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>> 1 // This code implementation is the intellectual property of
>> 2 // the GEANT4 collaboration.
1 // 3 //
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4 // * << 6 // and all its terms.
5 // * The Geant4 software is copyright of th << 7 //
6 // * the Geant4 Collaboration. It is provided << 8 // $Id: G4Integrator.icc,v 1.6 2000/06/15 17:31:23 gcosmo Exp $
7 // * conditions of the Geant4 Software License << 9 // GEANT4 tag $Name: geant4-03-01 $
8 // * LICENSE and available at http://cern.ch/ << 10 //
9 // * include a list of copyright holders. << 11 // Implementation of G4Integrator methods.
10 // * << 12 //
11 // * Neither the authors of this software syst << 13 //
12 // * institutes,nor the agencies providing fin <<
13 // * work make any representation or warran <<
14 // * regarding this software system or assum <<
15 // * use. Please see the license in the file <<
16 // * for the full disclaimer and the limitatio <<
17 // * <<
18 // * This code implementation is the result <<
19 // * technical work of the GEANT4 collaboratio <<
20 // * By using, copying, modifying or distri <<
21 // * any work based on the software) you ag <<
22 // * use in resulting scientific publicati <<
23 // * acceptance of all terms of the Geant4 Sof <<
24 // ******************************************* <<
25 // <<
26 // G4Integrator inline methods implementation <<
27 // <<
28 // Author: V.Grichine, 04.09.1999 - First impl <<
29 // G4SimpleIntegration class with H.P. <<
30 // E.TCherniaev advises <<
31 // ------------------------------------------- <<
32 14
33 ////////////////////////////////////////////// 15 /////////////////////////////////////////////////////////////////////
34 // 16 //
35 // Sympson integration method 17 // Sympson integration method
36 // 18 //
37 ////////////////////////////////////////////// 19 /////////////////////////////////////////////////////////////////////
38 // 20 //
39 // Integration of class member functions T::f << 21 // Integration of class member functions T::f by Simpson method.
40 22
41 template <class T, class F> << 23 template <class T, class F>
42 G4double G4Integrator<T, F>::Simpson(T& typeT, << 24 G4double G4Integrator<T,F>::Simpson( T& typeT,
43 G4double << 25 F f,
44 { << 26 G4double xInitial,
45 G4int i; << 27 G4double xFinal,
46 G4double step = (xFinal - xInitial) / itera << 28 G4int iterationNumber )
47 G4double x = xInitial; << 29 {
48 G4double xPlus = xInitial + 0.5 * step; << 30 G4int i ;
49 G4double mean = ((typeT.*f)(xInitial) + (ty << 31 G4double step = (xFinal - xInitial)/iterationNumber ;
50 G4double sum = (typeT.*f)(xPlus); << 32 G4double x = xInitial ;
>> 33 G4double xPlus = xInitial + 0.5*step ;
>> 34 G4double mean = ( (typeT.*f)(xInitial) + (typeT.*f)(xFinal) )*0.5 ;
>> 35 G4double sum = (typeT.*f)(xPlus) ;
>> 36
>> 37 for(i=1;i<iterationNumber;i++)
>> 38 {
>> 39 x += step ;
>> 40 xPlus += step ;
>> 41 mean += (typeT.*f)(x) ;
>> 42 sum += (typeT.*f)(xPlus) ;
>> 43 }
>> 44 mean += 2.0*sum ;
51 45
52 for(i = 1; i < iterationNumber; ++i) << 46 return mean*step/3.0 ;
53 { <<
54 x += step; <<
55 xPlus += step; <<
56 mean += (typeT.*f)(x); <<
57 sum += (typeT.*f)(xPlus); <<
58 } <<
59 mean += 2.0 * sum; <<
60 <<
61 return mean * step / 3.0; <<
62 } 47 }
63 48
64 ////////////////////////////////////////////// 49 /////////////////////////////////////////////////////////////////////
65 // 50 //
66 // Integration of class member functions T::f 51 // Integration of class member functions T::f by Simpson method.
67 // Convenient to use with 'this' pointer 52 // Convenient to use with 'this' pointer
68 53
69 template <class T, class F> << 54 template <class T, class F>
70 G4double G4Integrator<T, F>::Simpson(T* ptrT, << 55 G4double G4Integrator<T,F>::Simpson( T* ptrT,
71 G4double << 56 F f,
72 { << 57 G4double xInitial,
73 G4int i; << 58 G4double xFinal,
74 G4double step = (xFinal - xInitial) / itera << 59 G4int iterationNumber )
75 G4double x = xInitial; << 60 {
76 G4double xPlus = xInitial + 0.5 * step; << 61 G4int i ;
77 G4double mean = ((ptrT->*f)(xInitial) + (pt << 62 G4double step = (xFinal - xInitial)/iterationNumber ;
78 G4double sum = (ptrT->*f)(xPlus); << 63 G4double x = xInitial ;
>> 64 G4double xPlus = xInitial + 0.5*step ;
>> 65 G4double mean = ( (ptrT->*f)(xInitial) + (ptrT->*f)(xFinal) )*0.5 ;
>> 66 G4double sum = (ptrT->*f)(xPlus) ;
>> 67
>> 68 for(i=1;i<iterationNumber;i++)
>> 69 {
>> 70 x += step ;
>> 71 xPlus += step ;
>> 72 mean += (ptrT->*f)(x) ;
>> 73 sum += (ptrT->*f)(xPlus) ;
>> 74 }
>> 75 mean += 2.0*sum ;
79 76
80 for(i = 1; i < iterationNumber; ++i) << 77 return mean*step/3.0 ;
81 { <<
82 x += step; <<
83 xPlus += step; <<
84 mean += (ptrT->*f)(x); <<
85 sum += (ptrT->*f)(xPlus); <<
86 } <<
87 mean += 2.0 * sum; <<
88 <<
89 return mean * step / 3.0; <<
90 } 78 }
91 79
92 ////////////////////////////////////////////// 80 /////////////////////////////////////////////////////////////////////
93 // 81 //
94 // Integration of class member functions T::f 82 // Integration of class member functions T::f by Simpson method.
95 // Convenient to use, when function f is defin 83 // Convenient to use, when function f is defined in global scope, i.e. in main()
96 // program 84 // program
97 85
98 template <class T, class F> << 86 template <class T, class F>
99 G4double G4Integrator<T, F>::Simpson(G4double << 87 G4double G4Integrator<T,F>::Simpson( G4double (*f)(G4double),
100 G4double << 88 G4double xInitial,
101 { << 89 G4double xFinal,
102 G4int i; << 90 G4int iterationNumber )
103 G4double step = (xFinal - xInitial) / itera << 91 {
104 G4double x = xInitial; << 92 G4int i ;
105 G4double xPlus = xInitial + 0.5 * step; << 93 G4double step = (xFinal - xInitial)/iterationNumber ;
106 G4double mean = ((*f)(xInitial) + (*f)(xFin << 94 G4double x = xInitial ;
107 G4double sum = (*f)(xPlus); << 95 G4double xPlus = xInitial + 0.5*step ;
108 << 96 G4double mean = ( (*f)(xInitial) + (*f)(xFinal) )*0.5 ;
109 for(i = 1; i < iterationNumber; ++i) << 97 G4double sum = (*f)(xPlus) ;
110 { << 98
111 x += step; << 99 for(i=1;i<iterationNumber;i++)
112 xPlus += step; << 100 {
113 mean += (*f)(x); << 101 x += step ;
114 sum += (*f)(xPlus); << 102 xPlus += step ;
115 } << 103 mean += (*f)(x) ;
116 mean += 2.0 * sum; << 104 sum += (*f)(xPlus) ;
>> 105 }
>> 106 mean += 2.0*sum ;
117 107
118 return mean * step / 3.0; << 108 return mean*step/3.0 ;
119 } 109 }
120 110
121 ////////////////////////////////////////////// 111 //////////////////////////////////////////////////////////////////////////
122 // 112 //
123 // Adaptive Gauss method 113 // Adaptive Gauss method
124 // 114 //
125 ////////////////////////////////////////////// 115 //////////////////////////////////////////////////////////////////////////
126 // 116 //
127 // 117 //
128 118
129 template <class T, class F> << 119 template <class T, class F>
130 G4double G4Integrator<T, F>::Gauss(T& typeT, F << 120 G4double G4Integrator<T,F>::Gauss( T& typeT, F f,
131 G4double xF << 121 G4double xInitial, G4double xFinal )
132 { << 122 {
133 static const G4double root = 1.0 / std::sqrt << 123 static G4double root = 1.0/sqrt(3.0) ;
134 << 124
135 G4double xMean = (xInitial + xFinal) / 2.0; << 125 G4double xMean = (xInitial + xFinal)/2.0 ;
136 G4double Step = (xFinal - xInitial) / 2.0; << 126 G4double Step = (xFinal - xInitial)/2.0 ;
137 G4double delta = Step * root; << 127 G4double delta = Step*root ;
138 G4double sum = ((typeT.*f)(xMean + delta) << 128 G4double sum = ((typeT.*f)(xMean + delta) +
139 << 129 (typeT.*f)(xMean - delta)) ;
140 return sum * Step; << 130
>> 131 return sum*Step ;
141 } 132 }
142 133
143 ////////////////////////////////////////////// 134 //////////////////////////////////////////////////////////////////////
144 // 135 //
145 // 136 //
146 137
147 template <class T, class F> << 138 template <class T, class F> G4double
148 G4double G4Integrator<T, F>::Gauss(T* ptrT, F << 139 G4Integrator<T,F>::Gauss( T* ptrT, F f, G4double a, G4double b )
149 { 140 {
150 return Gauss(*ptrT, f, a, b); << 141 return Gauss(*ptrT,f,a,b) ;
151 } 142 }
152 143
153 ////////////////////////////////////////////// 144 ///////////////////////////////////////////////////////////////////////
154 // 145 //
155 // 146 //
156 147
157 template <class T, class F> 148 template <class T, class F>
158 G4double G4Integrator<T, F>::Gauss(G4double (* << 149 G4double G4Integrator<T,F>::Gauss( G4double (*f)(G4double),
159 G4double xF << 150 G4double xInitial, G4double xFinal)
160 { 151 {
161 static const G4double root = 1.0 / std::sqrt << 152 static G4double root = 1.0/sqrt(3.0) ;
162 << 153
163 G4double xMean = (xInitial + xFinal) / 2.0; << 154 G4double xMean = (xInitial + xFinal)/2.0 ;
164 G4double Step = (xFinal - xInitial) / 2.0; << 155 G4double Step = (xFinal - xInitial)/2.0 ;
165 G4double delta = Step * root; << 156 G4double delta = Step*root ;
166 G4double sum = ((*f)(xMean + delta) + (*f) << 157 G4double sum = ( (*f)(xMean + delta) + (*f)(xMean - delta) ) ;
167 << 158
168 return sum * Step; << 159 return sum*Step ;
169 } 160 }
170 161
171 ////////////////////////////////////////////// 162 ///////////////////////////////////////////////////////////////////////////
172 // 163 //
173 // 164 //
174 165
175 template <class T, class F> << 166 template <class T, class F>
176 void G4Integrator<T, F>::AdaptGauss(T& typeT, << 167 void G4Integrator<T,F>::AdaptGauss( T& typeT, F f, G4double xInitial,
177 G4double x << 168 G4double xFinal, G4double fTolerance,
178 G4double& << 169 G4double& sum,
179 { << 170 G4int& depth )
180 if(depth > 100) << 171 {
181 { << 172 if(depth > 100)
182 G4cout << "G4Integrator<T,F>::AdaptGauss: << 173 {
183 G4cout << "Function varies too rapidly to << 174 G4cout<<"G4Integrator<T,F>::AdaptGauss: WARNING !!!"<<G4endl ;
184 << G4endl; << 175 G4cout
185 << 176 <<"Function varies too rapidly to get stated accuracy in 100 steps "<<G4endl ;
186 return; << 177
187 } << 178 return ;
188 G4double xMean = (xInitial + xFinal) / 2 << 179 }
189 G4double leftHalf = Gauss(typeT, f, xInitia << 180 G4double xMean = (xInitial + xFinal)/2.0 ;
190 G4double rightHalf = Gauss(typeT, f, xMean, << 181 G4double leftHalf = Gauss(typeT,f,xInitial,xMean) ;
191 G4double full = Gauss(typeT, f, xInitia << 182 G4double rightHalf = Gauss(typeT,f,xMean,xFinal) ;
192 if(std::fabs(leftHalf + rightHalf - full) < << 183 G4double full = Gauss(typeT,f,xInitial,xFinal) ;
193 { << 184 if(fabs(leftHalf+rightHalf-full) < fTolerance)
194 sum += full; << 185 {
195 } << 186 sum += full ;
196 else << 187 }
197 { << 188 else
198 ++depth; << 189 {
199 AdaptGauss(typeT, f, xInitial, xMean, fTol << 190 depth++ ;
200 AdaptGauss(typeT, f, xMean, xFinal, fToler << 191 AdaptGauss(typeT,f,xInitial,xMean,fTolerance,sum,depth) ;
201 } << 192 AdaptGauss(typeT,f,xMean,xFinal,fTolerance,sum,depth) ;
202 } << 193 }
203 << 194 }
204 template <class T, class F> << 195
205 void G4Integrator<T, F>::AdaptGauss(T* ptrT, F << 196 template <class T, class F>
206 G4double x << 197 void G4Integrator<T,F>::AdaptGauss( T* ptrT, F f, G4double xInitial,
207 G4double& << 198 G4double xFinal, G4double fTolerance,
>> 199 G4double& sum,
>> 200 G4int& depth )
