f betw << 312 // Returns the integral of the function f between a and b, by 2*fNumber point
302 // Gauss-Legendre integration: the function is 313 // Gauss-Legendre integration: the function is evaluated exactly
303 // 2*fNumber times at interior points in the r << 314 // 2*fNumber times at interior points in the range of integration.
304 // Since the weights and abscissas are, in thi << 315 // Since the weights and abscissas are, in this case, symmetric around
305 // the midpoint of the range of integration, t << 316 // the midpoint of the range of integration, there are actually only
306 // fNumber distinct values of each. 317 // fNumber distinct values of each.
307 // Convenient for using with some class object 318 // Convenient for using with some class object dataT
308 319
309 template <class T, class F> 320 template <class T, class F>
310 G4double G4Integrator<T, F>::Legendre(T& typeT << 321 G4double G4Integrator<T,F>::Legendre( T& typeT, F f, G4double a, G4double b,
311 G4int nL << 322 G4int nLegendre )
312 { 323 {
313 G4double nwt, nwt1, temp1, temp2, temp3, tem << 324 G4double nwt, nwt1, temp1, temp2, temp3, temp ;
314 G4double xDiff, xMean, dx, integral; << 325 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 } <<
325 326
326 G4double* fAbscissa = new G4double[fNumber]; << 327 const G4double tolerance = 1.6e-10 ;
327 G4double* fWeight = new G4double[fNumber]; << 328 G4int i, j, k = nLegendre ;
328 << 329 G4int fNumber = (nLegendre + 1)/2 ;
329 for(i = 1; i <= fNumber; ++i) // Loop over << 330
330 { << 331 if(2*fNumber != k)
331 nwt = std::cos(CLHEP::pi * (i - 0.25) / << 332 {
332 (k + 0.5)); // Initial roo << 333 G4Exception("G4Integrator<T,F>::Legendre(T&,F, ...)", "InvalidCall",
333 << 334 FatalException, "Invalid (odd) nLegendre in constructor.");
334 do // loop of Newton's method << 335 }
335 { << 336
336 temp1 = 1.0; << 337 G4double* fAbscissa = new G4double[fNumber] ;
337 temp2 = 0.0; << 338 G4double* fWeight = new G4double[fNumber] ;
338 for(j = 1; j <= k; ++j) << 339
339 { << 340 for(i=1;i<=fNumber;i++) // Loop over the desired roots
340 temp3 = temp2; << 341 {
341 temp2 = temp1; << 342 nwt = std::cos(CLHEP::pi*(i - 0.25)/(k + 0.5)) ; // Initial root approximation
342 temp1 = ((2.0 * j - 1.0) * nwt * temp2 << 343
343 } << 344 do // loop of Newton's method
344 temp = k * (nwt * temp1 - temp2) / (nwt << 345 {
345 nwt1 = nwt; << 346 temp1 = 1.0 ;
346 nwt = nwt1 - temp1 / temp; // Newton's << 347 temp2 = 0.0 ;
347 } while(std::fabs(nwt - nwt1) > tolerance) << 348 for(j=1;j<=k;j++)
348 << 349 {
349 fAbscissa[fNumber - i] = nwt; << 350 temp3 = temp2 ;
350 fWeight[fNumber - i] = 2.0 / ((1.0 - nwt << 351 temp2 = temp1 ;
351 } << 352 temp1 = ((2.0*j - 1.0)*nwt*temp2 - (j - 1.0)*temp3)/j ;
352 << 353 }
353 // << 354 temp = k*(nwt*temp1 - temp2)/(nwt*nwt - 1.0) ;
354 // Now we ready to get integral << 355 nwt1 = nwt ;
355 // << 356 nwt = nwt1 - temp1/temp ; // Newton's method
356 << 357 }
357 xMean = 0.5 * (a + b); << 358 while(std::fabs(nwt - nwt1) > tolerance) ;
358 xDiff = 0.5 * (b - a); << 359
359 integral = 0.0; << 360 fAbscissa[fNumber-i] = nwt ;
360 for(i = 0; i < fNumber; ++i) << 361 fWeight[fNumber-i] = 2.0/((1.0 - nwt*nwt)*temp*temp) ;
361 { << 362 }
362 dx = xDiff * fAbscissa[i]; << 363
363 integral += fWeight[i] * ((typeT.*f)(xMean << 364 //
364 } << 365 // Now we ready to get integral
365 delete[] fAbscissa; << 366 //
366 delete[] fWeight; << 367
367 return integral *= xDiff; << 368 xMean = 0.5*(a + b) ;
368 } << 369 xDiff = 0.5*(b - a) ;
>> 370 integral = 0.0 ;
>> 371 for(i=0;i<fNumber;i++)
>> 372 {
>> 373 dx = xDiff*fAbscissa[i] ;
>> 374 integral += fWeight[i]*( (typeT.*f)(xMean + dx) +
>> 375 (typeT.*f)(xMean - dx) ) ;
>> 376 }
>> 377 delete[] fAbscissa;
>> 378 delete[] fWeight;
>> 379 return integral *= xDiff ;
>> 380 }
369 381
370 ////////////////////////////////////////////// 382 ///////////////////////////////////////////////////////////////////////
371 // 383 //
372 // Convenient for using with the pointer 'this 384 // Convenient for using with the pointer 'this'
373 385
374 template <class T, class F> 386 template <class T, class F>
375 G4double G4Integrator<T, F>::Legendre(T* ptrT, << 387 G4double G4Integrator<T,F>::Legendre( T* ptrT, F f, G4double a,
376 G4int nL << 388 G4double b, G4int nLegendre )
377 { 389 {
378 return Legendre(*ptrT, f, a, b, nLegendre); << 390 return Legendre(*ptrT,f,a,b,nLegendre) ;
379 } 391 }
380 392
381 ////////////////////////////////////////////// 393 ///////////////////////////////////////////////////////////////////////
382 // 394 //
383 // Convenient for using with global scope func 395 // Convenient for using with global scope function f
384 396
385 template <class T, class F> 397 template <class T, class F>
386 G4double G4Integrator<T, F>::Legendre(G4double << 398 G4double G4Integrator<T,F>::Legendre( G4double (*f)(G4double),
387 G4double << 399 G4double a, G4double b, G4int nLegendre)
388 { 400 {
389 G4double nwt, nwt1, temp1, temp2, temp3, tem << 401 G4double nwt, nwt1, temp1, temp2, temp3, temp ;
390 G4double xDiff, xMean, dx, integral; << 402 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 } <<
401 403
402 G4double* fAbscissa = new G4double[fNumber]; << 404 const G4double tolerance = 1.6e-10 ;
403 G4double* fWeight = new G4double[fNumber]; << 405 G4int i, j, k = nLegendre ;
404 << 406 G4int fNumber = (nLegendre + 1)/2 ;
405 for(i = 1; i <= fNumber; i++) // Loop over << 407
406 { << 408 if(2*fNumber != k)
407 nwt = std::cos(CLHEP::pi * (i - 0.25) / << 409 {
408 (k + 0.5)); // Initial roo << 410 G4Exception("G4Integrator<T,F>::Legendre(...)", "InvalidCall",
409 << 411 FatalException, "Invalid (odd) nLegendre in constructor.");
410 do // loop of Newton's method << 412 }
411 { << 413
412 temp1 = 1.0; << 414 G4double* fAbscissa = new G4double[fNumber] ;
413 temp2 = 0.0; << 415 G4double* fWeight = new G4double[fNumber] ;
414 for(j = 1; j <= k; ++j) << 416
415 { << 417 for(i=1;i<=fNumber;i++) // Loop over the desired roots
416 temp3 = temp2; << 418 {
417 temp2 = temp1; << 419 nwt = std::cos(CLHEP::pi*(i - 0.25)/(k + 0.5)) ; // Initial root approximation
418 temp1 = ((2.0 * j - 1.0) * nwt * temp2 << 420
419 } << 421 do // loop of Newton's method
420 temp = k * (nwt * temp1 - temp2) / (nwt << 422 {
421 nwt1 = nwt; << 423 temp1 = 1.0 ;
422 nwt = nwt1 - temp1 / temp; // Newton's << 424 temp2 = 0.0 ;
423 } while(std::fabs(nwt - nwt1) > tolerance) << 425 for(j=1;j<=k;j++)
424 << 426 {
425 fAbscissa[fNumber - i] = nwt; << 427 temp3 = temp2 ;
426 fWeight[fNumber - i] = 2.0 / ((1.0 - nwt << 428 temp2 = temp1 ;
427 } << 429 temp1 = ((2.0*j - 1.0)*nwt*temp2 - (j - 1.0)*temp3)/j ;
428 << 430 }
429 // << 431 temp = k*(nwt*temp1 - temp2)/(nwt*nwt - 1.0) ;
430 // Now we ready to get integral << 432 nwt1 = nwt ;
431 // << 433 nwt = nwt1 - temp1/temp ; // Newton's method
432 << 434 }
433 xMean = 0.5 * (a + b); << 435 while(std::fabs(nwt - nwt1) > tolerance) ;
434 xDiff = 0.5 * (b - a); << 436
435 integral = 0.0; << 437 fAbscissa[fNumber-i] = nwt ;
436 for(i = 0; i < fNumber; ++i) << 438 fWeight[fNumber-i] = 2.0/((1.0 - nwt*nwt)*temp*temp) ;
437 { << 439 }
438 dx = xDiff * fAbscissa[i]; << 440
439 integral += fWeight[i] * ((*f)(xMean + dx) << 441 //
440 } << 442 // Now we ready to get integral
441 delete[] fAbscissa; << 443 //
442 delete[] fWeight; << 444
>> 445 xMean = 0.5*(a + b) ;
>> 446 xDiff = 0.5*(b - a) ;
>> 447 integral = 0.0 ;
>> 448 for(i=0;i<fNumber;i++)
>> 449 {
>> 450 dx = xDiff*fAbscissa[i] ;
>> 451 integral += fWeight[i]*( (*f)(xMean + dx) + (*f)(xMean - dx) ) ;
>> 452 }
>> 453 delete[] fAbscissa;
>> 454 delete[] fWeight;
443 455
444 return integral *= xDiff; << 456 return integral *= xDiff ;
445 } << 457 }
446 458
447 ////////////////////////////////////////////// 459 ////////////////////////////////////////////////////////////////////////////
448 // 460 //
449 // Returns the integral of the function to be 461 // Returns the integral of the function to be pointed by T::f between a and b,
450 // by ten point Gauss-Legendre integration: th 462 // by ten point Gauss-Legendre integration: the function is evaluated exactly
451 // ten times at interior points in the range o 463 // ten times at interior points in the range of integration. Since the weights
452 // and abscissas are, in this case, symmetric << 464 // and abscissas are, in this case, symmetric around the midpoint of the
453 // range of integration, there are actually on 465 // range of integration, there are actually only five distinct values of each
454 // Convenient for using with class object type 466 // Convenient for using with class object typeT
455 467
456 template <class T, class F> << 468 template <class T, class F>
457 G4double G4Integrator<T, F>::Legendre10(T& typ << 469 G4double G4Integrator<T,F>::Legendre10( T& typeT, F f,G4double a, G4double b)
458 { 470 {
459 G4int i; << 471 G4int i ;
460 G4double xDiff, xMean, dx, integral; << 472 G4double xDiff, xMean, dx, integral ;
461 << 473
462 // From Abramowitz M., Stegan I.A. 1964 , Ha << 474 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 916
463 << 475
464 static const G4double abscissa[] = { 0.14887 << 476 static const G4double abscissa[] = { 0.148874338981631, 0.433395394129247,