208 { 201 {
209 AdaptGauss(*ptrT, f, xInitial, xFinal, fTole << 202 AdaptGauss(*ptrT,f,xInitial,xFinal,fTolerance,sum,depth) ;
210 } 203 }
211 204
212 ////////////////////////////////////////////// 205 /////////////////////////////////////////////////////////////////////////
213 // 206 //
214 // 207 //
215 template <class T, class F> 208 template <class T, class F>
216 void G4Integrator<T, F>::AdaptGauss(G4double ( << 209 void G4Integrator<T,F>::AdaptGauss( G4double (*f)(G4double),
217 G4double x << 210 G4double xInitial, G4double xFinal,
218 G4double& << 211 G4double fTolerance, G4double& sum,
219 { << 212 G4int& depth )
220 if(depth > 100) << 213 {
221 { << 214 if(depth > 100)
222 G4cout << "G4SimpleIntegration::AdaptGauss << 215 {
223 G4cout << "Function varies too rapidly to << 216 G4cout<<"G4SimpleIntegration::AdaptGauss: WARNING !!!"<<G4endl ;
224 << G4endl; << 217 G4cout<<"Function varies too rapidly to get stated accuracy in 100 steps "
225 << 218 <<G4endl ;
226 return; << 219
227 } << 220 return ;
228 G4double xMean = (xInitial + xFinal) / 2 << 221 }
229 G4double leftHalf = Gauss(f, xInitial, xMea << 222 G4double xMean = (xInitial + xFinal)/2.0 ;
230 G4double rightHalf = Gauss(f, xMean, xFinal) << 223 G4double leftHalf = Gauss(f,xInitial,xMean) ;
231 G4double full = Gauss(f, xInitial, xFin << 224 G4double rightHalf = Gauss(f,xMean,xFinal) ;
232 if(std::fabs(leftHalf + rightHalf - full) < << 225 G4double full = Gauss(f,xInitial,xFinal) ;
233 { << 226 if(fabs(leftHalf+rightHalf-full) < fTolerance)
234 sum += full; << 227 {
235 } << 228 sum += full ;
236 else << 229 }
237 { << 230 else
238 ++depth; << 231 {
239 AdaptGauss(f, xInitial, xMean, fTolerance, << 232 depth++ ;
240 AdaptGauss(f, xMean, xFinal, fTolerance, s << 233 AdaptGauss(f,xInitial,xMean,fTolerance,sum,depth) ;
241 } << 234 AdaptGauss(f,xMean,xFinal,fTolerance,sum,depth) ;
>> 235 }
242 } 236 }
243 237
>> 238
>> 239
>> 240
244 ////////////////////////////////////////////// 241 ////////////////////////////////////////////////////////////////////////
245 // 242 //
246 // Adaptive Gauss integration with accuracy 'e 243 // Adaptive Gauss integration with accuracy 'e'
247 // Convenient for using with class object type 244 // Convenient for using with class object typeT
248 << 245
249 template <class T, class F> << 246 template<class T, class F> G4double
250 G4double G4Integrator<T, F>::AdaptiveGauss(T& << 247 G4Integrator<T,F>::AdaptiveGauss( T& typeT, F f, G4double xInitial,
251 G4d << 248 G4double xFinal, G4double e )
252 { << 249 {
253 G4int depth = 0; << 250 G4int depth = 0 ;
254 G4double sum = 0.0; << 251 G4double sum = 0.0 ;
255 AdaptGauss(typeT, f, xInitial, xFinal, e, su << 252 AdaptGauss(typeT,f,xInitial,xFinal,e,sum,depth) ;
256 return sum; << 253 return sum ;
257 } 254 }
258 255
259 ////////////////////////////////////////////// 256 ////////////////////////////////////////////////////////////////////////
260 // 257 //
261 // Adaptive Gauss integration with accuracy 'e 258 // Adaptive Gauss integration with accuracy 'e'
262 // Convenient for using with 'this' pointer 259 // Convenient for using with 'this' pointer
263 << 260
264 template <class T, class F> << 261 template<class T, class F> G4double
265 G4double G4Integrator<T, F>::AdaptiveGauss(T* << 262 G4Integrator<T,F>::AdaptiveGauss( T* ptrT, F f, G4double xInitial,
266 G4d << 263 G4double xFinal, G4double e )
267 { 264 {
268 return AdaptiveGauss(*ptrT, f, xInitial, xFi << 265 return AdaptiveGauss(*ptrT,f,xInitial,xFinal,e) ;
269 } 266 }
270 267
271 ////////////////////////////////////////////// 268 ////////////////////////////////////////////////////////////////////////
272 // 269 //
273 // Adaptive Gauss integration with accuracy 'e 270 // Adaptive Gauss integration with accuracy 'e'
274 // Convenient for using with global scope func 271 // Convenient for using with global scope function f
275 << 272
276 template <class T, class F> << 273 template <class T, class F> G4double
277 G4double G4Integrator<T, F>::AdaptiveGauss(G4d << 274 G4Integrator<T,F>::AdaptiveGauss( G4double (*f)(G4double),
278 G4d << 275 G4double xInitial, G4double xFinal, G4double e )
279 G4d << 276 {
280 { << 277 G4int depth = 0 ;
281 G4int depth = 0; << 278 G4double sum = 0.0 ;
282 G4double sum = 0.0; << 279 AdaptGauss(f,xInitial,xFinal,e,sum,depth) ;
283 AdaptGauss(f, xInitial, xFinal, e, sum, dept << 280 return sum ;
284 return sum; <<
285 } 281 }
286 282
287 ////////////////////////////////////////////// 283 ////////////////////////////////////////////////////////////////////////////
288 // Gauss integration methods involving ortogon 284 // Gauss integration methods involving ortogonal polynomials
289 ////////////////////////////////////////////// 285 ////////////////////////////////////////////////////////////////////////////
290 // 286 //
291 // Methods involving Legendre polynomials << 287 // Methods involving Legendre polynomials
292 // 288 //
293 ////////////////////////////////////////////// 289 /////////////////////////////////////////////////////////////////////////
294 // 290 //
295 // The value nLegendre set the accuracy requir 291 // The value nLegendre set the accuracy required, i.e the number of points
296 // where the function pFunction will be evalua 292 // where the function pFunction will be evaluated during integration.
297 // The function creates the arrays for absciss << 293 // The function creates the arrays for abscissas and weights that used
298 // in Gauss-Legendre quadrature method. << 294 // in Gauss-Legendre quadrature method.
299 // The values a and b are the limits of integr 295 // The values a and b are the limits of integration of the function f .
300 // nLegendre MUST BE EVEN !!! 296 // nLegendre MUST BE EVEN !!!
301 // Returns the integral of the function f betw << 297 // Returns the integral of the function f between a and b, by 2*fNumber point
302 // Gauss-Legendre integration: the function is 298 // Gauss-Legendre integration: the function is evaluated exactly
303 // 2*fNumber times at interior points in the r << 299 // 2*fNumber times at interior points in the range of integration.
304 // Since the weights and abscissas are, in thi << 300 // Since the weights and abscissas are, in this case, symmetric around
305 // the midpoint of the range of integration, t << 301 // the midpoint of the range of integration, there are actually only
306 // fNumber distinct values of each. 302 // fNumber distinct values of each.
307 // Convenient for using with some class object 303 // Convenient for using with some class object dataT
308 304
309 template <class T, class F> << 305 template <class T, class F> G4double
310 G4double G4Integrator<T, F>::Legendre(T& typeT << 306 G4Integrator<T,F>::Legendre( T& typeT, F f, G4double a, G4double b, G4int nLegendre)
311 G4int nL <<
312 { 307 {
313 G4double nwt, nwt1, temp1, temp2, temp3, tem << 308 G4double newton, newton1, temp1, temp2, temp3, temp ;
314 G4double xDiff, xMean, dx, integral; << 309 G4double xDiff, xMean, dx, integral ;
315 310
316 const G4double tolerance = 1.6e-10; << 311 const G4double tolerance = 1.6e-10 ;
317 G4int i, j, k = nLegendre; << 312 G4int i, j, k = nLegendre ;
318 G4int fNumber = (nLegendre + 1) / 2; << 313 G4int fNumber = (nLegendre + 1)/2 ;
319 << 314
320 if(2 * fNumber != k) << 315 if(2*fNumber != k)
321 { << 316 {
322 G4Exception("G4Integrator<T,F>::Legendre(T << 317 G4Exception("Invalid (odd) n Legendre in G4Integrator<T,F>::Legendre") ;
323 FatalException, "Invalid (odd) << 318 }
324 } << 319
325 << 320 G4double* fAbscissa = new G4double[fNumber] ;
326 G4double* fAbscissa = new G4double[fNumber]; << 321 G4double* fWeight = new G4double[fNumber] ;
327 G4double* fWeight = new G4double[fNumber]; << 322
328 << 323 for(i=1;i<=fNumber;i++) // Loop over the desired roots
329 for(i = 1; i <= fNumber; ++i) // Loop over << 324 {
330 { << 325 newton = cos(pi*(i - 0.25)/(k + 0.5)) ; // Initial root approximation
331 nwt = std::cos(CLHEP::pi * (i - 0.25) / << 326
332 (k + 0.5)); // Initial roo << 327 do // loop of Newton's method
333 << 328 {
334 do // loop of Newton's method << 329 temp1 = 1.0 ;
335 { << 330 temp2 = 0.0 ;
336 temp1 = 1.0; << 331 for(j=1;j<=k;j++)
337 temp2 = 0.0; << 332 {
338 for(j = 1; j <= k; ++j) << 333 temp3 = temp2 ;
339 { << 334 temp2 = temp1 ;
340 temp3 = temp2; << 335 temp1 = ((2.0*j - 1.0)*newton*temp2 - (j - 1.0)*temp3)/j ;
341 temp2 = temp1; << 336 }
342 temp1 = ((2.0 * j - 1.0) * nwt * temp2 << 337 temp = k*(newton*temp1 - temp2)/(newton*newton - 1.0) ;
343 } << 338 newton1 = newton ;
344 temp = k * (nwt * temp1 - temp2) / (nwt << 339 newton = newton1 - temp1/temp ; // Newton's method
345 nwt1 = nwt; << 340 }
346 nwt = nwt1 - temp1 / temp; // Newton's << 341 while(fabs(newton - newton1) > tolerance) ;
347 } while(std::fabs(nwt - nwt1) > tolerance) << 342
348 << 343 fAbscissa[fNumber-i] = newton ;
349 fAbscissa[fNumber - i] = nwt; << 344 fWeight[fNumber-i] = 2.0/((1.0 - newton*newton)*temp*temp) ;
350 fWeight[fNumber - i] = 2.0 / ((1.0 - nwt << 345 }
351 } << 346 //
352 << 347 // Now we ready to get integral
353 // << 348 //
354 // Now we ready to get integral << 349
355 // << 350 xMean = 0.5*(a + b) ;
356 << 351 xDiff = 0.5*(b - a) ;
357 xMean = 0.5 * (a + b); << 352 integral = 0.0 ;
358 xDiff = 0.5 * (b - a); << 353 for(i=0;i<fNumber;i++)
359 integral = 0.0; << 354 {
360 for(i = 0; i < fNumber; ++i) << 355 dx = xDiff*fAbscissa[i] ;
361 { << 356 integral += fWeight[i]*( (typeT.*f)(xMean + dx) +
362 dx = xDiff * fAbscissa[i]; << 357 (typeT.*f)(xMean - dx) ) ;
363 integral += fWeight[i] * ((typeT.*f)(xMean << 358 }
364 } << 359 return integral *= xDiff ;
365 delete[] fAbscissa; << 360 }
366 delete[] fWeight; <<
367 return integral *= xDiff; <<
368 } <<
369 361
370 ////////////////////////////////////////////// 362 ///////////////////////////////////////////////////////////////////////
371 // 363 //
372 // Convenient for using with the pointer 'this 364 // Convenient for using with the pointer 'this'
373 365
374 template <class T, class F> << 366 template <class T, class F> G4double
375 G4double G4Integrator<T, F>::Legendre(T* ptrT, << 367 G4Integrator<T,F>::Legendre( T* ptrT, F f, G4double a, G4double b, G4int nLegendre)
376 G4int nL <<
377 { 368 {
378 return Legendre(*ptrT, f, a, b, nLegendre); << 369 return Legendre(*ptrT,f,a,b,nLegendre) ;
379 } 370 }
380 371
381 ////////////////////////////////////////////// 372 ///////////////////////////////////////////////////////////////////////
382 // 373 //
383 // Convenient for using with global scope func 374 // Convenient for using with global scope function f
384 375
385 template <class T, class F> 376 template <class T, class F>
386 G4double G4Integrator<T, F>::Legendre(G4double << 377 G4double G4Integrator<T,F>::
387 G4double << 378 Legendre( G4double (*f)(G4double), G4double a, G4double b, G4int nLegendre)
388 { 379 {
389 G4double nwt, nwt1, temp1, temp2, temp3, tem << 380 G4double newton, newton1, temp1, temp2, temp3, temp ;
390 G4double xDiff, xMean, dx, integral; << 381 G4double xDiff, xMean, dx, integral ;
391 <<
392 const G4double tolerance = 1.6e-10; <<
393 G4int i, j, k = nLegendre; <<
394 G4int fNumber = (nLegendre + 1) / 2; <<
395 <<
396 if(2 * fNumber != k) <<
397 { <<
398 G4Exception("G4Integrator<T,F>::Legendre(. <<
399 FatalException, "Invalid (odd) <<
400 } <<
401 <<
402 G4double* fAbscissa = new G4double[fNumber]; <<
403 G4double* fWeight = new G4double[fNumber]; <<
404 <<
405 for(i = 1; i <= fNumber; i++) // Loop over <<
406 { <<
407 nwt = std::cos(CLHEP::pi * (i - 0.25) / <<
408 (k + 0.5)); // Initial roo <<
409 382
410 do // loop of Newton's method << 383 const G4double tolerance = 1.6e-10 ;
411 { << 384 G4int i, j, k = nLegendre ;
412 temp1 = 1.0; << 385 G4int fNumber = (nLegendre + 1)/2 ;
413 temp2 = 0.0; << 386
414 for(j = 1; j <= k; ++j) << 387 if(2*fNumber != k)
415 { << 388 {
416 temp3 = temp2; << 389 G4Exception("Invalid (odd) n Legendre in G4Integrator<T,F>::Legendre") ;
417 temp2 = temp1; << 390 }
418 temp1 = ((2.0 * j - 1.0) * nwt * temp2 << 391
419 } << 392 G4double* fAbscissa = new G4double[fNumber] ;
420 temp = k * (nwt * temp1 - temp2) / (nwt << 393 G4double* fWeight = new G4double[fNumber] ;
421 nwt1 = nwt; << 394
422 nwt = nwt1 - temp1 / temp; // Newton's << 395 for(i=1;i<=fNumber;i++) // Loop over the desired roots
423 } while(std::fabs(nwt - nwt1) > tolerance) << 396 {
424 << 397 newton = cos(pi*(i - 0.25)/(k + 0.5)) ; // Initial root approximation
425 fAbscissa[fNumber - i] = nwt; << 398
426 fWeight[fNumber - i] = 2.0 / ((1.0 - nwt << 399 do // loop of Newton's method
427 } << 400 {
428 << 401 temp1 = 1.0 ;
429 // << 402 temp2 = 0.0 ;
430 // Now we ready to get integral << 403 for(j=1;j<=k;j++)
431 // << 404 {
432 << 405 temp3 = temp2 ;
433 xMean = 0.5 * (a + b); << 406 temp2 = temp1 ;
434 xDiff = 0.5 * (b - a); << 407 temp1 = ((2.0*j - 1.0)*newton*temp2 - (j - 1.0)*temp3)/j ;
435 integral = 0.0; << 408 }
436 for(i = 0; i < fNumber; ++i) << 409 temp = k*(newton*temp1 - temp2)/(newton*newton - 1.0) ;
437 { << 410 newton1 = newton ;
438 dx = xDiff * fAbscissa[i]; << 411 newton = newton1 - temp1/temp ; // Newton's method
439 integral += fWeight[i] * ((*f)(xMean + dx) << 412 }
440 } << 413 while(fabs(newton - newton1) > tolerance) ;
441 delete[] fAbscissa; << 414
442 delete[] fWeight; << 415 fAbscissa[fNumber-i] = newton ;
443 << 416 fWeight[fNumber-i] = 2.0/((1.0 - newton*newton)*temp*temp) ;
444 return integral *= xDiff; << 417 }
445 } << 418 //
>> 419 // Now we ready to get integral
>> 420 //
>> 421
>> 422 xMean = 0.5*(a + b) ;
>> 423 xDiff = 0.5*(b - a) ;
>> 424 integral = 0.0 ;
>> 425 for(i=0;i<fNumber;i++)
>> 426 {
>> 427 dx = xDiff*fAbscissa[i] ;
>> 428 integral += fWeight[i]*( (*f)(xMean + dx) + (*f)(xMean - dx) ) ;
>> 429 }
>> 430 return integral *= xDiff ;
>> 431 }
446 432
447 ////////////////////////////////////////////// 433 ////////////////////////////////////////////////////////////////////////////
448 // 434 //
449 // Returns the integral of the function to be 435 // Returns the integral of the function to be pointed by T::f between a and b,
450 // by ten point Gauss-Legendre integration: th 436 // by ten point Gauss-Legendre integration: the function is evaluated exactly
451 // ten times at interior points in the range o 437 // ten times at interior points in the range of integration. Since the weights
452 // and abscissas are, in this case, symmetric << 438 // and abscissas are, in this case, symmetric around the midpoint of the
453 // range of integration, there are actually on 439 // range of integration, there are actually only five distinct values of each
454 // Convenient for using with class object type 440 // Convenient for using with class object typeT
455 441
456 template <class T, class F> << 442 template <class T, class F>
457 G4double G4Integrator<T, F>::Legendre10(T& typ << 443 G4double G4Integrator<T,F>::Legendre10( T& typeT, F f,G4double a, G4double b)
458 { 444 {
459 G4int i; << 445 G4int i ;
460 G4double xDiff, xMean, dx, integral; << 446 G4double xDiff, xMean, dx, integral ;
461 << 447
462 // From Abramowitz M., Stegan I.A. 1964 , Ha << 448 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 916
463 << 449
464 static const G4double abscissa[] = { 0.14887 << 450 static G4double abscissa[] = { 0.148874338981631, 0.433395394129247,
465 0.67940 << 451 0.679409568299024, 0.865063366688985,
466 0.97390 << 452 0.973906528517172 } ;
467 << 453
468 static const G4double weight[] = { 0.2955242 << 454 static G4double weight[] = { 0.295524224714753, 0.269266719309996,
469 0.2190863 << 455 0.219086362515982, 0.149451349150581,
470 0.0666713 << 456 0.066671344308688 } ;
471 xMean = 0.5 * (a + << 457 xMean = 0.5*(a + b) ;
472 xDiff = 0.5 * (b - << 458 xDiff = 0.5*(b - a) ;
473 integral = 0.0; << 459 integral = 0.0 ;
474 for(i = 0; i < 5; ++i) << 460 for(i=0;i<5;i++)
475 { << 461 {
476 dx = xDiff * abscissa[i]; << 462 dx = xDiff*abscissa[i] ;
477 integral += weight[i] * ((typeT.*f)(xMean << 463 integral += weight[i]*( (typeT.*f)(xMean + dx) + (typeT.*f)(xMean - dx)) ;
478 } << 464 }
479 return integral *= xDiff; << 465 return integral *= xDiff ;
480 } 466 }
481 467
482 ////////////////////////////////////////////// 468 ///////////////////////////////////////////////////////////////////////////
483 // 469 //
484 // Convenient for using with the pointer 'this 470 // Convenient for using with the pointer 'this'
485 471
486 template <class T, class F> << 472 template <class T, class F>
487 G4double G4Integrator<T, F>::Legendre10(T* ptr << 473 G4double G4Integrator<T,F>::Legendre10( T* ptrT, F f,G4double a, G4double b)
488 { 474 {
489 return Legendre10(*ptrT, f, a, b); << 475 return Legendre10(*ptrT,f,a,b) ;
490 } << 476 }
491 477
492 ////////////////////////////////////////////// 478 //////////////////////////////////////////////////////////////////////////
493 // 479 //
494 // Convenient for using with global scope func 480 // Convenient for using with global scope functions
495 481
496 template <class T, class F> << 482 template <class T, class F> G4double
497 G4double G4Integrator<T, F>::Legendre10(G4doub << 483 G4Integrator<T,F>::Legendre10( G4double (*f)(G4double), G4double a, G4double b)
498 G4doub <<
499 { 484 {
500 G4int i; << 485 G4int i ;
501 G4double xDiff, xMean, dx, integral; << 486 G4double xDiff, xMean, dx, integral ;
502 << 487
503 // From Abramowitz M., Stegan I.A. 1964 , Ha << 488 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 916
504 << 489
505 static const G4double abscissa[] = { 0.14887 << 490 static G4double abscissa[] = { 0.148874338981631, 0.433395394129247,
506 0.67940 << 491 0.679409568299024, 0.865063366688985,
507 0.97390 << 492 0.973906528517172 } ;
508 << 493
509 static const G4double weight[] = { 0.2955242 << 494 static G4double weight[] = { 0.295524224714753, 0.269266719309996,
510 0.2190863 << 495 0.219086362515982, 0.149451349150581,
511 0.0666713 << 496 0.066671344308688 } ;
512 xMean = 0.5 * (a + << 497 xMean = 0.5*(a + b) ;
513 xDiff = 0.5 * (b - << 498 xDiff = 0.5*(b - a) ;
514 integral = 0.0; << 499 integral = 0.0 ;
515 for(i = 0; i < 5; ++i) << 500 for(i=0;i<5;i++)
516 { << 501 {
517 dx = xDiff * abscissa[i]; << 502 dx = xDiff*abscissa[i] ;
518 integral += weight[i] * ((*f)(xMean + dx) << 503 integral += weight[i]*( (*f)(xMean + dx) + (*f)(xMean - dx)) ;
519 } << 504 }
520 return integral *= xDiff; << 505 return integral *= xDiff ;
521 } 506 }
522 507
523 ////////////////////////////////////////////// 508 ///////////////////////////////////////////////////////////////////////
524 // 509 //
525 // Returns the integral of the function to be 510 // Returns the integral of the function to be pointed by T::f between a and b,
526 // by 96 point Gauss-Legendre integration: the 511 // by 96 point Gauss-Legendre integration: the function is evaluated exactly
527 // ten Times at interior points in the range o 512 // ten Times at interior points in the range of integration. Since the weights
528 // and abscissas are, in this case, symmetric << 513 // and abscissas are, in this case, symmetric around the midpoint of the
529 // range of integration, there are actually on 514 // range of integration, there are actually only five distinct values of each
530 // Convenient for using with some class object 515 // Convenient for using with some class object typeT
531 516
532 template <class T, class F> << 517 template <class T, class F>
533 G4double G4Integrator<T, F>::Legendre96(T& typ << 518 G4double G4Integrator<T,F>::Legendre96( T& typeT, F f,G4double a, G4double b)
534 { 519 {
535 G4int i; << 520 G4int i ;
536 G4double xDiff, xMean, dx, integral; << 521 G4double xDiff, xMean, dx, integral ;
537 << 522
538 // From Abramowitz M., Stegan I.A. 1964 , Ha << 523 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 919
539 << 524
540 static const G4double abscissa[] = { << 525 static G4double
541 0.016276744849602969579, 0.048812985136049 << 526 abscissa[] = {
542 0.081297495464425558994, 0.113695850110665 << 527 0.016276744849602969579, 0.048812985136049731112,
543 0.145973714654896941989, 0.178096882367618 << 528 0.081297495464425558994, 0.113695850110665920911,
544 << 529 0.145973714654896941989, 0.178096882367618602759, // 6
545 0.210031310460567203603, 0.241743156163840 << 530
546 0.273198812591049141487, 0.304364944354496 << 531 0.210031310460567203603, 0.241743156163840012328,
547 0.335208522892625422616, 0.365696861472313 << 532 0.273198812591049141487, 0.304364944354496353024,
548 << 533 0.335208522892625422616, 0.365696861472313635031, // 12
549 0.395797649828908603285, 0.425478988407300 << 534
550 0.454709422167743008636, 0.483457973920596 << 535 0.395797649828908603285, 0.425478988407300545365,
551 0.511694177154667673586, 0.539388108324357 << 536 0.454709422167743008636, 0.483457973920596359768,
552 << 537 0.511694177154667673586, 0.539388108324357436227, // 18
553 0.566510418561397168404, 0.593032364777572 << 538
554 0.618925840125468570386, 0.644163403784967 << 539 0.566510418561397168404, 0.593032364777572080684,
555 0.668718310043916153953, 0.692564536642171 << 540 0.618925840125468570386, 0.644163403784967106798,
556 << 541 0.668718310043916153953, 0.692564536642171561344, // 24
557 0.715676812348967626225, 0.738030643744400 << 542
558 0.759602341176647498703, 0.780369043867433 << 543 0.715676812348967626225, 0.738030643744400132851,
559 0.800308744139140817229, 0.819400310737931 << 544 0.759602341176647498703, 0.780369043867433217604,
560 << 545 0.800308744139140817229, 0.819400310737931675539, // 30
561 0.837623511228187121494, 0.854959033434601 << 546
562 0.871388505909296502874, 0.886894517402420 << 547 0.837623511228187121494, 0.854959033434601455463,
563 0.901460635315852341319, 0.915071423120898 << 548 0.871388505909296502874, 0.886894517402420416057,
564 << 549 0.901460635315852341319, 0.915071423120898074206, // 36
565 0.927712456722308690965, 0.939370339752755 << 550
566 0.950032717784437635756, 0.959688291448742 << 551 0.927712456722308690965, 0.939370339752755216932,
567 0.968326828463264212174, 0.975939174585136 << 552 0.950032717784437635756, 0.959688291448742539300,
568 << 553 0.968326828463264212174, 0.975939174585136466453, // 42
569 0.982517263563014677447, 0.988054126329623 << 554
570 0.992543900323762624572, 0.995981842987209 << 555 0.982517263563014677447, 0.988054126329623799481,
571 0.998364375863181677724, 0.999689503883230 << 556 0.992543900323762624572, 0.995981842987209290650,