465 0.67940 << 477 0.679409568299024, 0.865063366688985,
466 0.97390 << 478 0.973906528517172 } ;
467 << 479
468 static const G4double weight[] = { 0.2955242 << 480 static const G4double weight[] = { 0.295524224714753, 0.269266719309996,
469 0.2190863 << 481 0.219086362515982, 0.149451349150581,
470 0.0666713 << 482 0.066671344308688 } ;
471 xMean = 0.5 * (a + << 483 xMean = 0.5*(a + b) ;
472 xDiff = 0.5 * (b - << 484 xDiff = 0.5*(b - a) ;
473 integral = 0.0; << 485 integral = 0.0 ;
474 for(i = 0; i < 5; ++i) << 486 for(i=0;i<5;i++)
475 { << 487 {
476 dx = xDiff * abscissa[i]; << 488 dx = xDiff*abscissa[i] ;
477 integral += weight[i] * ((typeT.*f)(xMean << 489 integral += weight[i]*( (typeT.*f)(xMean + dx) + (typeT.*f)(xMean - dx)) ;
478 } << 490 }
479 return integral *= xDiff; << 491 return integral *= xDiff ;
480 } 492 }
481 493
482 ////////////////////////////////////////////// 494 ///////////////////////////////////////////////////////////////////////////
483 // 495 //
484 // Convenient for using with the pointer 'this 496 // Convenient for using with the pointer 'this'
485 497
486 template <class T, class F> << 498 template <class T, class F>
487 G4double G4Integrator<T, F>::Legendre10(T* ptr << 499 G4double G4Integrator<T,F>::Legendre10( T* ptrT, F f,G4double a, G4double b)
488 { 500 {
489 return Legendre10(*ptrT, f, a, b); << 501 return Legendre10(*ptrT,f,a,b) ;
490 } << 502 }
491 503
492 ////////////////////////////////////////////// 504 //////////////////////////////////////////////////////////////////////////
493 // 505 //
494 // Convenient for using with global scope func 506 // Convenient for using with global scope functions
495 507
496 template <class T, class F> 508 template <class T, class F>
497 G4double G4Integrator<T, F>::Legendre10(G4doub << 509 G4double G4Integrator<T,F>::Legendre10( G4double (*f)(G4double),
498 G4doub << 510 G4double a, G4double b )
499 { 511 {
500 G4int i; << 512 G4int i ;
501 G4double xDiff, xMean, dx, integral; << 513 G4double xDiff, xMean, dx, integral ;
502 << 514
503 // From Abramowitz M., Stegan I.A. 1964 , Ha << 515 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 916
504 << 516
505 static const G4double abscissa[] = { 0.14887 << 517 static const G4double abscissa[] = { 0.148874338981631, 0.433395394129247,
506 0.67940 << 518 0.679409568299024, 0.865063366688985,
507 0.97390 << 519 0.973906528517172 } ;
508 << 520
509 static const G4double weight[] = { 0.2955242 << 521 static const G4double weight[] = { 0.295524224714753, 0.269266719309996,
510 0.2190863 << 522 0.219086362515982, 0.149451349150581,
511 0.0666713 << 523 0.066671344308688 } ;
512 xMean = 0.5 * (a + << 524 xMean = 0.5*(a + b) ;
513 xDiff = 0.5 * (b - << 525 xDiff = 0.5*(b - a) ;
514 integral = 0.0; << 526 integral = 0.0 ;
515 for(i = 0; i < 5; ++i) << 527 for(i=0;i<5;i++)
516 { << 528 {
517 dx = xDiff * abscissa[i]; << 529 dx = xDiff*abscissa[i] ;
518 integral += weight[i] * ((*f)(xMean + dx) << 530 integral += weight[i]*( (*f)(xMean + dx) + (*f)(xMean - dx)) ;
519 } << 531 }
520 return integral *= xDiff; << 532 return integral *= xDiff ;
521 } 533 }
522 534
523 ////////////////////////////////////////////// 535 ///////////////////////////////////////////////////////////////////////
524 // 536 //
525 // Returns the integral of the function to be 537 // Returns the integral of the function to be pointed by T::f between a and b,
526 // by 96 point Gauss-Legendre integration: the 538 // by 96 point Gauss-Legendre integration: the function is evaluated exactly
527 // ten Times at interior points in the range o 539 // ten Times at interior points in the range of integration. Since the weights
528 // and abscissas are, in this case, symmetric << 540 // and abscissas are, in this case, symmetric around the midpoint of the
529 // range of integration, there are actually on 541 // range of integration, there are actually only five distinct values of each
530 // Convenient for using with some class object 542 // Convenient for using with some class object typeT
531 543
532 template <class T, class F> << 544 template <class T, class F>
533 G4double G4Integrator<T, F>::Legendre96(T& typ << 545 G4double G4Integrator<T,F>::Legendre96( T& typeT, F f,G4double a, G4double b)
534 { 546 {
535 G4int i; << 547 G4int i ;
536 G4double xDiff, xMean, dx, integral; << 548 G4double xDiff, xMean, dx, integral ;
537 << 549
538 // From Abramowitz M., Stegan I.A. 1964 , Ha << 550 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 919
539 << 551
540 static const G4double abscissa[] = { << 552 static const G4double
541 0.016276744849602969579, 0.048812985136049 << 553 abscissa[] = {
542 0.081297495464425558994, 0.113695850110665 << 554 0.016276744849602969579, 0.048812985136049731112,
543 0.145973714654896941989, 0.178096882367618 << 555 0.081297495464425558994, 0.113695850110665920911,
544 << 556 0.145973714654896941989, 0.178096882367618602759, // 6
545 0.210031310460567203603, 0.241743156163840 << 557
546 0.273198812591049141487, 0.304364944354496 << 558 0.210031310460567203603, 0.241743156163840012328,
547 0.335208522892625422616, 0.365696861472313 << 559 0.273198812591049141487, 0.304364944354496353024,
548 << 560 0.335208522892625422616, 0.365696861472313635031, // 12
549 0.395797649828908603285, 0.425478988407300 << 561
550 0.454709422167743008636, 0.483457973920596 << 562 0.395797649828908603285, 0.425478988407300545365,
551 0.511694177154667673586, 0.539388108324357 << 563 0.454709422167743008636, 0.483457973920596359768,
552 << 564 0.511694177154667673586, 0.539388108324357436227, // 18
553 0.566510418561397168404, 0.593032364777572 << 565
554 0.618925840125468570386, 0.644163403784967 << 566 0.566510418561397168404, 0.593032364777572080684,
555 0.668718310043916153953, 0.692564536642171 << 567 0.618925840125468570386, 0.644163403784967106798,
556 << 568 0.668718310043916153953, 0.692564536642171561344, // 24
557 0.715676812348967626225, 0.738030643744400 << 569
558 0.759602341176647498703, 0.780369043867433 << 570 0.715676812348967626225, 0.738030643744400132851,
559 0.800308744139140817229, 0.819400310737931 << 571 0.759602341176647498703, 0.780369043867433217604,
560 << 572 0.800308744139140817229, 0.819400310737931675539, // 30
561 0.837623511228187121494, 0.854959033434601 << 573
562 0.871388505909296502874, 0.886894517402420 << 574 0.837623511228187121494, 0.854959033434601455463,
563 0.901460635315852341319, 0.915071423120898 << 575 0.871388505909296502874, 0.886894517402420416057,
564 << 576 0.901460635315852341319, 0.915071423120898074206, // 36
565 0.927712456722308690965, 0.939370339752755 << 577
566 0.950032717784437635756, 0.959688291448742 << 578 0.927712456722308690965, 0.939370339752755216932,
567 0.968326828463264212174, 0.975939174585136 << 579 0.950032717784437635756, 0.959688291448742539300,
568 << 580 0.968326828463264212174, 0.975939174585136466453, // 42
569 0.982517263563014677447, 0.988054126329623 << 581
570 0.992543900323762624572, 0.995981842987209 << 582 0.982517263563014677447, 0.988054126329623799481,
571 0.998364375863181677724, 0.999689503883230 << 583 0.992543900323762624572, 0.995981842987209290650,
572 }; << 584 0.998364375863181677724, 0.999689503883230766828 // 48
573 << 585 } ;
574 static const G4double weight[] = { << 586
575 0.032550614492363166242, 0.032516118713868 << 587 static const G4double
576 0.032447163714064269364, 0.032343822568575 << 588 weight[] = {
577 0.032206204794030250669, 0.032034456231992 << 589 0.032550614492363166242, 0.032516118713868835987,
578 << 590 0.032447163714064269364, 0.032343822568575928429,
579 0.031828758894411006535, 0.031589330770727 << 591 0.032206204794030250669, 0.032034456231992663218, // 6
580 0.031316425596862355813, 0.031010332586313 << 592
581 0.030671376123669149014, 0.030299915420827 << 593 0.031828758894411006535, 0.031589330770727168558,
582 << 594 0.031316425596862355813, 0.031010332586313837423,
583 0.029896344136328385984, 0.029461089958167 << 595 0.030671376123669149014, 0.030299915420827593794, // 12
584 0.028994614150555236543, 0.028497411065085 << 596
585 0.027970007616848334440, 0.027412962726029 << 597 0.029896344136328385984, 0.029461089958167905970,
586 << 598 0.028994614150555236543, 0.028497411065085385646,
587 0.026826866725591762198, 0.026212340735672 << 599 0.027970007616848334440, 0.027412962726029242823, // 18
588 0.025570036005349361499, 0.024900633222483 << 600
589 0.024204841792364691282, 0.023483399085926 << 601 0.026826866725591762198, 0.026212340735672413913,
590 << 602 0.025570036005349361499, 0.024900633222483610288,
591 0.022737069658329374001, 0.021966644438744 << 603 0.024204841792364691282, 0.023483399085926219842, // 24
592 0.021172939892191298988, 0.020356797154333 << 604
593 0.019519081140145022410, 0.018660679627411 << 605 0.022737069658329374001, 0.021966644438744349195,
594 << 606 0.021172939892191298988, 0.020356797154333324595,
595 0.017782502316045260838, 0.016885479864245 << 607 0.019519081140145022410, 0.018660679627411467385, // 30
596 0.015970562902562291381, 0.015038721026994 << 608
597 0.014090941772314860916, 0.013128229566961 << 609 0.017782502316045260838, 0.016885479864245172450,