572 }; << 557 0.998364375863181677724, 0.999689503883230766828 // 48
573 << 558 } ;
574 static const G4double weight[] = { << 559
575 0.032550614492363166242, 0.032516118713868 << 560 static G4double
576 0.032447163714064269364, 0.032343822568575 << 561 weight[] = {
577 0.032206204794030250669, 0.032034456231992 << 562 0.032550614492363166242, 0.032516118713868835987,
578 << 563 0.032447163714064269364, 0.032343822568575928429,
579 0.031828758894411006535, 0.031589330770727 << 564 0.032206204794030250669, 0.032034456231992663218, // 6
580 0.031316425596862355813, 0.031010332586313 << 565
581 0.030671376123669149014, 0.030299915420827 << 566 0.031828758894411006535, 0.031589330770727168558,
582 << 567 0.031316425596862355813, 0.031010332586313837423,
583 0.029896344136328385984, 0.029461089958167 << 568 0.030671376123669149014, 0.030299915420827593794, // 12
584 0.028994614150555236543, 0.028497411065085 << 569
585 0.027970007616848334440, 0.027412962726029 << 570 0.029896344136328385984, 0.029461089958167905970,
586 << 571 0.028994614150555236543, 0.028497411065085385646,
587 0.026826866725591762198, 0.026212340735672 << 572 0.027970007616848334440, 0.027412962726029242823, // 18
588 0.025570036005349361499, 0.024900633222483 << 573
589 0.024204841792364691282, 0.023483399085926 << 574 0.026826866725591762198, 0.026212340735672413913,
590 << 575 0.025570036005349361499, 0.024900633222483610288,
591 0.022737069658329374001, 0.021966644438744 << 576 0.024204841792364691282, 0.023483399085926219842, // 24
592 0.021172939892191298988, 0.020356797154333 << 577
593 0.019519081140145022410, 0.018660679627411 << 578 0.022737069658329374001, 0.021966644438744349195,
594 << 579 0.021172939892191298988, 0.020356797154333324595,
595 0.017782502316045260838, 0.016885479864245 << 580 0.019519081140145022410, 0.018660679627411467385, // 30
596 0.015970562902562291381, 0.015038721026994 << 581
597 0.014090941772314860916, 0.013128229566961 << 582 0.017782502316045260838, 0.016885479864245172450,
598 << 583 0.015970562902562291381, 0.015038721026994938006,
599 0.012151604671088319635, 0.011162102099838 << 584 0.014090941772314860916, 0.013128229566961572637, // 36
600 0.010160770535008415758, 0.009148671230783 << 585
601 0.008126876925698759217, 0.007096470791153 << 586 0.012151604671088319635, 0.011162102099838498591,
602 << 587 0.010160770535008415758, 0.009148671230783386633,
603 0.006058545504235961683, 0.005014202742927 << 588 0.008126876925698759217, 0.007096470791153865269, // 42
604 0.003964554338444686674, 0.002910731817934 << 589
605 0.001853960788946921732, 0.000796792065552 << 590 0.006058545504235961683, 0.005014202742927517693,
606 }; << 591 0.003964554338444686674, 0.002910731817934946408,
607 xMean = 0.5 * (a + b); << 592 0.001853960788946921732, 0.000796792065552012429 // 48
608 xDiff = 0.5 * (b - a); << 593 } ;
609 integral = 0.0; << 594 xMean = 0.5*(a + b) ;
610 for(i = 0; i < 48; ++i) << 595 xDiff = 0.5*(b - a) ;
611 { << 596 integral = 0.0 ;
612 dx = xDiff * abscissa[i]; << 597 for(i=0;i<48;i++)
613 integral += weight[i] * ((typeT.*f)(xMean << 598 {
614 } << 599 dx = xDiff*abscissa[i] ;
615 return integral *= xDiff; << 600 integral += weight[i]*((typeT.*f)(xMean + dx) + (typeT.*f)(xMean - dx)) ;
>> 601 }
>> 602 return integral *= xDiff ;
616 } 603 }
617 604
618 ////////////////////////////////////////////// 605 ///////////////////////////////////////////////////////////////////////
619 // 606 //
620 // Convenient for using with the pointer 'this 607 // Convenient for using with the pointer 'this'
621 608
622 template <class T, class F> << 609 template <class T, class F>
623 G4double G4Integrator<T, F>::Legendre96(T* ptr << 610 G4double G4Integrator<T,F>::Legendre96( T* ptrT, F f,G4double a, G4double b)
624 { 611 {
625 return Legendre96(*ptrT, f, a, b); << 612 return Legendre96(*ptrT,f,a,b) ;
626 } << 613 }
627 614
628 ////////////////////////////////////////////// 615 ///////////////////////////////////////////////////////////////////////
629 // 616 //
630 // Convenient for using with global scope func << 617 // Convenient for using with global scope function f
631 618
632 template <class T, class F> << 619 template <class T, class F> G4double
633 G4double G4Integrator<T, F>::Legendre96(G4doub << 620 G4Integrator<T,F>::Legendre96( G4double (*f)(G4double), G4double a, G4double b)
634 G4doub <<
635 { 621 {
636 G4int i; << 622 G4int i ;
637 G4double xDiff, xMean, dx, integral; << 623 G4double xDiff, xMean, dx, integral ;
638 << 624
639 // From Abramowitz M., Stegan I.A. 1964 , Ha << 625 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 919
640 << 626
641 static const G4double abscissa[] = { << 627 static G4double
642 0.016276744849602969579, 0.048812985136049 << 628 abscissa[] = {
643 0.081297495464425558994, 0.113695850110665 << 629 0.016276744849602969579, 0.048812985136049731112,
644 0.145973714654896941989, 0.178096882367618 << 630 0.081297495464425558994, 0.113695850110665920911,
645 << 631 0.145973714654896941989, 0.178096882367618602759, // 6
646 0.210031310460567203603, 0.241743156163840 << 632
647 0.273198812591049141487, 0.304364944354496 << 633 0.210031310460567203603, 0.241743156163840012328,
648 0.335208522892625422616, 0.365696861472313 << 634 0.273198812591049141487, 0.304364944354496353024,
649 << 635 0.335208522892625422616, 0.365696861472313635031, // 12
650 0.395797649828908603285, 0.425478988407300 << 636
651 0.454709422167743008636, 0.483457973920596 << 637 0.395797649828908603285, 0.425478988407300545365,
652 0.511694177154667673586, 0.539388108324357 << 638 0.454709422167743008636, 0.483457973920596359768,
653 << 639 0.511694177154667673586, 0.539388108324357436227, // 18
654 0.566510418561397168404, 0.593032364777572 << 640
655 0.618925840125468570386, 0.644163403784967 << 641 0.566510418561397168404, 0.593032364777572080684,
656 0.668718310043916153953, 0.692564536642171 << 642 0.618925840125468570386, 0.644163403784967106798,
657 << 643 0.668718310043916153953, 0.692564536642171561344, // 24
658 0.715676812348967626225, 0.738030643744400 << 644
659 0.759602341176647498703, 0.780369043867433 << 645 0.715676812348967626225, 0.738030643744400132851,
660 0.800308744139140817229, 0.819400310737931 << 646 0.759602341176647498703, 0.780369043867433217604,
661 << 647 0.800308744139140817229, 0.819400310737931675539, // 30
662 0.837623511228187121494, 0.854959033434601 << 648
663 0.871388505909296502874, 0.886894517402420 << 649 0.837623511228187121494, 0.854959033434601455463,
664 0.901460635315852341319, 0.915071423120898 << 650 0.871388505909296502874, 0.886894517402420416057,
665 << 651 0.901460635315852341319, 0.915071423120898074206, // 36
666 0.927712456722308690965, 0.939370339752755 << 652
667 0.950032717784437635756, 0.959688291448742 << 653 0.927712456722308690965, 0.939370339752755216932,
668 0.968326828463264212174, 0.975939174585136 << 654 0.950032717784437635756, 0.959688291448742539300,
669 << 655 0.968326828463264212174, 0.975939174585136466453, // 42
670 0.982517263563014677447, 0.988054126329623 << 656
671 0.992543900323762624572, 0.995981842987209 << 657 0.982517263563014677447, 0.988054126329623799481,
672 0.998364375863181677724, 0.999689503883230 << 658 0.992543900323762624572, 0.995981842987209290650,
673 }; << 659 0.998364375863181677724, 0.999689503883230766828 // 48
674 << 660 } ;
675 static const G4double weight[] = { << 661
676 0.032550614492363166242, 0.032516118713868 << 662 static G4double
677 0.032447163714064269364, 0.032343822568575 << 663 weight[] = {
678 0.032206204794030250669, 0.032034456231992 << 664 0.032550614492363166242, 0.032516118713868835987,
679 << 665 0.032447163714064269364, 0.032343822568575928429,
680 0.031828758894411006535, 0.031589330770727 << 666 0.032206204794030250669, 0.032034456231992663218, // 6
681 0.031316425596862355813, 0.031010332586313 << 667
682 0.030671376123669149014, 0.030299915420827 << 668 0.031828758894411006535, 0.031589330770727168558,
683 << 669 0.031316425596862355813, 0.031010332586313837423,
684 0.029896344136328385984, 0.029461089958167 << 670 0.030671376123669149014, 0.030299915420827593794, // 12
685 0.028994614150555236543, 0.028497411065085 << 671
686 0.027970007616848334440, 0.027412962726029 << 672 0.029896344136328385984, 0.029461089958167905970,
687 << 673 0.028994614150555236543, 0.028497411065085385646,
688 0.026826866725591762198, 0.026212340735672 << 674 0.027970007616848334440, 0.027412962726029242823, // 18
689 0.025570036005349361499, 0.024900633222483 << 675
690 0.024204841792364691282, 0.023483399085926 << 676 0.026826866725591762198, 0.026212340735672413913,
691 << 677 0.025570036005349361499, 0.024900633222483610288,
692 0.022737069658329374001, 0.021966644438744 << 678 0.024204841792364691282, 0.023483399085926219842, // 24
693 0.021172939892191298988, 0.020356797154333 << 679
694 0.019519081140145022410, 0.018660679627411 << 680 0.022737069658329374001, 0.021966644438744349195,
695 << 681 0.021172939892191298988, 0.020356797154333324595,
696 0.017782502316045260838, 0.016885479864245 << 682 0.019519081140145022410, 0.018660679627411467385, // 30
697 0.015970562902562291381, 0.015038721026994 << 683
698 0.014090941772314860916, 0.013128229566961 << 684 0.017782502316045260838, 0.016885479864245172450,
699 << 685 0.015970562902562291381, 0.015038721026994938006,
700 0.012151604671088319635, 0.011162102099838 << 686 0.014090941772314860916, 0.013128229566961572637, // 36
701 0.010160770535008415758, 0.009148671230783 << 687
702 0.008126876925698759217, 0.007096470791153 << 688 0.012151604671088319635, 0.011162102099838498591,
703 << 689 0.010160770535008415758, 0.009148671230783386633,
704 0.006058545504235961683, 0.005014202742927 << 690 0.008126876925698759217, 0.007096470791153865269, // 42
705 0.003964554338444686674, 0.002910731817934 << 691
706 0.001853960788946921732, 0.000796792065552 << 692 0.006058545504235961683, 0.005014202742927517693,
707 }; << 693 0.003964554338444686674, 0.002910731817934946408,
708 xMean = 0.5 * (a + b); << 694 0.001853960788946921732, 0.000796792065552012429 // 48
709 xDiff = 0.5 * (b - a); << 695 } ;
710 integral = 0.0; << 696 xMean = 0.5*(a + b) ;
711 for(i = 0; i < 48; ++i) << 697 xDiff = 0.5*(b - a) ;
712 { << 698 integral = 0.0 ;
713 dx = xDiff * abscissa[i]; << 699 for(i=0;i<48;i++)
714 integral += weight[i] * ((*f)(xMean + dx) << 700 {
715 } << 701 dx = xDiff*abscissa[i] ;
716 return integral *= xDiff; << 702 integral += weight[i]*((*f)(xMean + dx) + (*f)(xMean - dx)) ;
>> 703 }
>> 704 return integral *= xDiff ;
717 } 705 }
718 706
719 ////////////////////////////////////////////// 707 //////////////////////////////////////////////////////////////////////////////
720 // 708 //
721 // Methods involving Chebyshev polynomials << 709 // Methods involving Chebyshev polynomials
722 // 710 //
723 ////////////////////////////////////////////// 711 ///////////////////////////////////////////////////////////////////////////
724 // 712 //
725 // Integrates function pointed by T::f from a << 713 // Integrates function pointed by T::f from a to b by Gauss-Chebyshev
726 // quadrature method. 714 // quadrature method.