598 << 610 0.015970562902562291381, 0.015038721026994938006,
599 0.012151604671088319635, 0.011162102099838 << 611 0.014090941772314860916, 0.013128229566961572637, // 36
600 0.010160770535008415758, 0.009148671230783 << 612
601 0.008126876925698759217, 0.007096470791153 << 613 0.012151604671088319635, 0.011162102099838498591,
602 << 614 0.010160770535008415758, 0.009148671230783386633,
603 0.006058545504235961683, 0.005014202742927 << 615 0.008126876925698759217, 0.007096470791153865269, // 42
604 0.003964554338444686674, 0.002910731817934 << 616
605 0.001853960788946921732, 0.000796792065552 << 617 0.006058545504235961683, 0.005014202742927517693,
606 }; << 618 0.003964554338444686674, 0.002910731817934946408,
607 xMean = 0.5 * (a + b); << 619 0.001853960788946921732, 0.000796792065552012429 // 48
608 xDiff = 0.5 * (b - a); << 620 } ;
609 integral = 0.0; << 621 xMean = 0.5*(a + b) ;
610 for(i = 0; i < 48; ++i) << 622 xDiff = 0.5*(b - a) ;
611 { << 623 integral = 0.0 ;
612 dx = xDiff * abscissa[i]; << 624 for(i=0;i<48;i++)
613 integral += weight[i] * ((typeT.*f)(xMean << 625 {
614 } << 626 dx = xDiff*abscissa[i] ;
615 return integral *= xDiff; << 627 integral += weight[i]*((typeT.*f)(xMean + dx) + (typeT.*f)(xMean - dx)) ;
>> 628 }
>> 629 return integral *= xDiff ;
616 } 630 }
617 631
618 ////////////////////////////////////////////// 632 ///////////////////////////////////////////////////////////////////////
619 // 633 //
620 // Convenient for using with the pointer 'this 634 // Convenient for using with the pointer 'this'
621 635
622 template <class T, class F> << 636 template <class T, class F>
623 G4double G4Integrator<T, F>::Legendre96(T* ptr << 637 G4double G4Integrator<T,F>::Legendre96( T* ptrT, F f,G4double a, G4double b)
624 { 638 {
625 return Legendre96(*ptrT, f, a, b); << 639 return Legendre96(*ptrT,f,a,b) ;
626 } << 640 }
627 641
628 ////////////////////////////////////////////// 642 ///////////////////////////////////////////////////////////////////////
629 // 643 //
630 // Convenient for using with global scope func << 644 // Convenient for using with global scope function f
631 645
632 template <class T, class F> 646 template <class T, class F>
633 G4double G4Integrator<T, F>::Legendre96(G4doub << 647 G4double G4Integrator<T,F>::Legendre96( G4double (*f)(G4double),
634 G4doub << 648 G4double a, G4double b )
635 { 649 {
636 G4int i; << 650 G4int i ;
637 G4double xDiff, xMean, dx, integral; << 651 G4double xDiff, xMean, dx, integral ;
638 << 652
639 // From Abramowitz M., Stegan I.A. 1964 , Ha << 653 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 919
640 << 654
641 static const G4double abscissa[] = { << 655 static const G4double
642 0.016276744849602969579, 0.048812985136049 << 656 abscissa[] = {
643 0.081297495464425558994, 0.113695850110665 << 657 0.016276744849602969579, 0.048812985136049731112,
644 0.145973714654896941989, 0.178096882367618 << 658 0.081297495464425558994, 0.113695850110665920911,
645 << 659 0.145973714654896941989, 0.178096882367618602759, // 6
646 0.210031310460567203603, 0.241743156163840 << 660
647 0.273198812591049141487, 0.304364944354496 << 661 0.210031310460567203603, 0.241743156163840012328,
648 0.335208522892625422616, 0.365696861472313 << 662 0.273198812591049141487, 0.304364944354496353024,
649 << 663 0.335208522892625422616, 0.365696861472313635031, // 12
650 0.395797649828908603285, 0.425478988407300 << 664
651 0.454709422167743008636, 0.483457973920596 << 665 0.395797649828908603285, 0.425478988407300545365,
652 0.511694177154667673586, 0.539388108324357 << 666 0.454709422167743008636, 0.483457973920596359768,
653 << 667 0.511694177154667673586, 0.539388108324357436227, // 18
654 0.566510418561397168404, 0.593032364777572 << 668
655 0.618925840125468570386, 0.644163403784967 << 669 0.566510418561397168404, 0.593032364777572080684,
656 0.668718310043916153953, 0.692564536642171 << 670 0.618925840125468570386, 0.644163403784967106798,
657 << 671 0.668718310043916153953, 0.692564536642171561344, // 24
658 0.715676812348967626225, 0.738030643744400 << 672
659 0.759602341176647498703, 0.780369043867433 << 673 0.715676812348967626225, 0.738030643744400132851,
660 0.800308744139140817229, 0.819400310737931 << 674 0.759602341176647498703, 0.780369043867433217604,
661 << 675 0.800308744139140817229, 0.819400310737931675539, // 30
662 0.837623511228187121494, 0.854959033434601 << 676
663 0.871388505909296502874, 0.886894517402420 << 677 0.837623511228187121494, 0.854959033434601455463,
664 0.901460635315852341319, 0.915071423120898 << 678 0.871388505909296502874, 0.886894517402420416057,
665 << 679 0.901460635315852341319, 0.915071423120898074206, // 36
666 0.927712456722308690965, 0.939370339752755 << 680
667 0.950032717784437635756, 0.959688291448742 << 681 0.927712456722308690965, 0.939370339752755216932,
668 0.968326828463264212174, 0.975939174585136 << 682 0.950032717784437635756, 0.959688291448742539300,
669 << 683 0.968326828463264212174, 0.975939174585136466453, // 42
670 0.982517263563014677447, 0.988054126329623 << 684
671 0.992543900323762624572, 0.995981842987209 << 685 0.982517263563014677447, 0.988054126329623799481,
672 0.998364375863181677724, 0.999689503883230 << 686 0.992543900323762624572, 0.995981842987209290650,
673 }; << 687 0.998364375863181677724, 0.999689503883230766828 // 48
674 << 688 } ;
675 static const G4double weight[] = { << 689
676 0.032550614492363166242, 0.032516118713868 << 690 static const G4double
677 0.032447163714064269364, 0.032343822568575 << 691 weight[] = {
678 0.032206204794030250669, 0.032034456231992 << 692 0.032550614492363166242, 0.032516118713868835987,
679 << 693 0.032447163714064269364, 0.032343822568575928429,
680 0.031828758894411006535, 0.031589330770727 << 694 0.032206204794030250669, 0.032034456231992663218, // 6
681 0.031316425596862355813, 0.031010332586313 << 695
682 0.030671376123669149014, 0.030299915420827 << 696 0.031828758894411006535, 0.031589330770727168558,
683 << 697 0.031316425596862355813, 0.031010332586313837423,
684 0.029896344136328385984, 0.029461089958167 << 698 0.030671376123669149014, 0.030299915420827593794, // 12
685 0.028994614150555236543, 0.028497411065085 << 699
686 0.027970007616848334440, 0.027412962726029 << 700 0.029896344136328385984, 0.029461089958167905970,
687 << 701 0.028994614150555236543, 0.028497411065085385646,
688 0.026826866725591762198, 0.026212340735672 << 702 0.027970007616848334440, 0.027412962726029242823, // 18
689 0.025570036005349361499, 0.024900633222483 << 703
690 0.024204841792364691282, 0.023483399085926 << 704 0.026826866725591762198, 0.026212340735672413913,
691 << 705 0.025570036005349361499, 0.024900633222483610288,
692 0.022737069658329374001, 0.021966644438744 << 706 0.024204841792364691282, 0.023483399085926219842, // 24
693 0.021172939892191298988, 0.020356797154333 << 707
694 0.019519081140145022410, 0.018660679627411 << 708 0.022737069658329374001, 0.021966644438744349195,
695 << 709 0.021172939892191298988, 0.020356797154333324595,
696 0.017782502316045260838, 0.016885479864245 << 710 0.019519081140145022410, 0.018660679627411467385, // 30
697 0.015970562902562291381, 0.015038721026994 << 711
698 0.014090941772314860916, 0.013128229566961 << 712 0.017782502316045260838, 0.016885479864245172450,
699 << 713 0.015970562902562291381, 0.015038721026994938006,
700 0.012151604671088319635, 0.011162102099838 << 714 0.014090941772314860916, 0.013128229566961572637, // 36
701 0.010160770535008415758, 0.009148671230783 << 715
702 0.008126876925698759217, 0.007096470791153 << 716 0.012151604671088319635, 0.011162102099838498591,
703 << 717 0.010160770535008415758, 0.009148671230783386633,
704 0.006058545504235961683, 0.005014202742927 << 718 0.008126876925698759217, 0.007096470791153865269, // 42
705 0.003964554338444686674, 0.002910731817934 << 719
706 0.001853960788946921732, 0.000796792065552 << 720 0.006058545504235961683, 0.005014202742927517693,
707 }; << 721 0.003964554338444686674, 0.002910731817934946408,
708 xMean = 0.5 * (a + b); << 722 0.001853960788946921732, 0.000796792065552012429 // 48
709 xDiff = 0.5 * (b - a); << 723 } ;
710 integral = 0.0; << 724 xMean = 0.5*(a + b) ;
711 for(i = 0; i < 48; ++i) << 725 xDiff = 0.5*(b - a) ;
712 { << 726 integral = 0.0 ;
713 dx = xDiff * abscissa[i]; << 727 for(i=0;i<48;i++)
714 integral += weight[i] * ((*f)(xMean + dx) << 728 {
715 } << 729 dx = xDiff*abscissa[i] ;
716 return integral *= xDiff; << 730 integral += weight[i]*((*f)(xMean + dx) + (*f)(xMean - dx)) ;
>> 731 }
>> 732 return integral *= xDiff ;
717 } 733 }
718 734
719 ////////////////////////////////////////////// 735 //////////////////////////////////////////////////////////////////////////////
720 // 736 //
721 // Methods involving Chebyshev polynomials << 737 // Methods involving Chebyshev polynomials
722 // 738 //
723 ////////////////////////////////////////////// 739 ///////////////////////////////////////////////////////////////////////////
724 // 740 //
725 // Integrates function pointed by T::f from a << 741 // Integrates function pointed by T::f from a to b by Gauss-Chebyshev
726 // quadrature method. 742 // quadrature method.
727 // Convenient for using with class object type 743 // Convenient for using with class object typeT
728 744
729 template <class T, class F> 745 template <class T, class F>
730 G4double G4Integrator<T, F>::Chebyshev(T& type << 746 G4double G4Integrator<T,F>::Chebyshev( T& typeT, F f, G4double a,
731 G4int n << 747 G4double b, G4int nChebyshev )
732 { 748 {
733 G4int i; << 749 G4int i ;
734 G4double xDiff, xMean, dx, integral = 0.0; << 750 G4double xDiff, xMean, dx, integral = 0.0 ;
735 << 751
736 G4int fNumber = nChebyshev; // Try to << 752 G4int fNumber = nChebyshev ; // Try to reduce fNumber twice ??
737 G4double cof = CLHEP::pi / fNumber; << 753 G4double cof = CLHEP::pi/fNumber ;
738 G4double* fAbscissa = new G4double[fNumber]; << 754 G4double* fAbscissa = new G4double[fNumber] ;
739 G4double* fWeight = new G4double[fNumber]; << 755 G4double* fWeight = new G4double[fNumber] ;
740 for(i = 0; i < fNumber; ++i) << 756 for(i=0;i<fNumber;i++)
741 { << 757 {
742 fAbscissa[i] = std::cos(cof * (i + 0.5)); << 758 fAbscissa[i] = std::cos(cof*(i + 0.5)) ;
743 fWeight[i] = cof * std::sqrt(1 - fAbscis << 759 fWeight[i] = cof*std::sqrt(1 - fAbscissa[i]*fAbscissa[i]) ;
744 } << 760 }
745 << 761
746 // << 762 //
747 // Now we ready to estimate the integral << 763 // Now we ready to estimate the integral
748 // << 764 //
749 << 765
750 xMean = 0.5 * (a + b); << 766 xMean = 0.5*(a + b) ;
751 xDiff = 0.5 * (b - a); << 767 xDiff = 0.5*(b - a) ;
752 for(i = 0; i < fNumber; ++i) << 768 for(i=0;i<fNumber;i++)
753 { << 769 {
754 dx = xDiff * fAbscissa[i]; << 770 dx = xDiff*fAbscissa[i] ;
755 integral += fWeight[i] * (typeT.*f)(xMean << 771 integral += fWeight[i]*(typeT.*f)(xMean + dx) ;
756 } << 772 }
757 delete[] fAbscissa; << 773 delete[] fAbscissa;
758 delete[] fWeight; << 774 delete[] fWeight;
759 return integral *= xDiff; << 775 return integral *= xDiff ;
760 } 776 }
761 777
762 ////////////////////////////////////////////// 778 ///////////////////////////////////////////////////////////////////////
763 // 779 //
764 // Convenient for using with 'this' pointer 780 // Convenient for using with 'this' pointer
765 781
766 template <class T, class F> 782 template <class T, class F>
767 G4double G4Integrator<T, F>::Chebyshev(T* ptrT << 783 G4double G4Integrator<T,F>::Chebyshev( T* ptrT, F f, G4double a,
768 G4int n << 784 G4double b, G4int n )
769 { 785 {
770 return Chebyshev(*ptrT, f, a, b, n); << 786 return Chebyshev(*ptrT,f,a,b,n) ;
771 } << 787 }
772 788
773 ////////////////////////////////////////////// 789 ////////////////////////////////////////////////////////////////////////
774 // 790 //
775 // For use with global scope functions f << 791 // For use with global scope functions f
776 792
777 template <class T, class F> 793 template <class T, class F>
778 G4double G4Integrator<T, F>::Chebyshev(G4doubl << 794 G4double G4Integrator<T,F>::Chebyshev( G4double (*f)(G4double),
779 G4doubl << 795 G4double a, G4double b, G4int nChebyshev )
780 { 796 {
781 G4int i; << 797 G4int i ;
782 G4double xDiff, xMean, dx, integral = 0.0; << 798 G4double xDiff, xMean, dx, integral = 0.0 ;
783 << 799
784 G4int fNumber = nChebyshev; // Try to << 800 G4int fNumber = nChebyshev ; // Try to reduce fNumber twice ??
785 G4double cof = CLHEP::pi / fNumber; << 801 G4double cof = CLHEP::pi/fNumber ;
786 G4double* fAbscissa = new G4double[fNumber]; << 802 G4double* fAbscissa = new G4double[fNumber] ;
787 G4double* fWeight = new G4double[fNumber]; << 803 G4double* fWeight = new G4double[fNumber] ;
788 for(i = 0; i < fNumber; ++i) << 804 for(i=0;i<fNumber;i++)
789 { << 805 {
790 fAbscissa[i] = std::cos(cof * (i + 0.5)); << 806 fAbscissa[i] = std::cos(cof*(i + 0.5)) ;
791 fWeight[i] = cof * std::sqrt(1 - fAbscis << 807 fWeight[i] = cof*std::sqrt(1 - fAbscissa[i]*fAbscissa[i]) ;
792 } << 808 }
793 << 809
794 // << 810 //
795 // Now we ready to estimate the integral << 811 // Now we ready to estimate the integral
796 // << 812 //
797 << 813
798 xMean = 0.5 * (a + b); << 814 xMean = 0.5*(a + b) ;
799 xDiff = 0.5 * (b - a); << 815 xDiff = 0.5*(b - a) ;
800 for(i = 0; i < fNumber; ++i) << 816 for(i=0;i<fNumber;i++)
801 { << 817 {
802 dx = xDiff * fAbscissa[i]; << 818 dx = xDiff*fAbscissa[i] ;
803 integral += fWeight[i] * (*f)(xMean + dx); << 819 integral += fWeight[i]*(*f)(xMean + dx) ;
804 } << 820 }
805 delete[] fAbscissa; << 821 delete[] fAbscissa;
806 delete[] fWeight; << 822 delete[] fWeight;
807 return integral *= xDiff; << 823 return integral *= xDiff ;
808 } 824 }
809 825
810 ////////////////////////////////////////////// 826 //////////////////////////////////////////////////////////////////////
811 // 827 //
812 // Method involving Laguerre polynomials 828 // Method involving Laguerre polynomials
813 // 829 //
814 ////////////////////////////////////////////// 830 //////////////////////////////////////////////////////////////////////
815 // 831 //
816 // Integral from zero to infinity of std::pow( << 832 // Integral from zero to infinity of std::pow(x,alpha)*std::exp(-x)*f(x).
817 // The value of nLaguerre sets the accuracy. 833 // The value of nLaguerre sets the accuracy.
818 // The function creates arrays fAbscissa[0,.., << 834 // The function creates arrays fAbscissa[0,..,nLaguerre-1] and
819 // fWeight[0,..,nLaguerre-1] . << 835 // fWeight[0,..,nLaguerre-1] .
820 // Convenient for using with class object 'typ 836 // Convenient for using with class object 'typeT' and (typeT.*f) function
821 // (T::f) 837 // (T::f)
822 838
823 template <class T, class F> 839 template <class T, class F>
824 G4double G4Integrator<T, F>::Laguerre(T& typeT << 840 G4double G4Integrator<T,F>::Laguerre( T& typeT, F f, G4double alpha,
825 G4int nL << 841 G4int nLaguerre )
826 { 842 {
827 const G4double tolerance = 1.0e-10; << 843 const G4double tolerance = 1.0e-10 ;
828 const G4int maxNumber = 12; << 844 const G4int maxNumber = 12 ;
829 G4int i, j, k; << 845 G4int i, j, k ;
830 G4double nwt = 0., nwt1, temp1, temp2, << 846 G4double nwt=0., nwt1, temp1, temp2, temp3, temp, cofi ;
831 G4double integral = 0.0; << 847 G4double integral = 0.0 ;
832 << 848
833 G4int fNumber = nLaguerre; << 849 G4int fNumber = nLaguerre ;
834 G4double* fAbscissa = new G4double[fNumber]; << 850 G4double* fAbscissa = new G4double[fNumber] ;
835 G4double* fWeight = new G4double[fNumber]; << 851 G4double* fWeight = new G4double[fNumber] ;
>> 852
>> 853 for(i=1;i<=fNumber;i++) // Loop over the desired roots
>> 854 {
>> 855 if(i == 1)
>> 856 {
>> 857 nwt = (1.0 + alpha)*(3.0 + 0.92*alpha)
>> 858 / (1.0 + 2.4*fNumber + 1.8*alpha) ;
>> 859 }
>> 860 else if(i == 2)
>> 861 {
>> 862 nwt += (15.0 + 6.25*alpha)/(1.0 + 0.9*alpha + 2.5*fNumber) ;
>> 863 }
>> 864 else
>> 865 {
>> 866 cofi = i - 2 ;
>> 867 nwt += ((1.0+2.55*cofi)/(1.9*cofi)
>> 868 + 1.26*cofi*alpha/(1.0+3.5*cofi))
>> 869 * (nwt - fAbscissa[i-3])/(1.0 + 0.3*alpha) ;
>> 870 }
>> 871 for(k=1;k<=maxNumber;k++)
>> 872 {
>> 873 temp1 = 1.0 ;
>> 874 temp2 = 0.0 ;
836 875
837 for(i = 1; i <= fNumber; ++i) // Loop over << 876 for(j=1;j<=fNumber;j++)
838 { << 877 {
839 if(i == 1) << 878 temp3 = temp2 ;
840 { << 879 temp2 = temp1 ;