727 // Convenient for using with class object type 715 // Convenient for using with class object typeT
728 716
729 template <class T, class F> << 717 template <class T, class F> G4double
730 G4double G4Integrator<T, F>::Chebyshev(T& type << 718 G4Integrator<T,F>::Chebyshev( T& typeT, F f, G4double a,
731 G4int n << 719 G4double b, G4int nChebyshev )
732 { << 720 {
733 G4int i; << 721 G4int i ;
734 G4double xDiff, xMean, dx, integral = 0.0; << 722 G4double xDiff, xMean, dx, integral = 0.0 ;
735 << 723
736 G4int fNumber = nChebyshev; // Try to << 724 G4int fNumber = nChebyshev ; // Try to reduce fNumber twice ??
737 G4double cof = CLHEP::pi / fNumber; << 725 G4double cof = pi/fNumber ;
738 G4double* fAbscissa = new G4double[fNumber]; << 726 G4double* fAbscissa = new G4double[fNumber] ;
739 G4double* fWeight = new G4double[fNumber]; << 727 G4double* fWeight = new G4double[fNumber] ;
740 for(i = 0; i < fNumber; ++i) << 728 for(i=0;i<fNumber;i++)
741 { << 729 {
742 fAbscissa[i] = std::cos(cof * (i + 0.5)); << 730 fAbscissa[i] = cos(cof*(i + 0.5)) ;
743 fWeight[i] = cof * std::sqrt(1 - fAbscis << 731 fWeight[i] = cof*sqrt(1 - fAbscissa[i]*fAbscissa[i]) ;
744 } << 732 }
745 << 733 //
746 // << 734 // Now we ready to estimate the integral
747 // Now we ready to estimate the integral << 735 //
748 // << 736 xMean = 0.5*(a + b) ;
749 << 737 xDiff = 0.5*(b - a) ;
750 xMean = 0.5 * (a + b); << 738 for(i=0;i<fNumber;i++)
751 xDiff = 0.5 * (b - a); << 739 {
752 for(i = 0; i < fNumber; ++i) << 740 dx = xDiff*fAbscissa[i] ;
753 { << 741 integral += fWeight[i]*(typeT.*f)(xMean + dx) ;
754 dx = xDiff * fAbscissa[i]; << 742 }
755 integral += fWeight[i] * (typeT.*f)(xMean << 743 return integral *= xDiff ;
756 } <<
757 delete[] fAbscissa; <<
758 delete[] fWeight; <<
759 return integral *= xDiff; <<
760 } 744 }
761 745
762 ////////////////////////////////////////////// 746 ///////////////////////////////////////////////////////////////////////
763 // 747 //
764 // Convenient for using with 'this' pointer 748 // Convenient for using with 'this' pointer
765 749
766 template <class T, class F> << 750 template <class T, class F> G4double
767 G4double G4Integrator<T, F>::Chebyshev(T* ptrT << 751 G4Integrator<T,F>::Chebyshev( T* ptrT, F f, G4double a, G4double b, G4int n)
768 G4int n <<
769 { 752 {
770 return Chebyshev(*ptrT, f, a, b, n); << 753 return Chebyshev(*ptrT,f,a,b,n) ;
771 } << 754 }
772 755
773 ////////////////////////////////////////////// 756 ////////////////////////////////////////////////////////////////////////
774 // 757 //
775 // For use with global scope functions f << 758 // For use with global scope functions f
776 <<
777 template <class T, class F> <<
778 G4double G4Integrator<T, F>::Chebyshev(G4doubl <<
779 G4doubl <<
780 { <<
781 G4int i; <<
782 G4double xDiff, xMean, dx, integral = 0.0; <<
783 759
784 G4int fNumber = nChebyshev; // Try to << 760 template <class T, class F> G4double
785 G4double cof = CLHEP::pi / fNumber; << 761 G4Integrator<T,F>::Chebyshev( G4double (*f)(G4double),
786 G4double* fAbscissa = new G4double[fNumber]; << 762 G4double a, G4double b, G4int nChebyshev)
787 G4double* fWeight = new G4double[fNumber]; << 763 {
788 for(i = 0; i < fNumber; ++i) << 764 G4int i ;
789 { << 765 G4double xDiff, xMean, dx, integral = 0.0 ;
790 fAbscissa[i] = std::cos(cof * (i + 0.5)); << 766
791 fWeight[i] = cof * std::sqrt(1 - fAbscis << 767 G4int fNumber = nChebyshev ; // Try to reduce fNumber twice ??
792 } << 768 G4double cof = pi/fNumber ;
793 << 769 G4double* fAbscissa = new G4double[fNumber] ;
794 // << 770 G4double* fWeight = new G4double[fNumber] ;
795 // Now we ready to estimate the integral << 771 for(i=0;i<fNumber;i++)
796 // << 772 {
797 << 773 fAbscissa[i] = cos(cof*(i + 0.5)) ;
798 xMean = 0.5 * (a + b); << 774 fWeight[i] = cof*sqrt(1 - fAbscissa[i]*fAbscissa[i]) ;
799 xDiff = 0.5 * (b - a); << 775 }
800 for(i = 0; i < fNumber; ++i) << 776 //
801 { << 777 // Now we ready to estimate the integral
802 dx = xDiff * fAbscissa[i]; << 778 //
803 integral += fWeight[i] * (*f)(xMean + dx); << 779 xMean = 0.5*(a + b) ;
804 } << 780 xDiff = 0.5*(b - a) ;
805 delete[] fAbscissa; << 781 for(i=0;i<fNumber;i++)
806 delete[] fWeight; << 782 {
807 return integral *= xDiff; << 783 dx = xDiff*fAbscissa[i] ;
>> 784 integral += fWeight[i]*(*f)(xMean + dx) ;
>> 785 }
>> 786 return integral *= xDiff ;
808 } 787 }
809 788
810 ////////////////////////////////////////////// 789 //////////////////////////////////////////////////////////////////////
811 // 790 //
812 // Method involving Laguerre polynomials 791 // Method involving Laguerre polynomials
813 // 792 //
814 ////////////////////////////////////////////// 793 //////////////////////////////////////////////////////////////////////
815 // 794 //
816 // Integral from zero to infinity of std::pow( << 795 // Integral from zero to infinity of pow(x,alpha)*exp(-x)*f(x).
817 // The value of nLaguerre sets the accuracy. 796 // The value of nLaguerre sets the accuracy.
818 // The function creates arrays fAbscissa[0,.., << 797 // The function creates arrays fAbscissa[0,..,nLaguerre-1] and
819 // fWeight[0,..,nLaguerre-1] . << 798 // fWeight[0,..,nLaguerre-1] .
820 // Convenient for using with class object 'typ 799 // Convenient for using with class object 'typeT' and (typeT.*f) function
821 // (T::f) 800 // (T::f)
822 801
823 template <class T, class F> << 802 template <class T, class F> G4double
824 G4double G4Integrator<T, F>::Laguerre(T& typeT << 803 G4Integrator<T,F>::Laguerre( T& typeT, F f, G4double alpha, G4int nLaguerre )
825 G4int nL <<
826 { 804 {
827 const G4double tolerance = 1.0e-10; << 805 const G4double tolerance = 1.0e-10 ;
828 const G4int maxNumber = 12; << 806 const G4int maxNumber = 12 ;
829 G4int i, j, k; << 807 G4int i, j, k ;
830 G4double nwt = 0., nwt1, temp1, temp2, << 808 G4double newton, newton1, temp1, temp2, temp3, temp, cofi ;
831 G4double integral = 0.0; << 809 G4double integral = 0.0 ;
832 << 810
833 G4int fNumber = nLaguerre; << 811 G4int fNumber = nLaguerre ;
834 G4double* fAbscissa = new G4double[fNumber]; << 812 G4double* fAbscissa = new G4double[fNumber] ;
835 G4double* fWeight = new G4double[fNumber]; << 813 G4double* fWeight = new G4double[fNumber] ;
>> 814
>> 815 for(i=1;i<=fNumber;i++) // Loop over the desired roots
>> 816 {
>> 817 if(i == 1)
>> 818 {
>> 819 newton = (1.0 + alpha)*(3.0 + 0.92*alpha)/(1.0 + 2.4*fNumber + 1.8*alpha) ;
>> 820 }
>> 821 else if(i == 2)
>> 822 {
>> 823 newton += (15.0 + 6.25*alpha)/(1.0 + 0.9*alpha + 2.5*fNumber) ;
>> 824 }
>> 825 else
>> 826 {
>> 827 cofi = i - 2 ;
>> 828 newton += ((1.0+2.55*cofi)/(1.9*cofi) + 1.26*cofi*alpha/(1.0+3.5*cofi))*
>> 829 (newton - fAbscissa[i-3])/(1.0 + 0.3*alpha) ;
>> 830 }
>> 831 for(k=1;k<=maxNumber;k++)
>> 832 {
>> 833 temp1 = 1.0 ;
>> 834 temp2 = 0.0 ;
836 835
837 for(i = 1; i <= fNumber; ++i) // Loop over << 836 for(j=1;j<=fNumber;j++)
838 { << 837 {
839 if(i == 1) << 838 temp3 = temp2 ;
840 { << 839 temp2 = temp1 ;
841 nwt = (1.0 + alpha) * (3.0 + 0.92 * alph << 840 temp1 = ((2*j - 1 + alpha - newton)*temp2 - (j - 1 + alpha)*temp3)/j ;
842 (1.0 + 2.4 * fNumber + 1.8 * alpha << 841 }
843 } << 842 temp = (fNumber*temp1 - (fNumber +alpha)*temp2)/newton ;
844 else if(i == 2) << 843 newton1 = newton ;
845 { << 844 newton = newton1 - temp1/temp ;
846 nwt += (15.0 + 6.25 * alpha) / (1.0 + 0. << 845
847 } << 846 if(fabs(newton - newton1) <= tolerance)
848 else << 847 {
849 { << 848 break ;
850 cofi = i - 2; << 849 }
851 nwt += ((1.0 + 2.55 * cofi) / (1.9 * cof << 850 }
852 1.26 * cofi * alpha / (1.0 + 3.5 << 851 if(k > maxNumber)
853 (nwt - fAbscissa[i - 3]) / (1.0 + << 852 {
854 } << 853 G4Exception("Too many (>12) iterations in G4Integration::Laguerre") ;
855 for(k = 1; k <= maxNumber; ++k) << 854 }
856 { << 855
857 temp1 = 1.0; << 856 fAbscissa[i-1] = newton ;
858 temp2 = 0.0; << 857 fWeight[i-1] = -exp(GammaLogarithm(alpha + fNumber) -
859 << 858 GammaLogarithm((G4double)fNumber))/(temp*fNumber*temp2) ;
860 for(j = 1; j <= fNumber; ++j) << 859 }
861 { << 860 //
862 temp3 = temp2; << 861 // Integral evaluation
863 temp2 = temp1; << 862 //