841 nwt = (1.0 + alpha) * (3.0 + 0.92 * alph << 880 temp1 = ((2*j - 1 + alpha - nwt)*temp2 - (j - 1 + alpha)*temp3)/j ;
842 (1.0 + 2.4 * fNumber + 1.8 * alpha << 881 }
843 } << 882 temp = (fNumber*temp1 - (fNumber +alpha)*temp2)/nwt ;
844 else if(i == 2) << 883 nwt1 = nwt ;
845 { << 884 nwt = nwt1 - temp1/temp ;
846 nwt += (15.0 + 6.25 * alpha) / (1.0 + 0. << 885
847 } << 886 if(std::fabs(nwt - nwt1) <= tolerance)
848 else << 887 {
849 { << 888 break ;
850 cofi = i - 2; << 889 }
851 nwt += ((1.0 + 2.55 * cofi) / (1.9 * cof << 890 }
852 1.26 * cofi * alpha / (1.0 + 3.5 << 891 if(k > maxNumber)
853 (nwt - fAbscissa[i - 3]) / (1.0 + << 892 {
854 } << 893 G4Exception("G4Integrator<T,F>::Laguerre(T,F, ...)", "Error",
855 for(k = 1; k <= maxNumber; ++k) << 894 FatalException, "Too many (>12) iterations.");
856 { << 895 }
857 temp1 = 1.0; << 896
858 temp2 = 0.0; << 897 fAbscissa[i-1] = nwt ;
859 << 898 fWeight[i-1] = -std::exp(GammaLogarithm(alpha + fNumber) -
860 for(j = 1; j <= fNumber; ++j) << 899 GammaLogarithm((G4double)fNumber))/(temp*fNumber*temp2) ;
861 { << 900 }
862 temp3 = temp2; << 901
863 temp2 = temp1; << 902 //
864 temp1 = << 903 // Integral evaluation
865 ((2 * j - 1 + alpha - nwt) * temp2 - << 904 //
866 } << 905
867 temp = (fNumber * temp1 - (fNumber + alp << 906 for(i=0;i<fNumber;i++)
868 nwt1 = nwt; << 907 {
869 nwt = nwt1 - temp1 / temp; << 908 integral += fWeight[i]*(typeT.*f)(fAbscissa[i]) ;
870 << 909 }
871 if(std::fabs(nwt - nwt1) <= tolerance) << 910 delete[] fAbscissa;
872 { << 911 delete[] fWeight;
873 break; << 912 return integral ;
874 } << 913 }
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 } <<
887 914
888 // <<
889 // Integral evaluation <<
890 // <<
891 915
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 } <<
900 916
901 ////////////////////////////////////////////// 917 //////////////////////////////////////////////////////////////////////
902 // 918 //
903 // 919 //
904 920
905 template <class T, class F> << 921 template <class T, class F> G4double
906 G4double G4Integrator<T, F>::Laguerre(T* ptrT, << 922 G4Integrator<T,F>::Laguerre( T* ptrT, F f, G4double alpha, G4int nLaguerre )
907 G4int nL <<
908 { 923 {
909 return Laguerre(*ptrT, f, alpha, nLaguerre); << 924 return Laguerre(*ptrT,f,alpha,nLaguerre) ;
910 } 925 }
911 926
912 ////////////////////////////////////////////// 927 ////////////////////////////////////////////////////////////////////////
913 // 928 //
914 // For use with global scope functions f << 929 // 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 } <<
980 930
981 // << 931 template <class T, class F> G4double
982 // Integral evaluation << 932 G4Integrator<T,F>::Laguerre( G4double (*f)(G4double),
983 // << 933 G4double alpha, G4int nLaguerre )
>> 934 {
>> 935 const G4double tolerance = 1.0e-10 ;
>> 936 const G4int maxNumber = 12 ;
>> 937 G4int i, j, k ;
>> 938 G4double nwt=0., nwt1, temp1, temp2, temp3, temp, cofi ;
>> 939 G4double integral = 0.0 ;
>> 940
>> 941 G4int fNumber = nLaguerre ;
>> 942 G4double* fAbscissa = new G4double[fNumber] ;
>> 943 G4double* fWeight = new G4double[fNumber] ;
>> 944
>> 945 for(i=1;i<=fNumber;i++) // Loop over the desired roots
>> 946 {
>> 947 if(i == 1)
>> 948 {
>> 949 nwt = (1.0 + alpha)*(3.0 + 0.92*alpha)
>> 950 / (1.0 + 2.4*fNumber + 1.8*alpha) ;
>> 951 }
>> 952 else if(i == 2)
>> 953 {
>> 954 nwt += (15.0 + 6.25*alpha)/(1.0 + 0.9*alpha + 2.5*fNumber) ;
>> 955 }
>> 956 else
>> 957 {
>> 958 cofi = i - 2 ;
>> 959 nwt += ((1.0+2.55*cofi)/(1.9*cofi)
>> 960 + 1.26*cofi*alpha/(1.0+3.5*cofi))
>> 961 * (nwt - fAbscissa[i-3])/(1.0 + 0.3*alpha) ;
>> 962 }
>> 963 for(k=1;k<=maxNumber;k++)
>> 964 {
>> 965 temp1 = 1.0 ;
>> 966 temp2 = 0.0 ;
984 967
985 for(i = 0; i < fNumber; i++) << 968 for(j=1;j<=fNumber;j++)
986 { << 969 {
987 integral += fWeight[i] * (*f)(fAbscissa[i] << 970 temp3 = temp2 ;
988 } << 971 temp2 = temp1 ;
989 delete[] fAbscissa; << 972 temp1 = ((2*j - 1 + alpha - nwt)*temp2 - (j - 1 + alpha)*temp3)/j ;
990 delete[] fWeight; << 973 }
991 return integral; << 974 temp = (fNumber*temp1 - (fNumber +alpha)*temp2)/nwt ;
>> 975 nwt1 = nwt ;
>> 976 nwt = nwt1 - temp1/temp ;
>> 977
>> 978 if(std::fabs(nwt - nwt1) <= tolerance)
>> 979 {
>> 980 break ;
>> 981 }
>> 982 }
>> 983 if(k > maxNumber)
>> 984 {
>> 985 G4Exception("G4Integrator<T,F>::Laguerre( ...)", "Error",
>> 986 FatalException, "Too many (>12) iterations.");
>> 987 }
>> 988
>> 989 fAbscissa[i-1] = nwt ;
>> 990 fWeight[i-1] = -std::exp(GammaLogarithm(alpha + fNumber) -
>> 991 GammaLogarithm((G4double)fNumber))/(temp*fNumber*temp2) ;
>> 992 }
>> 993
>> 994 //
>> 995 // Integral evaluation
>> 996 //
>> 997
>> 998 for(i=0;i<fNumber;i++)
>> 999 {
>> 1000 integral += fWeight[i]*(*f)(fAbscissa[i]) ;
>> 1001 }
>> 1002 delete[] fAbscissa;
>> 1003 delete[] fWeight;
>> 1004 return integral ;
992 } 1005 }
993 1006
994 ////////////////////////////////////////////// 1007 ///////////////////////////////////////////////////////////////////////
995 // 1008 //
996 // Auxiliary function which returns the value 1009 // Auxiliary function which returns the value of std::log(gamma-function(x))
997 // Returns the value ln(Gamma(xx) for xx > 0. << 1010 // Returns the value ln(Gamma(xx) for xx > 0. Full accuracy is obtained for
998 // xx > 1. For 0 < xx < 1. the reflection form 1011 // xx > 1. For 0 < xx < 1. the reflection formula (6.1.4) can be used first.
999 // (Adapted from Numerical Recipes in C) 1012 // (Adapted from Numerical Recipes in C)
1000 // 1013 //
1001 1014
1002 template <class T, class F> 1015 template <class T, class F>
1003 G4double G4Integrator<T, F>::GammaLogarithm(G << 1016 G4double G4Integrator<T,F>::GammaLogarithm(G4double xx)
1004 { 1017 {
1005 static const G4double cof[6] = { 76.1800917 1018 static const G4double cof[6] = { 76.18009172947146, -86.50532032941677,
1006 24.0140982 << 1019 24.01409824083091, -1.231739572450155,
1007 0.12086509 1020 0.1208650973866179e-2, -0.5395239384953e-5 };
1008 G4int j; 1021 G4int j;
1009 G4double x = xx - 1.0; << 1022 G4double x = xx - 1.0 ;
1010 G4double tmp = x + 5.5; << 1023 G4double tmp = x + 5.5 ;
1011 tmp -= (x + 0.5) * std::log(tmp); << 1024 tmp -= (x + 0.5) * std::log(tmp) ;
1012 G4double ser = 1.000000000190015; << 1025 G4double ser = 1.000000000190015 ;
1013 1026
1014 for(j = 0; j <= 5; ++j) << 1027 for ( j = 0; j <= 5; j++ )
1015 { 1028 {
1016 x += 1.0; << 1029 x += 1.0 ;
1017 ser += cof[j] / x; << 1030 ser += cof[j]/x ;
1018 } 1031 }
1019 return -tmp + std::log(2.5066282746310005 * << 1032 return -tmp + std::log(2.5066282746310005*ser) ;
1020 } 1033 }
1021 1034
1022 ///////////////////////////////////////////// 1035 ///////////////////////////////////////////////////////////////////////
1023 // 1036 //
1024 // Method involving Hermite polynomials 1037 // Method involving Hermite polynomials
1025 // 1038 //
1026 ///////////////////////////////////////////// 1039 ///////////////////////////////////////////////////////////////////////
1027 // 1040 //
1028 // 1041 //
1029 // Gauss-Hermite method for integration of st << 1042 // Gauss-Hermite method for integration of std::exp(-x*x)*f(x)
1030 // from minus infinity to plus infinity . << 1043 // from minus infinity to plus infinity .
1031 // 1044 //
1032 1045
1033 template <class T, class F> << 1046 template <class T, class F>
1034 G4double G4Integrator<T, F>::Hermite(T& typeT << 1047 G4double G4Integrator<T,F>::Hermite( T& typeT, F f, G4int nHermite )
1035 { 1048 {
1036 const G4double tolerance = 1.0e-12; << 1049 const G4double tolerance = 1.0e-12 ;
1037 const G4int maxNumber = 12; << 1050 const G4int maxNumber = 12 ;
1038 << 1051
1039 G4int i, j, k; << 1052 G4int i, j, k ;
1040 G4double integral = 0.0; << 1053 G4double integral = 0.0 ;
1041 G4double nwt = 0., nwt1, temp1, temp2, << 1054 G4double nwt=0., nwt1, temp1, temp2, temp3, temp ;
1042 1055
1043 G4double piInMinusQ = << 1056 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 1057
1046 G4int fNumber = (nHermite + 1) / 2; << 1058 G4int fNumber = (nHermite +1)/2 ;
1047 G4double* fAbscissa = new G4double[fNumber] << 1059 G4double* fAbscissa = new G4double[fNumber] ;
1048 G4double* fWeight = new G4double[fNumber] << 1060 G4double* fWeight = new G4double[fNumber] ;
1049 1061
1050 for(i = 1; i <= fNumber; ++i) << 1062 for(i=1;i<=fNumber;i++)
1051 { << 1063 {
1052 if(i == 1) << 1064 if(i == 1)
1053 { << 1065 {
1054 nwt = std::sqrt((G4double)(2 * nHermite << 1066 nwt = std::sqrt((G4double)(2*nHermite + 1)) -
1055 1.85575001 * std::pow((G4double)( << 1067 1.85575001*std::pow((G4double)(2*nHermite + 1),-0.16666999) ;
1056 } << 1068 }
1057 else if(i == 2) << 1069 else if(i == 2)
1058 { << 1070 {
1059 nwt -= 1.14001 * std::pow((G4double) nH << 1071 nwt -= 1.14001*std::pow((G4double)nHermite,0.425999)/nwt ;
1060 } << 1072 }
1061 else if(i == 3) << 1073 else if(i == 3)
1062 { << 1074 {
1063 nwt = 1.86002 * nwt - 0.86002 * fAbscis << 1075 nwt = 1.86002*nwt - 0.86002*fAbscissa[0] ;
1064 } << 1076 }
1065 else if(i == 4) << 1077 else if(i == 4)
1066 { << 1078 {
1067 nwt = 1.91001 * nwt - 0.91001 * fAbscis << 1079 nwt = 1.91001*nwt - 0.91001*fAbscissa[1] ;
1068 } << 1080 }
1069 else << 1081 else
1070 { << 1082 {
1071 nwt = 2.0 * nwt - fAbscissa[i - 3]; << 1083 nwt = 2.0*nwt - fAbscissa[i - 3] ;
1072 } << 1084 }
1073 for(k = 1; k <= maxNumber; ++k) << 1085 for(k=1;k<=maxNumber;k++)
1074 { << 1086 {
1075 temp1 = piInMinusQ; << 1087 temp1 = piInMinusQ ;
1076 temp2 = 0.0; << 1088 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 1089
1107 for(i = 0; i < fNumber; ++i) << 1090 for(j=1;j<=nHermite;j++)
1108 { << 1091 {
1109 integral += << 1092 temp3 = temp2 ;
1110 fWeight[i] * ((typeT.*f)(fAbscissa[i]) << 1093 temp2 = temp1 ;
1111 } << 1094 temp1 = nwt*std::sqrt(2.0/j)*temp2 -
1112 delete[] fAbscissa; << 1095 std::sqrt(((G4double)(j - 1))/j)*temp3 ;
1113 delete[] fWeight; << 1096 }
1114 return integral; << 1097 temp = std::sqrt((G4double)2*nHermite)*temp2 ;
>> 1098 nwt1 = nwt ;
>> 1099 nwt = nwt1 - temp1/temp ;
>> 1100
>> 1101 if(std::fabs(nwt - nwt1) <= tolerance)
>> 1102 {
>> 1103 break ;
>> 1104 }
>> 1105 }
>> 1106 if(k > maxNumber)
>> 1107 {
>> 1108 G4Exception("G4Integrator<T,F>::Hermite(T,F, ...)", "Error",
>> 1109 FatalException, "Too many (>12) iterations.");
>> 1110 }
>> 1111 fAbscissa[i-1] = nwt ;
>> 1112 fWeight[i-1] = 2.0/(temp*temp) ;
>> 1113 }
>> 1114
>> 1115 //
>> 1116 // Integral calculation
>> 1117 //
>> 1118
>> 1119 for(i=0;i<fNumber;i++)
>> 1120 {
>> 1121 integral += fWeight[i]*( (typeT.*f)(fAbscissa[i]) +
>> 1122 (typeT.*f)(-fAbscissa[i]) ) ;
>> 1123 }
>> 1124 delete[] fAbscissa;
>> 1125 delete[] fWeight;
>> 1126 return integral ;
1115 } 1127 }
1116 1128
>> 1129
1117 ///////////////////////////////////////////// 1130 ////////////////////////////////////////////////////////////////////////
1118 // 1131 //
1119 // For use with 'this' pointer 1132 // For use with 'this' pointer
1120 1133
1121 template <class T, class F> << 1134 template <class T, class F>
1122 G4double G4Integrator<T, F>::Hermite(T* ptrT, << 1135 G4double G4Integrator<T,F>::Hermite( T* ptrT, F f, G4int n )
1123 { 1136 {
1124 return Hermite(*ptrT, f, n); << 1137 return Hermite(*ptrT,f,n) ;
1125 } << 1138 }
1126 1139
1127 ///////////////////////////////////////////// 1140 ////////////////////////////////////////////////////////////////////////
1128 // 1141 //
1129 // For use with global scope f 1142 // For use with global scope f
1130 1143
1131 template <class T, class F> 1144 template <class T, class F>
1132 G4double G4Integrator<T, F>::Hermite(G4double << 1145 G4double G4Integrator<T,F>::Hermite( G4double (*f)(G4double), G4int nHermite)
1133 { 1146 {
1134 const G4double tolerance = 1.0e-12; << 1147 const G4double tolerance = 1.0e-12 ;
1135 const G4int maxNumber = 12; << 1148 const G4int maxNumber = 12 ;
1136 << 1149
1137 G4int i, j, k; << 1150 G4int i, j, k ;
1138 G4double integral = 0.0; << 1151 G4double integral = 0.0 ;
1139 G4double nwt = 0., nwt1, temp1, temp2, << 1152 G4double nwt=0., nwt1, temp1, temp2, temp3, temp ;
1140 << 1153
1141 G4double piInMinusQ = << 1154 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:: << 1155
1143 << 1156 G4int fNumber = (nHermite +1)/2 ;
1144 G4int fNumber = (nHermite + 1) / 2; << 1157 G4double* fAbscissa = new G4double[fNumber] ;
1145 G4double* fAbscissa = new G4double[fNumber] << 1158 G4double* fWeight = new G4double[fNumber] ;
1146 G4double* fWeight = new G4double[fNumber] << 1159
1147 << 1160 for(i=1;i<=fNumber;i++)
1148 for(i = 1; i <= fNumber; ++i) << 1161 {
1149 { << 1162 if(i == 1)
1150 if(i == 1) << 1163 {
1151 { << 1164 nwt = std::sqrt((G4double)(2*nHermite + 1)) -
1152 nwt = std::sqrt((G4double)(2 * nHermite << 1165 1.85575001*std::pow((G4double)(2*nHermite + 1),-0.16666999) ;
1153 1.85575001 * std::pow((G4double)( << 1166 }
1154 } << 1167 else if(i == 2)
1155 else if(i == 2) << 1168 {
1156 { << 1169 nwt -= 1.14001*std::pow((G4double)nHermite,0.425999)/nwt ;
1157 nwt -= 1.14001 * std::pow((G4double) nH << 1170 }
1158 } << 1171 else if(i == 3)
1159 else if(i == 3) << 1172 {
1160 { << 1173 nwt = 1.86002*nwt - 0.86002*fAbscissa[0] ;
1161 nwt = 1.86002 * nwt - 0.86002 * fAbscis << 1174 }
1162 } << 1175 else if(i == 4)
1163 else if(i == 4) << 1176 {
1164 { << 1177 nwt = 1.91001*nwt - 0.91001*fAbscissa[1] ;
1165 nwt = 1.91001 * nwt - 0.91001 * fAbscis << 1178 }
1166 } << 1179 else
1167 else << 1180 {
1168 { << 1181 nwt = 2.0*nwt - fAbscissa[i - 3] ;
1169 nwt = 2.0 * nwt - fAbscissa[i - 3]; << 1182 }
1170 } << 1183 for(k=1;k<=maxNumber;k++)
1171 for(k = 1; k <= maxNumber; ++k) << 1184 {
1172 { << 1185 temp1 = piInMinusQ ;
1173 temp1 = piInMinusQ; << 1186 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 1187
1205 for(i = 0; i < fNumber; ++i) << 1188 for(j=1;j<=nHermite;j++)
1206 { << 1189 {
1207 integral += fWeight[i] * ((*f)(fAbscissa[ << 1190 temp3 = temp2 ;
1208 } << 1191 temp2 = temp1 ;
1209 delete[] fAbscissa; << 1192 temp1 = nwt*std::sqrt(2.0/j)*temp2 -
1210 delete[] fWeight; << 1193 std::sqrt(((G4double)(j - 1))/j)*temp3 ;
1211 return integral; << 1194 }
>> 1195 temp = std::sqrt((G4double)2*nHermite)*temp2 ;
>> 1196 nwt1 = nwt ;
>> 1197 nwt = nwt1 - temp1/temp ;
>> 1198
>> 1199 if(std::fabs(nwt - nwt1) <= tolerance)
>> 1200 {
>> 1201 break ;
>> 1202 }
>> 1203 }
>> 1204 if(k > maxNumber)
>> 1205 {
>> 1206 G4Exception("G4Integrator<T,F>::Hermite(...)", "Error",
>> 1207 FatalException, "Too many (>12) iterations.");
>> 1208 }
>> 1209 fAbscissa[i-1] = nwt ;
>> 1210 fWeight[i-1] = 2.0/(temp*temp) ;
>> 1211 }
>> 1212
>> 1213 //
>> 1214 // Integral calculation
>> 1215 //
>> 1216
>> 1217 for(i=0;i<fNumber;i++)
>> 1218 {
>> 1219 integral += fWeight[i]*( (*f)(fAbscissa[i]) + (*f)(-fAbscissa[i]) ) ;
>> 1220 }
>> 1221 delete[] fAbscissa;
>> 1222 delete[] fWeight;
>> 1223 return integral ;
1212 } 1224 }
1213 1225
1214 ///////////////////////////////////////////// 1226 ////////////////////////////////////////////////////////////////////////////
1215 // 1227 //
1216 // Method involving Jacobi polynomials 1228 // Method involving Jacobi polynomials
1217 // 1229 //
1218 ///////////////////////////////////////////// 1230 ////////////////////////////////////////////////////////////////////////////
1219 // 1231 //
1220 // Gauss-Jacobi method for integration of ((1 1232 // Gauss-Jacobi method for integration of ((1-x)^alpha)*((1+x)^beta)*f(x)
1221 // from minus unit to plus unit . 1233 // from minus unit to plus unit .