864 temp1 = << 863 for(i=0;i<fNumber;i++)
865 ((2 * j - 1 + alpha - nwt) * temp2 - << 864 {
866 } << 865 integral += fWeight[i]*(typeT.*f)(fAbscissa[i]) ;
867 temp = (fNumber * temp1 - (fNumber + alp << 866 }
868 nwt1 = nwt; << 867 return integral ;
869 nwt = nwt1 - temp1 / temp; << 868 }
870 <<
871 if(std::fabs(nwt - nwt1) <= tolerance) <<
872 { <<
873 break; <<
874 } <<
875 } <<
876 if(k > maxNumber) <<
877 { <<
878 G4Exception("G4Integrator<T,F>::Laguerre <<
879 FatalException, "Too many (> <<
880 } <<
881 <<
882 fAbscissa[i - 1] = nwt; <<
883 fWeight[i - 1] = -std::exp(GammaLogarith <<
884 GammaLogarithm( <<
885 (temp * fNumber * temp2); <<
886 } <<
887 869
888 // <<
889 // Integral evaluation <<
890 // <<
891 870
892 for(i = 0; i < fNumber; ++i) <<
893 { <<
894 integral += fWeight[i] * (typeT.*f)(fAbsci <<
895 } <<
896 delete[] fAbscissa; <<
897 delete[] fWeight; <<
898 return integral; <<
899 } <<
900 871
901 ////////////////////////////////////////////// 872 //////////////////////////////////////////////////////////////////////
902 // 873 //
903 // 874 //
904 875
905 template <class T, class F> << 876 template <class T, class F> G4double
906 G4double G4Integrator<T, F>::Laguerre(T* ptrT, << 877 G4Integrator<T,F>::Laguerre( T* ptrT, F f, G4double alpha, G4int nLaguerre )
907 G4int nL <<
908 { 878 {
909 return Laguerre(*ptrT, f, alpha, nLaguerre); << 879 return Laguerre(*ptrT,f,alpha,nLaguerre) ;
910 } 880 }
911 881
912 ////////////////////////////////////////////// 882 ////////////////////////////////////////////////////////////////////////
913 // 883 //
914 // For use with global scope functions f << 884 // For use with global scope functions f
915 885
916 template <class T, class F> << 886 template <class T, class F> G4double
917 G4double G4Integrator<T, F>::Laguerre(G4double << 887 G4Integrator<T,F>::Laguerre( G4double (*f)(G4double),
918 G4int nL << 888 G4double alpha, G4int nLaguerre)
919 { << 889 {
920 const G4double tolerance = 1.0e-10; << 890 const G4double tolerance = 1.0e-10 ;
921 const G4int maxNumber = 12; << 891 const G4int maxNumber = 12 ;
922 G4int i, j, k; << 892 G4int i, j, k ;
923 G4double nwt = 0., nwt1, temp1, temp2, << 893 G4double newton, newton1, temp1, temp2, temp3, temp, cofi ;
924 G4double integral = 0.0; << 894 G4double integral = 0.0 ;
925 << 895
926 G4int fNumber = nLaguerre; << 896 G4int fNumber = nLaguerre ;
927 G4double* fAbscissa = new G4double[fNumber]; << 897 G4double* fAbscissa = new G4double[fNumber] ;
928 G4double* fWeight = new G4double[fNumber]; << 898 G4double* fWeight = new G4double[fNumber] ;
929 << 899
930 for(i = 1; i <= fNumber; ++i) // Loop over << 900 for(i=1;i<=fNumber;i++) // Loop over the desired roots
931 { << 901 {
932 if(i == 1) << 902 if(i == 1)
933 { << 903 {
934 nwt = (1.0 + alpha) * (3.0 + 0.92 * alph << 904 newton = (1.0 + alpha)*(3.0 + 0.92*alpha)/(1.0 + 2.4*fNumber + 1.8*alpha) ;
935 (1.0 + 2.4 * fNumber + 1.8 * alpha << 905 }
936 } << 906 else if(i == 2)
937 else if(i == 2) << 907 {
938 { << 908 newton += (15.0 + 6.25*alpha)/(1.0 + 0.9*alpha + 2.5*fNumber) ;
939 nwt += (15.0 + 6.25 * alpha) / (1.0 + 0. << 909 }
940 } << 910 else
941 else << 911 {
942 { << 912 cofi = i - 2 ;
943 cofi = i - 2; << 913 newton += ((1.0+2.55*cofi)/(1.9*cofi) + 1.26*cofi*alpha/(1.0+3.5*cofi))*
944 nwt += ((1.0 + 2.55 * cofi) / (1.9 * cof << 914 (newton - fAbscissa[i-3])/(1.0 + 0.3*alpha) ;
945 1.26 * cofi * alpha / (1.0 + 3.5 << 915 }
946 (nwt - fAbscissa[i - 3]) / (1.0 + << 916 for(k=1;k<=maxNumber;k++)
947 } << 917 {
948 for(k = 1; k <= maxNumber; ++k) << 918 temp1 = 1.0 ;
949 { << 919 temp2 = 0.0 ;
950 temp1 = 1.0; <<
951 temp2 = 0.0; <<
952 <<
953 for(j = 1; j <= fNumber; ++j) <<
954 { <<
955 temp3 = temp2; <<
956 temp2 = temp1; <<
957 temp1 = <<
958 ((2 * j - 1 + alpha - nwt) * temp2 - <<
959 } <<
960 temp = (fNumber * temp1 - (fNumber + alp <<
961 nwt1 = nwt; <<
962 nwt = nwt1 - temp1 / temp; <<
963 <<
964 if(std::fabs(nwt - nwt1) <= tolerance) <<
965 { <<
966 break; <<
967 } <<
968 } <<
969 if(k > maxNumber) <<
970 { <<
971 G4Exception("G4Integrator<T,F>::Laguerre <<
972 "Too many (>12) iterations." <<
973 } <<
974 <<
975 fAbscissa[i - 1] = nwt; <<
976 fWeight[i - 1] = -std::exp(GammaLogarith <<
977 GammaLogarithm( <<
978 (temp * fNumber * temp2); <<
979 } <<
980 <<
981 // <<
982 // Integral evaluation <<
983 // <<
984 920
985 for(i = 0; i < fNumber; i++) << 921 for(j=1;j<=fNumber;j++)
986 { << 922 {
987 integral += fWeight[i] * (*f)(fAbscissa[i] << 923 temp3 = temp2 ;
988 } << 924 temp2 = temp1 ;
989 delete[] fAbscissa; << 925 temp1 = ((2*j - 1 + alpha - newton)*temp2 - (j - 1 + alpha)*temp3)/j ;
990 delete[] fWeight; << 926 }
991 return integral; << 927 temp = (fNumber*temp1 - (fNumber +alpha)*temp2)/newton ;
>> 928 newton1 = newton ;
>> 929 newton = newton1 - temp1/temp ;
>> 930
>> 931 if(fabs(newton - newton1) <= tolerance)
>> 932 {
>> 933 break ;
>> 934 }
>> 935 }
>> 936 if(k > maxNumber)
>> 937 {
>> 938 G4Exception("Too many (>12) iterations in G4Integration::Laguerre") ;
>> 939 }
>> 940
>> 941 fAbscissa[i-1] = newton ;
>> 942 fWeight[i-1] = -exp(GammaLogarithm(alpha + fNumber) -
>> 943 GammaLogarithm((G4double)fNumber))/(temp*fNumber*temp2) ;
>> 944 }
>> 945 //
>> 946 // Integral evaluation
>> 947 //
>> 948 for(i=0;i<fNumber;i++)
>> 949 {
>> 950 integral += fWeight[i]*(*f)(fAbscissa[i]) ;
>> 951 }
>> 952 return integral ;
992 } 953 }
993 954
994 ////////////////////////////////////////////// 955 ///////////////////////////////////////////////////////////////////////
995 // 956 //
996 // Auxiliary function which returns the value << 957 // Auxiliary function which returns the value of log(gamma-function(x))
997 // Returns the value ln(Gamma(xx) for xx > 0. << 958 // Returns the value ln(Gamma(xx) for xx > 0. Full accuracy is obtained for
998 // xx > 1. For 0 < xx < 1. the reflection form 959 // xx > 1. For 0 < xx < 1. the reflection formula (6.1.4) can be used first.
999 // (Adapted from Numerical Recipes in C) 960 // (Adapted from Numerical Recipes in C)
1000 // 961 //
1001 962
1002 template <class T, class F> 963 template <class T, class F>
1003 G4double G4Integrator<T, F>::GammaLogarithm(G << 964 G4double G4Integrator<T,F>::GammaLogarithm(G4double xx)
1004 { 965 {
1005 static const G4double cof[6] = { 76.1800917 << 966 static G4double cof[6] = { 76.18009172947146, -86.50532032941677,
1006 24.0140982 << 967 24.01409824083091, -1.231739572450155,
1007 0.12086509 << 968 0.1208650973866179e-2, -0.5395239384953e-5 } ;
1008 G4int j; << 969 register HepInt j;
1009 G4double x = xx - 1.0; << 970 G4double x = xx - 1.0 ;
1010 G4double tmp = x + 5.5; << 971 G4double tmp = x + 5.5 ;
1011 tmp -= (x + 0.5) * std::log(tmp); << 972 tmp -= (x + 0.5) * log(tmp) ;
1012 G4double ser = 1.000000000190015; << 973 G4double ser = 1.000000000190015 ;
1013 974
1014 for(j = 0; j <= 5; ++j) << 975 for ( j = 0; j <= 5; j++ )
1015 { 976 {
1016 x += 1.0; << 977 x += 1.0 ;
1017 ser += cof[j] / x; << 978 ser += cof[j]/x ;
1018 } 979 }
1019 return -tmp + std::log(2.5066282746310005 * << 980 return -tmp + log(2.5066282746310005*ser) ;
1020 } 981 }
1021 982
1022 ///////////////////////////////////////////// 983 ///////////////////////////////////////////////////////////////////////
1023 // 984 //
1024 // Method involving Hermite polynomials 985 // Method involving Hermite polynomials
1025 // 986 //
1026 ///////////////////////////////////////////// 987 ///////////////////////////////////////////////////////////////////////
1027 // 988 //
1028 // 989 //
1029 // Gauss-Hermite method for integration of st << 990 // Gauss-Hermite method for integration of exp(-x*x)*f(x)
1030 // from minus infinity to plus infinity . << 991 // from minus infinity to plus infinity .
1031 // 992 //
1032 993
1033 template <class T, class F> << 994 template <class T, class F>
1034 G4double G4Integrator<T, F>::Hermite(T& typeT << 995 G4double G4Integrator<T,F>::Hermite( T& typeT, F f, G4int nHermite)
1035 { 996 {
1036 const G4double tolerance = 1.0e-12; << 997 const G4double tolerance = 1.0e-12 ;
1037 const G4int maxNumber = 12; << 998 const G4int maxNumber = 12 ;