1222 // 1234 //
1223 1235
1224 template <class T, class F> << 1236 template <class T, class F>
1225 G4double G4Integrator<T, F>::Jacobi(T& typeT, << 1237 G4double G4Integrator<T,F>::Jacobi( T& typeT, F f, G4double alpha,
1226 G4double << 1238 G4double beta, G4int nJacobi)
1227 { << 1239 {
1228 const G4double tolerance = 1.0e-12; << 1240 const G4double tolerance = 1.0e-12 ;
1229 const G4double maxNumber = 12; << 1241 const G4double maxNumber = 12 ;
1230 G4int i, k, j; << 1242 G4int i, k, j ;
1231 G4double alphaBeta, alphaReduced, betaReduc << 1243 G4double alphaBeta, alphaReduced, betaReduced, root1=0., root2=0., root3=0. ;
1232 << 1244 G4double a, b, c, nwt1, nwt2, nwt3, nwt, temp, root=0., rootTemp ;
1233 G4double a, b, c, nwt1, nwt2, nwt3, nwt, te << 1245
1234 << 1246 G4int fNumber = nJacobi ;
1235 G4int fNumber = nJacobi; << 1247 G4double* fAbscissa = new G4double[fNumber] ;
1236 G4double* fAbscissa = new G4double[fNumber] << 1248 G4double* fWeight = new G4double[fNumber] ;
1237 G4double* fWeight = new G4double[fNumber] << 1249
1238 << 1250 for (i=1;i<=nJacobi;i++)
1239 for(i = 1; i <= nJacobi; ++i) << 1251 {
1240 { << 1252 if (i == 1)
1241 if(i == 1) << 1253 {
1242 { << 1254 alphaReduced = alpha/nJacobi ;
1243 alphaReduced = alpha / nJacobi; << 1255 betaReduced = beta/nJacobi ;
1244 betaReduced = beta / nJacobi; << 1256 root1 = (1.0+alpha)*(2.78002/(4.0+nJacobi*nJacobi)+
1245 root1 = (1.0 + alpha) * (2.78002 << 1257 0.767999*alphaReduced/nJacobi) ;
1246 0.767999 * alp << 1258 root2 = 1.0+1.48*alphaReduced+0.96002*betaReduced +
1247 root2 = 1.0 + 1.48 * alphaReduce << 1259 0.451998*alphaReduced*alphaReduced +
1248 0.451998 * alphaReduced * alpha << 1260 0.83001*alphaReduced*betaReduced ;
1249 0.83001 * alphaReduced * betaRe << 1261 root = 1.0-root1/root2 ;
1250 root = 1.0 - root1 / root2; << 1262 }
1251 } << 1263 else if (i == 2)
1252 else if(i == 2) << 1264 {
1253 { << 1265 root1=(4.1002+alpha)/((1.0+alpha)*(1.0+0.155998*alpha)) ;
1254 root1 = (4.1002 + alpha) / ((1.0 + alph << 1266 root2=1.0+0.06*(nJacobi-8.0)*(1.0+0.12*alpha)/nJacobi ;
1255 root2 = 1.0 + 0.06 * (nJacobi - 8.0) * << 1267 root3=1.0+0.012002*beta*(1.0+0.24997*std::fabs(alpha))/nJacobi ;
1256 root3 = << 1268 root -= (1.0-root)*root1*root2*root3 ;
1257 1.0 + 0.012002 * beta * (1.0 + 0.2499 << 1269 }
1258 root -= (1.0 - root) * root1 * root2 * << 1270 else if (i == 3)
1259 } << 1271 {
1260 else if(i == 3) << 1272 root1=(1.67001+0.27998*alpha)/(1.0+0.37002*alpha) ;
1261 { << 1273 root2=1.0+0.22*(nJacobi-8.0)/nJacobi ;
1262 root1 = (1.67001 + 0.27998 * alpha) / ( << 1274 root3=1.0+8.0*beta/((6.28001+beta)*nJacobi*nJacobi) ;
1263 root2 = 1.0 + 0.22 * (nJacobi - 8.0) / << 1275 root -= (fAbscissa[0]-root)*root1*root2*root3 ;
1264 root3 = 1.0 + 8.0 * beta / ((6.28001 + << 1276 }
1265 root -= (fAbscissa[0] - root) * root1 * << 1277 else if (i == nJacobi-1)
1266 } << 1278 {
1267 else if(i == nJacobi - 1) << 1279 root1=(1.0+0.235002*beta)/(0.766001+0.118998*beta) ;
1268 { << 1280 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 << 1281 root3=1.0/(1.0+20.0*alpha/((7.5+alpha)*nJacobi*nJacobi)) ;
1270 root2 = 1.0 / (1.0 + 0.639002 * (nJacob << 1282 root += (root-fAbscissa[nJacobi-4])*root1*root2*root3 ;
1271 (1.0 + 0.71001 * << 1283 }
1272 root3 = 1.0 / (1.0 + 20.0 * alpha / ((7 << 1284 else if (i == nJacobi)
1273 root += (root - fAbscissa[nJacobi - 4]) << 1285 {
1274 } << 1286 root1 = (1.0+0.37002*beta)/(1.67001+0.27998*beta) ;
1275 else if(i == nJacobi) << 1287 root2 = 1.0/(1.0+0.22*(nJacobi-8.0)/nJacobi) ;
1276 { << 1288 root3 = 1.0/(1.0+8.0*alpha/((6.28002+alpha)*nJacobi*nJacobi)) ;
1277 root1 = (1.0 + 0.37002 * beta) / (1.670 << 1289 root += (root-fAbscissa[nJacobi-3])*root1*root2*root3 ;
1278 root2 = 1.0 / (1.0 + 0.22 * (nJacobi - << 1290 }
1279 root3 = << 1291 else
1280 1.0 / (1.0 + 8.0 * alpha / ((6.28002 << 1292 {
1281 root += (root - fAbscissa[nJacobi - 3]) << 1293 root = 3.0*fAbscissa[i-2]-3.0*fAbscissa[i-3]+fAbscissa[i-4] ;
1282 } << 1294 }
1283 else << 1295 alphaBeta = alpha + beta ;
1284 { << 1296 for (k=1;k<=maxNumber;k++)
1285 root = 3.0 * fAbscissa[i - 2] - 3.0 * f << 1297 {
1286 } << 1298 temp = 2.0 + alphaBeta ;
1287 alphaBeta = alpha + beta; << 1299 nwt1 = (alpha-beta+temp*root)/2.0 ;
1288 for(k = 1; k <= maxNumber; ++k) << 1300 nwt2 = 1.0 ;
1289 { << 1301 for (j=2;j<=nJacobi;j++)
1290 temp = 2.0 + alphaBeta; << 1302 {
1291 nwt1 = (alpha - beta + temp * root) / 2 << 1303 nwt3 = nwt2 ;
1292 nwt2 = 1.0; << 1304 nwt2 = nwt1 ;
1293 for(j = 2; j <= nJacobi; ++j) << 1305 temp = 2*j+alphaBeta ;
1294 { << 1306 a = 2*j*(j+alphaBeta)*(temp-2.0) ;
1295 nwt3 = nwt2; << 1307 b = (temp-1.0)*(alpha*alpha-beta*beta+temp*(temp-2.0)*root) ;
1296 nwt2 = nwt1; << 1308 c = 2.0*(j-1+alpha)*(j-1+beta)*temp ;
1297 temp = 2 * j + alphaBeta; << 1309 nwt1 = (b*nwt2-c*nwt3)/a ;
1298 a = 2 * j * (j + alphaBeta) * (tem << 1310 }
1299 b = (temp - 1.0) * << 1311 nwt = (nJacobi*(alpha - beta - temp*root)*nwt1 +
1300 (alpha * alpha - beta * beta + te << 1312 2.0*(nJacobi + alpha)*(nJacobi + beta)*nwt2)/
1301 c = 2.0 * (j - 1 + alpha) * (j - 1 << 1313 (temp*(1.0 - root*root)) ;
1302 nwt1 = (b * nwt2 - c * nwt3) / a; << 1314 rootTemp = root ;
1303 } << 1315 root = rootTemp - nwt1/nwt ;
1304 nwt = (nJacobi * (alpha - beta - temp * << 1316 if (std::fabs(root-rootTemp) <= tolerance)
1305 2.0 * (nJacobi + alpha) * (nJaco << 1317 {
1306 (temp * (1.0 - root * root)); << 1318 break ;
1307 rootTemp = root; << 1319 }
1308 root = rootTemp - nwt1 / nwt; << 1320 }
1309 if(std::fabs(root - rootTemp) <= tolera << 1321 if (k > maxNumber)
1310 { << 1322 {
1311 break; << 1323 G4Exception("G4Integrator<T,F>::Jacobi(T,F, ...)", "Error",
1312 } << 1324 FatalException, "Too many (>12) iterations.");
1313 } << 1325 }
1314 if(k > maxNumber) << 1326 fAbscissa[i-1] = root ;
1315 { << 1327 fWeight[i-1] = std::exp(GammaLogarithm((G4double)(alpha+nJacobi)) +
1316 G4Exception("G4Integrator<T,F>::Jacobi( << 1328 GammaLogarithm((G4double)(beta+nJacobi)) -
1317 FatalException, "Too many ( << 1329 GammaLogarithm((G4double)(nJacobi+1.0)) -
1318 } << 1330 GammaLogarithm((G4double)(nJacobi + alphaBeta + 1.0)))
1319 fAbscissa[i - 1] = root; << 1331 *temp*std::pow(2.0,alphaBeta)/(nwt*nwt2) ;
1320 fWeight[i - 1] = << 1332 }
1321 std::exp(GammaLogarithm((G4double)(alph << 1333
1322 GammaLogarithm((G4double)(beta << 1334 //
1323 GammaLogarithm((G4double)(nJac << 1335 // Calculation of the integral
1324 GammaLogarithm((G4double)(nJac << 1336 //
1325 temp * std::pow(2.0, alphaBeta) / (nwt << 1337
1326 } << 1338 G4double integral = 0.0 ;
1327 << 1339 for(i=0;i<fNumber;i++)
1328 // << 1340 {
1329 // Calculation of the integral << 1341 integral += fWeight[i]*(typeT.*f)(fAbscissa[i]) ;
1330 // << 1342 }
1331 << 1343 delete[] fAbscissa;
1332 G4double integral = 0.0; << 1344 delete[] fWeight;
1333 for(i = 0; i < fNumber; ++i) << 1345 return integral ;
1334 { <<
1335 integral += fWeight[i] * (typeT.*f)(fAbsc <<
1336 } <<
1337 delete[] fAbscissa; <<
1338 delete[] fWeight; <<
1339 return integral; <<
1340 } 1346 }
1341 1347
>> 1348
1342 ///////////////////////////////////////////// 1349 /////////////////////////////////////////////////////////////////////////
1343 // 1350 //
1344 // For use with 'this' pointer 1351 // For use with 'this' pointer
1345 1352
1346 template <class T, class F> << 1353 template <class T, class F>
1347 G4double G4Integrator<T, F>::Jacobi(T* ptrT, << 1354 G4double G4Integrator<T,F>::Jacobi( T* ptrT, F f, G4double alpha,
1348 G4int n) << 1355 G4double beta, G4int n)
1349 { 1356 {
1350 return Jacobi(*ptrT, f, alpha, beta, n); << 1357 return Jacobi(*ptrT,f,alpha,beta,n) ;
1351 } << 1358 }
1352 1359
1353 ///////////////////////////////////////////// 1360 /////////////////////////////////////////////////////////////////////////
1354 // 1361 //
1355 // For use with global scope f << 1362 // For use with global scope f
1356 1363
1357 template <class T, class F> 1364 template <class T, class F>
1358 G4double G4Integrator<T, F>::Jacobi(G4double << 1365 G4double G4Integrator<T,F>::Jacobi( G4double (*f)(G4double), G4double alpha,
1359 G4double << 1366 G4double beta, G4int nJacobi)
1360 { 1367 {
1361 const G4double tolerance = 1.0e-12; << 1368 const G4double tolerance = 1.0e-12 ;
1362 const G4double maxNumber = 12; << 1369 const G4double maxNumber = 12 ;
1363 G4int i, k, j; << 1370 G4int i, k, j ;
1364 G4double alphaBeta, alphaReduced, betaReduc << 1371 G4double alphaBeta, alphaReduced, betaReduced, root1=0., root2=0., root3=0. ;
1365 << 1372 G4double a, b, c, nwt1, nwt2, nwt3, nwt, temp, root=0., rootTemp ;
1366 G4double a, b, c, nwt1, nwt2, nwt3, nwt, te << 1373
1367 << 1374 G4int fNumber = nJacobi ;
1368 G4int fNumber = nJacobi; << 1375 G4double* fAbscissa = new G4double[fNumber] ;
1369 G4double* fAbscissa = new G4double[fNumber] << 1376 G4double* fWeight = new G4double[fNumber] ;
1370 G4double* fWeight = new G4double[fNumber] << 1377
1371 << 1378 for (i=1;i<=nJacobi;i++)
1372 for(i = 1; i <= nJacobi; ++i) << 1379 {
1373 { << 1380 if (i == 1)
1374 if(i == 1) << 1381 {
1375 { << 1382 alphaReduced = alpha/nJacobi ;
1376 alphaReduced = alpha / nJacobi; << 1383 betaReduced = beta/nJacobi ;
1377 betaReduced = beta / nJacobi; << 1384 root1 = (1.0+alpha)*(2.78002/(4.0+nJacobi*nJacobi)+
1378 root1 = (1.0 + alpha) * (2.78002 << 1385 0.767999*alphaReduced/nJacobi) ;
1379 0.767999 * alp << 1386 root2 = 1.0+1.48*alphaReduced+0.96002*betaReduced +
1380 root2 = 1.0 + 1.48 * alphaReduce << 1387 0.451998*alphaReduced*alphaReduced +
1381 0.451998 * alphaReduced * alpha << 1388 0.83001*alphaReduced*betaReduced ;
1382 0.83001 * alphaReduced * betaRe << 1389 root = 1.0-root1/root2 ;
1383 root = 1.0 - root1 / root2; << 1390 }
1384 } << 1391 else if (i == 2)
1385 else if(i == 2) << 1392 {
1386 { << 1393 root1=(4.1002+alpha)/((1.0+alpha)*(1.0+0.155998*alpha)) ;
1387 root1 = (4.1002 + alpha) / ((1.0 + alph << 1394 root2=1.0+0.06*(nJacobi-8.0)*(1.0+0.12*alpha)/nJacobi ;
1388 root2 = 1.0 + 0.06 * (nJacobi - 8.0) * << 1395 root3=1.0+0.012002*beta*(1.0+0.24997*std::fabs(alpha))/nJacobi ;
1389 root3 = << 1396 root -= (1.0-root)*root1*root2*root3 ;
1390 1.0 + 0.012002 * beta * (1.0 + 0.2499 << 1397 }
1391 root -= (1.0 - root) * root1 * root2 * << 1398 else if (i == 3)
1392 } << 1399 {
1393 else if(i == 3) << 1400 root1=(1.67001+0.27998*alpha)/(1.0+0.37002*alpha) ;
1394 { << 1401 root2=1.0+0.22*(nJacobi-8.0)/nJacobi ;
1395 root1 = (1.67001 + 0.27998 * alpha) / ( << 1402 root3=1.0+8.0*beta/((6.28001+beta)*nJacobi*nJacobi) ;
1396 root2 = 1.0 + 0.22 * (nJacobi - 8.0) / << 1403 root -= (fAbscissa[0]-root)*root1*root2*root3 ;
1397 root3 = 1.0 + 8.0 * beta / ((6.28001 + << 1404 }
1398 root -= (fAbscissa[0] - root) * root1 * << 1405 else if (i == nJacobi-1)
1399 } << 1406 {
1400 else if(i == nJacobi - 1) << 1407 root1=(1.0+0.235002*beta)/(0.766001+0.118998*beta) ;
1401 { << 1408 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 << 1409 root3=1.0/(1.0+20.0*alpha/((7.5+alpha)*nJacobi*nJacobi)) ;
1403 root2 = 1.0 / (1.0 + 0.639002 * (nJacob << 1410 root += (root-fAbscissa[nJacobi-4])*root1*root2*root3 ;
1404 (1.0 + 0.71001 * << 1411 }
1405 root3 = 1.0 / (1.0 + 20.0 * alpha / ((7 << 1412 else if (i == nJacobi)
1406 root += (root - fAbscissa[nJacobi - 4]) << 1413 {
1407 } << 1414 root1 = (1.0+0.37002*beta)/(1.67001+0.27998*beta) ;
1408 else if(i == nJacobi) << 1415 root2 = 1.0/(1.0+0.22*(nJacobi-8.0)/nJacobi) ;
1409 { << 1416 root3 = 1.0/(1.0+8.0*alpha/((6.28002+alpha)*nJacobi*nJacobi)) ;
1410 root1 = (1.0 + 0.37002 * beta) / (1.670 << 1417 root += (root-fAbscissa[nJacobi-3])*root1*root2*root3 ;
1411 root2 = 1.0 / (1.0 + 0.22 * (nJacobi - << 1418 }
1412 root3 = << 1419 else
1413 1.0 / (1.0 + 8.0 * alpha / ((6.28002 << 1420 {
1414 root += (root - fAbscissa[nJacobi - 3]) << 1421 root = 3.0*fAbscissa[i-2]-3.0*fAbscissa[i-3]+fAbscissa[i-4] ;
1415 } << 1422 }
1416 else << 1423 alphaBeta = alpha + beta ;
1417 { << 1424 for (k=1;k<=maxNumber;k++)
1418 root = 3.0 * fAbscissa[i - 2] - 3.0 * f << 1425 {
1419 } << 1426 temp = 2.0 + alphaBeta ;
1420 alphaBeta = alpha + beta; << 1427 nwt1 = (alpha-beta+temp*root)/2.0 ;
1421 for(k = 1; k <= maxNumber; ++k) << 1428 nwt2 = 1.0 ;
1422 { << 1429 for (j=2;j<=nJacobi;j++)
1423 temp = 2.0 + alphaBeta; << 1430 {
1424 nwt1 = (alpha - beta + temp * root) / 2 << 1431 nwt3 = nwt2 ;
1425 nwt2 = 1.0; << 1432 nwt2 = nwt1 ;
1426 for(j = 2; j <= nJacobi; ++j) << 1433 temp = 2*j+alphaBeta ;
1427 { << 1434 a = 2*j*(j+alphaBeta)*(temp-2.0) ;
1428 nwt3 = nwt2; << 1435 b = (temp-1.0)*(alpha*alpha-beta*beta+temp*(temp-2.0)*root) ;
1429 nwt2 = nwt1; << 1436 c = 2.0*(j-1+alpha)*(j-1+beta)*temp ;
1430 temp = 2 * j + alphaBeta; << 1437 nwt1 = (b*nwt2-c*nwt3)/a ;
1431 a = 2 * j * (j + alphaBeta) * (tem << 1438 }
1432 b = (temp - 1.0) * << 1439 nwt = (nJacobi*(alpha - beta - temp*root)*nwt1 +
1433 (alpha * alpha - beta * beta + te << 1440 2.0*(nJacobi + alpha)*(nJacobi + beta)*nwt2) /
1434 c = 2.0 * (j - 1 + alpha) * (j - 1 << 1441 (temp*(1.0 - root*root)) ;
1435 nwt1 = (b * nwt2 - c * nwt3) / a; << 1442 rootTemp = root ;
1436 } << 1443 root = rootTemp - nwt1/nwt ;
1437 nwt = (nJacobi * (alpha - beta - temp * << 1444 if (std::fabs(root-rootTemp) <= tolerance)
1438 2.0 * (nJacobi + alpha) * (nJaco << 1445 {
1439 (temp * (1.0 - root * root)); << 1446 break ;
1440 rootTemp = root; << 1447 }
1441 root = rootTemp - nwt1 / nwt; << 1448 }
1442 if(std::fabs(root - rootTemp) <= tolera << 1449 if (k > maxNumber)
1443 { << 1450 {
1444 break; << 1451 G4Exception("G4Integrator<T,F>::Jacobi(...)", "Error",
1445 } << 1452 FatalException, "Too many (>12) iterations.");
1446 } << 1453 }
1447 if(k > maxNumber) << 1454 fAbscissa[i-1] = root ;
1448 { << 1455 fWeight[i-1] =
1449 G4Exception("G4Integrator<T,F>::Jacobi( << 1456 std::exp(GammaLogarithm((G4double)(alpha+nJacobi)) +
1450 "Too many (>12) iterations. << 1457 GammaLogarithm((G4double)(beta+nJacobi)) -
1451 } << 1458 GammaLogarithm((G4double)(nJacobi+1.0)) -
1452 fAbscissa[i - 1] = root; << 1459 GammaLogarithm((G4double)(nJacobi + alphaBeta + 1.0)))
1453 fWeight[i - 1] = << 1460 *temp*std::pow(2.0,alphaBeta)/(nwt*nwt2);
1454 std::exp(GammaLogarithm((G4double)(alph << 1461 }
1455 GammaLogarithm((G4double)(beta << 1462
1456 GammaLogarithm((G4double)(nJac << 1463 //
1457 GammaLogarithm((G4double)(nJac << 1464 // Calculation of the integral
1458 temp * std::pow(2.0, alphaBeta) / (nwt << 1465 //
1459 } << 1466
1460 << 1467 G4double integral = 0.0 ;
1461 // << 1468 for(i=0;i<fNumber;i++)
1462 // Calculation of the integral << 1469 {
1463 // << 1470 integral += fWeight[i]*(*f)(fAbscissa[i]) ;
1464 << 1471 }
1465 G4double integral = 0.0; << 1472 delete[] fAbscissa;
1466 for(i = 0; i < fNumber; ++i) << 1473 delete[] fWeight;
1467 { << 1474 return integral ;
1468 integral += fWeight[i] * (*f)(fAbscissa[i <<
1469 } <<
1470 delete[] fAbscissa; <<
1471 delete[] fWeight; <<
1472 return integral; <<
1473 } 1475 }
1474 1476
1475 // 1477 //
1476 // 1478 //
1477 ///////////////////////////////////////////// 1479 ///////////////////////////////////////////////////////////////////
1478 1480