1038 << 999
1039 G4int i, j, k; << 1000 G4int i, j, k ;
1040 G4double integral = 0.0; << 1001 G4double integral = 0.0 ;
1041 G4double nwt = 0., nwt1, temp1, temp2, << 1002 G4double newton, newton1, temp1, temp2, temp3, temp ;
1042 1003
1043 G4double piInMinusQ = << 1004 G4double piInMinusQ = pow(pi,-0.25) ; // 1.0/sqrt(sqrt(pi)) ??
1044 std::pow(CLHEP::pi, -0.25); // 1.0/std:: <<
1045 1005
1046 G4int fNumber = (nHermite + 1) / 2; << 1006 G4int fNumber = (nHermite +1)/2 ;
1047 G4double* fAbscissa = new G4double[fNumber] << 1007 G4double* fAbscissa = new G4double[fNumber] ;
1048 G4double* fWeight = new G4double[fNumber] << 1008 G4double* fWeight = new G4double[fNumber] ;
1049 <<
1050 for(i = 1; i <= fNumber; ++i) <<
1051 { <<
1052 if(i == 1) <<
1053 { <<
1054 nwt = std::sqrt((G4double)(2 * nHermite <<
1055 1.85575001 * std::pow((G4double)( <<
1056 } <<
1057 else if(i == 2) <<
1058 { <<
1059 nwt -= 1.14001 * std::pow((G4double) nH <<
1060 } <<
1061 else if(i == 3) <<
1062 { <<
1063 nwt = 1.86002 * nwt - 0.86002 * fAbscis <<
1064 } <<
1065 else if(i == 4) <<
1066 { <<
1067 nwt = 1.91001 * nwt - 0.91001 * fAbscis <<
1068 } <<
1069 else <<
1070 { <<
1071 nwt = 2.0 * nwt - fAbscissa[i - 3]; <<
1072 } <<
1073 for(k = 1; k <= maxNumber; ++k) <<
1074 { <<
1075 temp1 = piInMinusQ; <<
1076 temp2 = 0.0; <<
1077 <<
1078 for(j = 1; j <= nHermite; ++j) <<
1079 { <<
1080 temp3 = temp2; <<
1081 temp2 = temp1; <<
1082 temp1 = nwt * std::sqrt(2.0 / j) * te <<
1083 std::sqrt(((G4double)(j - 1)) <<
1084 } <<
1085 temp = std::sqrt((G4double) 2 * nHermit <<
1086 nwt1 = nwt; <<
1087 nwt = nwt1 - temp1 / temp; <<
1088 <<
1089 if(std::fabs(nwt - nwt1) <= tolerance) <<
1090 { <<
1091 break; <<
1092 } <<
1093 } <<
1094 if(k > maxNumber) <<
1095 { <<
1096 G4Exception("G4Integrator<T,F>::Hermite <<
1097 FatalException, "Too many ( <<
1098 } <<
1099 fAbscissa[i - 1] = nwt; <<
1100 fWeight[i - 1] = 2.0 / (temp * temp); <<
1101 } <<
1102 1009
1103 // << 1010 for(i=1;i<=fNumber;i++)
1104 // Integral calculation << 1011 {
1105 // << 1012 if(i == 1)
>> 1013 {
>> 1014 newton = sqrt((G4double)(2*nHermite + 1)) -
>> 1015 1.85575001*pow((G4double)(2*nHermite + 1),-0.16666999) ;
>> 1016 }
>> 1017 else if(i == 2)
>> 1018 {
>> 1019 newton -= 1.14001*pow((G4double)nHermite,0.425999)/newton ;
>> 1020 }
>> 1021 else if(i == 3)
>> 1022 {
>> 1023 newton = 1.86002*newton - 0.86002*fAbscissa[0] ;
>> 1024 }
>> 1025 else if(i == 4)
>> 1026 {
>> 1027 newton = 1.91001*newton - 0.91001*fAbscissa[1] ;
>> 1028 }
>> 1029 else
>> 1030 {
>> 1031 newton = 2.0*newton - fAbscissa[i - 3] ;
>> 1032 }
>> 1033 for(k=1;k<=maxNumber;k++)
>> 1034 {
>> 1035 temp1 = piInMinusQ ;
>> 1036 temp2 = 0.0 ;
1106 1037
1107 for(i = 0; i < fNumber; ++i) << 1038 for(j=1;j<=nHermite;j++)
1108 { << 1039 {
1109 integral += << 1040 temp3 = temp2 ;
1110 fWeight[i] * ((typeT.*f)(fAbscissa[i]) << 1041 temp2 = temp1 ;
1111 } << 1042 temp1 = newton*sqrt(2.0/j)*temp2 -
1112 delete[] fAbscissa; << 1043 sqrt(((G4double)(j - 1))/j)*temp3 ;
1113 delete[] fWeight; << 1044 }
1114 return integral; << 1045 temp = sqrt((G4double)2*nHermite)*temp2 ;
>> 1046 newton1 = newton ;
>> 1047 newton = newton1 - temp1/temp ;
>> 1048
>> 1049 if(fabs(newton - newton1) <= tolerance)
>> 1050 {
>> 1051 break ;
>> 1052 }
>> 1053 }
>> 1054 if(k > maxNumber)
>> 1055 {
>> 1056 G4Exception("Too many (>12) iterations in G4Integrator<T,F>::Hermite") ;
>> 1057 }
>> 1058 fAbscissa[i-1] = newton ;
>> 1059 fWeight[i-1] = 2.0/(temp*temp) ;
>> 1060 }
>> 1061 //
>> 1062 // Integral calculation
>> 1063 //
>> 1064 for(i=0;i<fNumber;i++)
>> 1065 {
>> 1066 integral += fWeight[i]*( (typeT.*f)(fAbscissa[i]) +
>> 1067 (typeT.*f)(-fAbscissa[i]) ) ;
>> 1068 }
>> 1069 return integral ;
1115 } 1070 }
1116 1071
>> 1072
1117 ///////////////////////////////////////////// 1073 ////////////////////////////////////////////////////////////////////////
1118 // 1074 //
1119 // For use with 'this' pointer 1075 // For use with 'this' pointer
1120 1076
1121 template <class T, class F> << 1077 template <class T, class F>
1122 G4double G4Integrator<T, F>::Hermite(T* ptrT, << 1078 G4double G4Integrator<T,F>::Hermite( T* ptrT, F f, G4int n)
1123 { 1079 {
1124 return Hermite(*ptrT, f, n); << 1080 return Hermite(*ptrT,f,n) ;
1125 } << 1081 }
1126 1082
1127 ///////////////////////////////////////////// 1083 ////////////////////////////////////////////////////////////////////////
1128 // 1084 //
1129 // For use with global scope f 1085 // For use with global scope f
1130 1086
1131 template <class T, class F> 1087 template <class T, class F>
1132 G4double G4Integrator<T, F>::Hermite(G4double << 1088 G4double G4Integrator<T,F>::Hermite( G4double (*f)(G4double), G4int nHermite)
1133 { 1089 {
1134 const G4double tolerance = 1.0e-12; << 1090 const G4double tolerance = 1.0e-12 ;
1135 const G4int maxNumber = 12; << 1091 const G4int maxNumber = 12 ;
1136 << 1092
1137 G4int i, j, k; << 1093 G4int i, j, k ;
1138 G4double integral = 0.0; << 1094 G4double integral = 0.0 ;
1139 G4double nwt = 0., nwt1, temp1, temp2, << 1095 G4double newton, newton1, temp1, temp2, temp3, temp ;
1140 << 1096
1141 G4double piInMinusQ = << 1097 G4double piInMinusQ = pow(pi,-0.25) ; // 1.0/sqrt(sqrt(pi)) ??
1142 std::pow(CLHEP::pi, -0.25); // 1.0/std:: << 1098
1143 << 1099 G4int fNumber = (nHermite +1)/2 ;
1144 G4int fNumber = (nHermite + 1) / 2; << 1100 G4double* fAbscissa = new G4double[fNumber] ;
1145 G4double* fAbscissa = new G4double[fNumber] << 1101 G4double* fWeight = new G4double[fNumber] ;
1146 G4double* fWeight = new G4double[fNumber] << 1102
1147 << 1103 for(i=1;i<=fNumber;i++)
1148 for(i = 1; i <= fNumber; ++i) << 1104 {
1149 { << 1105 if(i == 1)
1150 if(i == 1) << 1106 {
1151 { << 1107 newton = sqrt((G4double)(2*nHermite + 1)) -
1152 nwt = std::sqrt((G4double)(2 * nHermite << 1108 1.85575001*pow((G4double)(2*nHermite + 1),-0.16666999) ;
1153 1.85575001 * std::pow((G4double)( << 1109 }
1154 } << 1110 else if(i == 2)
1155 else if(i == 2) << 1111 {
1156 { << 1112 newton -= 1.14001*pow((G4double)nHermite,0.425999)/newton ;
1157 nwt -= 1.14001 * std::pow((G4double) nH << 1113 }
1158 } << 1114 else if(i == 3)
1159 else if(i == 3) << 1115 {
1160 { << 1116 newton = 1.86002*newton - 0.86002*fAbscissa[0] ;
1161 nwt = 1.86002 * nwt - 0.86002 * fAbscis << 1117 }
1162 } << 1118 else if(i == 4)
1163 else if(i == 4) << 1119 {
1164 { << 1120 newton = 1.91001*newton - 0.91001*fAbscissa[1] ;
1165 nwt = 1.91001 * nwt - 0.91001 * fAbscis << 1121 }
1166 } << 1122 else
1167 else << 1123 {
1168 { << 1124 newton = 2.0*newton - fAbscissa[i - 3] ;
1169 nwt = 2.0 * nwt - fAbscissa[i - 3]; << 1125 }
1170 } << 1126 for(k=1;k<=maxNumber;k++)
1171 for(k = 1; k <= maxNumber; ++k) << 1127 {
1172 { << 1128 temp1 = piInMinusQ ;
1173 temp1 = piInMinusQ; << 1129 temp2 = 0.0 ;
1174 temp2 = 0.0; <<
1175 <<
1176 for(j = 1; j <= nHermite; ++j) <<
1177 { <<
1178 temp3 = temp2; <<
1179 temp2 = temp1; <<
1180 temp1 = nwt * std::sqrt(2.0 / j) * te <<
1181 std::sqrt(((G4double)(j - 1)) <<
1182 } <<
1183 temp = std::sqrt((G4double) 2 * nHermit <<
1184 nwt1 = nwt; <<
1185 nwt = nwt1 - temp1 / temp; <<
1186 <<
1187 if(std::fabs(nwt - nwt1) <= tolerance) <<
1188 { <<
1189 break; <<
1190 } <<
1191 } <<
1192 if(k > maxNumber) <<
1193 { <<
1194 G4Exception("G4Integrator<T,F>::Hermite <<
1195 "Too many (>12) iterations. <<
1196 } <<
1197 fAbscissa[i - 1] = nwt; <<
1198 fWeight[i - 1] = 2.0 / (temp * temp); <<
1199 } <<
1200 <<
1201 // <<
1202 // Integral calculation <<
1203 // <<
1204 1130
1205 for(i = 0; i < fNumber; ++i) << 1131 for(j=1;j<=nHermite;j++)
1206 { << 1132 {
1207 integral += fWeight[i] * ((*f)(fAbscissa[ << 1133 temp3 = temp2 ;
1208 } << 1134 temp2 = temp1 ;
1209 delete[] fAbscissa; << 1135 temp1 = newton*sqrt(2.0/j)*temp2 -
1210 delete[] fWeight; << 1136 sqrt(((G4double)(j - 1))/j)*temp3 ;
1211 return integral; << 1137 }
>> 1138 temp = sqrt((G4double)2*nHermite)*temp2 ;
>> 1139 newton1 = newton ;
>> 1140 newton = newton1 - temp1/temp ;
>> 1141
>> 1142 if(fabs(newton - newton1) <= tolerance)
>> 1143 {
>> 1144 break ;
>> 1145 }
>> 1146 }
>> 1147 if(k > maxNumber)
>> 1148 {
>> 1149 G4Exception("Too many (>12) iterations in G4Integrator<T,F>::Hermite") ;
>> 1150 }
>> 1151 fAbscissa[i-1] = newton ;
>> 1152 fWeight[i-1] = 2.0/(temp*temp) ;
>> 1153 }
>> 1154 //
>> 1155 // Integral calculation
>> 1156 //
>> 1157 for(i=0;i<fNumber;i++)
>> 1158 {
>> 1159 integral += fWeight[i]*( (*f)(fAbscissa[i]) + (*f)(-fAbscissa[i]) ) ;
>> 1160 }
>> 1161 return integral ;
1212 } 1162 }
1213 1163
1214 ///////////////////////////////////////////// 1164 ////////////////////////////////////////////////////////////////////////////
1215 // 1165 //
1216 // Method involving Jacobi polynomials 1166 // Method involving Jacobi polynomials
1217 // 1167 //
1218 ///////////////////////////////////////////// 1168 ////////////////////////////////////////////////////////////////////////////
1219 // 1169 //
1220 // Gauss-Jacobi method for integration of ((1 1170 // Gauss-Jacobi method for integration of ((1-x)^alpha)*((1+x)^beta)*f(x)
1221 // from minus unit to plus unit . 1171 // from minus unit to plus unit .
1222 // 1172 //
1223 1173
1224 template <class T, class F> << 1174 template <class T, class F>
1225 G4double G4Integrator<T, F>::Jacobi(T& typeT, << 1175 G4double G4Integrator<T,F>::Jacobi( T& typeT, F f, G4double alpha,
1226 G4double << 1176 G4double beta, G4int nJacobi)
1227 { << 1177 {
1228 const G4double tolerance = 1.0e-12; << 1178 const G4double tolerance = 1.0e-12 ;
1229 const G4double maxNumber = 12; << 1179 const G4double maxNumber = 12 ;
1230 G4int i, k, j; << 1180 G4int i, k, j ;
1231 G4double alphaBeta, alphaReduced, betaReduc << 1181 G4double alphaBeta, alphaReduced, betaReduced, root1, root2, root3 ;
1232 << 1182 G4double a, b, c, newton1, newton2, newton3, newton, temp, root, rootTemp ;
1233 G4double a, b, c, nwt1, nwt2, nwt3, nwt, te << 1183
1234 << 1184 G4int fNumber = nJacobi ;
1235 G4int fNumber = nJacobi; << 1185 G4double* fAbscissa = new G4double[fNumber] ;
1236 G4double* fAbscissa = new G4double[fNumber] << 1186 G4double* fWeight = new G4double[fNumber] ;
1237 G4double* fWeight = new G4double[fNumber] << 1187
1238 << 1188 for (i=1;i<=nJacobi;i++)
1239 for(i = 1; i <= nJacobi; ++i) << 1189 {
1240 { << 1190 if (i == 1)
1241 if(i == 1) << 1191 {
1242 { << 1192 alphaReduced = alpha/nJacobi ;
1243 alphaReduced = alpha / nJacobi; << 1193 betaReduced = beta/nJacobi ;
1244 betaReduced = beta / nJacobi; << 1194 root1 = (1.0+alpha)*(2.78002/(4.0+nJacobi*nJacobi)+
1245 root1 = (1.0 + alpha) * (2.78002 << 1195 0.767999*alphaReduced/nJacobi) ;
1246 0.767999 * alp << 1196 root2 = 1.0+1.48*alphaReduced+0.96002*betaReduced +
1247 root2 = 1.0 + 1.48 * alphaReduce << 1197 0.451998*alphaReduced*alphaReduced +
1248 0.451998 * alphaReduced * alpha << 1198 0.83001*alphaReduced*betaReduced ;
1249 0.83001 * alphaReduced * betaRe << 1199 root = 1.0-root1/root2 ;
1250 root = 1.0 - root1 / root2; << 1200 }
1251 } << 1201 else if (i == 2)
1252 else if(i == 2) << 1202 {
1253 { << 1203 root1=(4.1002+alpha)/((1.0+alpha)*(1.0+0.155998*alpha)) ;
1254 root1 = (4.1002 + alpha) / ((1.0 + alph << 1204 root2=1.0+0.06*(nJacobi-8.0)*(1.0+0.12*alpha)/nJacobi ;
1255 root2 = 1.0 + 0.06 * (nJacobi - 8.0) * << 1205 root3=1.0+0.012002*beta*(1.0+0.24997*fabs(alpha))/nJacobi ;
1256 root3 = << 1206 root -= (1.0-root)*root1*root2*root3 ;
1257 1.0 + 0.012002 * beta * (1.0 + 0.2499 << 1207 }
1258 root -= (1.0 - root) * root1 * root2 * << 1208 else if (i == 3)
1259 } << 1209 {
1260 else if(i == 3) << 1210 root1=(1.67001+0.27998*alpha)/(1.0+0.37002*alpha) ;
1261 { << 1211 root2=1.0+0.22*(nJacobi-8.0)/nJacobi ;
1262 root1 = (1.67001 + 0.27998 * alpha) / ( << 1212 root3=1.0+8.0*beta/((6.28001+beta)*nJacobi*nJacobi) ;
1263 root2 = 1.0 + 0.22 * (nJacobi - 8.0) / << 1213 root -= (fAbscissa[0]-root)*root1*root2*root3 ;
1264 root3 = 1.0 + 8.0 * beta / ((6.28001 + << 1214 }
1265 root -= (fAbscissa[0] - root) * root1 * << 1215 else if (i == nJacobi-1)
1266 } << 1216 {
1267 else if(i == nJacobi - 1) << 1217 root1=(1.0+0.235002*beta)/(0.766001+0.118998*beta) ;
1268 { << 1218 root2=1.0/(1.0+0.639002*(nJacobi-4.0)/(1.0+0.71001*(nJacobi-4.0))) ;
1269 root1 = (1.0 + 0.235002 * beta) / (0.76 << 1219 root3=1.0/(1.0+20.0*alpha/((7.5+alpha)*nJacobi*nJacobi)) ;
1270 root2 = 1.0 / (1.0 + 0.639002 * (nJacob << 1220 root += (root-fAbscissa[nJacobi-4])*root1*root2*root3 ;
1271 (1.0 + 0.71001 * << 1221 }
1272 root3 = 1.0 / (1.0 + 20.0 * alpha / ((7 << 1222 else if (i == nJacobi)
1273 root += (root - fAbscissa[nJacobi - 4]) << 1223 {
1274 } << 1224 root1 = (1.0+0.37002*beta)/(1.67001+0.27998*beta) ;
1275 else if(i == nJacobi) << 1225 root2 = 1.0/(1.0+0.22*(nJacobi-8.0)/nJacobi) ;
1276 { << 1226 root3 = 1.0/(1.0+8.0*alpha/((6.28002+alpha)*nJacobi*nJacobi)) ;
1277 root1 = (1.0 + 0.37002 * beta) / (1.670 << 1227 root += (root-fAbscissa[nJacobi-3])*root1*root2*root3 ;
1278 root2 = 1.0 / (1.0 + 0.22 * (nJacobi - << 1228 }
1279 root3 = << 1229 else
1280 1.0 / (1.0 + 8.0 * alpha / ((6.28002 << 1230 {
1281 root += (root - fAbscissa[nJacobi - 3]) << 1231 root = 3.0*fAbscissa[i-2]-3.0*fAbscissa[i-3]+fAbscissa[i-4] ;
1282 } << 1232 }
1283 else << 1233 alphaBeta = alpha + beta ;
1284 { << 1234 for (k=1;k<=maxNumber;k++)
1285 root = 3.0 * fAbscissa[i - 2] - 3.0 * f << 1235 {
1286 } << 1236 temp = 2.0 + alphaBeta ;
1287 alphaBeta = alpha + beta; << 1237 newton1 = (alpha-beta+temp*root)/2.0 ;
1288 for(k = 1; k <= maxNumber; ++k) << 1238 newton2 = 1.0 ;
1289 { << 1239 for (j=2;j<=nJacobi;j++)
1290 temp = 2.0 + alphaBeta; << 1240 {
1291 nwt1 = (alpha - beta + temp * root) / 2 << 1241 newton3 = newton2 ;
1292 nwt2 = 1.0; << 1242 newton2 = newton1 ;
1293 for(j = 2; j <= nJacobi; ++j) << 1243 temp = 2*j+alphaBeta ;
1294 { << 1244 a = 2*j*(j+alphaBeta)*(temp-2.0) ;
1295 nwt3 = nwt2; << 1245 b = (temp-1.0)*(alpha*alpha-beta*beta+temp*(temp-2.0)*root) ;
1296 nwt2 = nwt1; << 1246 c = 2.0*(j-1+alpha)*(j-1+beta)*temp ;
1297 temp = 2 * j + alphaBeta; << 1247 newton1 = (b*newton2-c*newton3)/a ;
1298 a = 2 * j * (j + alphaBeta) * (tem << 1248 }
1299 b = (temp - 1.0) * << 1249 newton = (nJacobi*(alpha - beta - temp*root)*newton1 +
1300 (alpha * alpha - beta * beta + te << 1250 2.0*(nJacobi + alpha)*(nJacobi + beta)*newton2)/
1301 c = 2.0 * (j - 1 + alpha) * (j - 1 << 1251 (temp*(1.0 - root*root)) ;
1302 nwt1 = (b * nwt2 - c * nwt3) / a; << 1252 rootTemp = root ;
1303 } << 1253 root = rootTemp - newton1/newton ;
1304 nwt = (nJacobi * (alpha - beta - temp * << 1254 if (fabs(root-rootTemp) <= tolerance)
1305 2.0 * (nJacobi + alpha) * (nJaco << 1255 {
1306 (temp * (1.0 - root * root)); << 1256 break ;
1307 rootTemp = root; << 1257 }
1308 root = rootTemp - nwt1 / nwt; << 1258 }
1309 if(std::fabs(root - rootTemp) <= tolera << 1259 if (k > maxNumber)
1310 { << 1260 {
1311 break; << 1261 G4Exception("Too many iterations (>12) in G4Integrator<T,F>::Jacobi") ;
1312 } << 1262 }
1313 } << 1263 fAbscissa[i-1] = root ;
1314 if(k > maxNumber) << 1264 fWeight[i-1] = exp(GammaLogarithm((G4double)(alpha+nJacobi)) +
1315 { << 1265 GammaLogarithm((G4double)(beta+nJacobi)) -
1316 G4Exception("G4Integrator<T,F>::Jacobi( << 1266 GammaLogarithm((G4double)(nJacobi+1.0)) -
1317 FatalException, "Too many ( << 1267 GammaLogarithm((G4double)(nJacobi + alphaBeta + 1.0)))
1318 } << 1268 *temp*pow(2.0,alphaBeta)/(newton*newton2) ;
1319 fAbscissa[i - 1] = root; << 1269 }
1320 fWeight[i - 1] = << 1270 //
1321 std::exp(GammaLogarithm((G4double)(alph << 1271 // Calculation of the integral
1322 GammaLogarithm((G4double)(beta << 1272 //
1323 GammaLogarithm((G4double)(nJac << 1273 G4double integral = 0.0 ;
1324 GammaLogarithm((G4double)(nJac << 1274 for(i=0;i<fNumber;i++)
1325 temp * std::pow(2.0, alphaBeta) / (nwt << 1275 {
1326 } << 1276 integral += fWeight[i]*(typeT.*f)(fAbscissa[i]) ;
1327 << 1277 }
1328 // << 1278 return integral ;
1329 // Calculation of the integral <<
1330 // <<
1331 <<
1332 G4double integral = 0.0; <<
1333 for(i = 0; i < fNumber; ++i) <<
1334 { <<
1335 integral += fWeight[i] * (typeT.*f)(fAbsc <<
1336 } <<
1337 delete[] fAbscissa; <<
1338 delete[] fWeight; <<
1339 return integral; <<
1340 } 1279 }
1341 1280
>> 1281
1342 ///////////////////////////////////////////// 1282 /////////////////////////////////////////////////////////////////////////
1343 // 1283 //
1344 // For use with 'this' pointer 1284 // For use with 'this' pointer
1345 1285
1346 template <class T, class F> << 1286 template <class T, class F>
1347 G4double G4Integrator<T, F>::Jacobi(T* ptrT, << 1287 G4double G4Integrator<T,F>::Jacobi( T* ptrT, F f, G4double alpha,
1348 G4int n) << 1288 G4double beta, G4int n)
1349 { 1289 {
1350 return Jacobi(*ptrT, f, alpha, beta, n); << 1290 return Jacobi(*ptrT,f,alpha,beta,n) ;
1351 } << 1291 }
1352 1292
1353 ///////////////////////////////////////////// 1293 /////////////////////////////////////////////////////////////////////////
1354 // 1294 //
1355 // For use with global scope f << 1295 // For use with global scope f
1356 1296
1357 template <class T, class F> 1297 template <class T, class F>
1358 G4double G4Integrator<T, F>::Jacobi(G4double << 1298 G4double G4Integrator<T,F>::Jacobi( G4double (*f)(G4double), G4double alpha,
1359 G4double << 1299 G4double beta, G4int nJacobi)
1360 { 1300 {
1361 const G4double tolerance = 1.0e-12; << 1301 const G4double tolerance = 1.0e-12 ;
1362 const G4double maxNumber = 12; << 1302 const G4double maxNumber = 12 ;
1363 G4int i, k, j; << 1303 G4int i, k, j ;
1364 G4double alphaBeta, alphaReduced, betaReduc << 1304 G4double alphaBeta, alphaReduced, betaReduced, root1, root2, root3 ;
1365 << 1305 G4double a, b, c, newton1, newton2, newton3, newton, temp, root, rootTemp ;
1366 G4double a, b, c, nwt1, nwt2, nwt3, nwt, te << 1306
1367 << 1307 G4int fNumber = nJacobi ;
1368 G4int fNumber = nJacobi; << 1308 G4double* fAbscissa = new G4double[fNumber] ;
1369 G4double* fAbscissa = new G4double[fNumber] << 1309 G4double* fWeight = new G4double[fNumber] ;
1370 G4double* fWeight = new G4double[fNumber] << 1310
1371 << 1311 for (i=1;i<=nJacobi;i++)
1372 for(i = 1; i <= nJacobi; ++i) << 1312 {
1373 { << 1313 if (i == 1)
1374 if(i == 1) << 1314 {
1375 { << 1315 alphaReduced = alpha/nJacobi ;
1376 alphaReduced = alpha / nJacobi; << 1316 betaReduced = beta/nJacobi ;
1377 betaReduced = beta / nJacobi; << 1317 root1 = (1.0+alpha)*(2.78002/(4.0+nJacobi*nJacobi)+
1378 root1 = (1.0 + alpha) * (2.78002 << 1318 0.767999*alphaReduced/nJacobi) ;
1379 0.767999 * alp << 1319 root2 = 1.0+1.48*alphaReduced+0.96002*betaReduced +
1380 root2 = 1.0 + 1.48 * alphaReduce << 1320 0.451998*alphaReduced*alphaReduced +
1381 0.451998 * alphaReduced * alpha << 1321 0.83001*alphaReduced*betaReduced ;
1382 0.83001 * alphaReduced * betaRe << 1322 root = 1.0-root1/root2 ;
1383 root = 1.0 - root1 / root2; << 1323 }
1384 } << 1324 else if (i == 2)
1385 else if(i == 2) << 1325 {
1386 { << 1326 root1=(4.1002+alpha)/((1.0+alpha)*(1.0+0.155998*alpha)) ;
1387 root1 = (4.1002 + alpha) / ((1.0 + alph << 1327 root2=1.0+0.06*(nJacobi-8.0)*(1.0+0.12*alpha)/nJacobi ;
1388 root2 = 1.0 + 0.06 * (nJacobi - 8.0) * << 1328 root3=1.0+0.012002*beta*(1.0+0.24997*fabs(alpha))/nJacobi ;
1389 root3 = << 1329 root -= (1.0-root)*root1*root2*root3 ;
1390 1.0 + 0.012002 * beta * (1.0 + 0.2499 << 1330 }
1391 root -= (1.0 - root) * root1 * root2 * << 1331 else if (i == 3)
1392 } << 1332 {
1393 else if(i == 3) << 1333 root1=(1.67001+0.27998*alpha)/(1.0+0.37002*alpha) ;
1394 { << 1334 root2=1.0+0.22*(nJacobi-8.0)/nJacobi ;
1395 root1 = (1.67001 + 0.27998 * alpha) / ( << 1335 root3=1.0+8.0*beta/((6.28001+beta)*nJacobi*nJacobi) ;
1396 root2 = 1.0 + 0.22 * (nJacobi - 8.0) / << 1336 root -= (fAbscissa[0]-root)*root1*root2*root3 ;
1397 root3 = 1.0 + 8.0 * beta / ((6.28001 + << 1337 }
1398 root -= (fAbscissa[0] - root) * root1 * << 1338 else if (i == nJacobi-1)
1399 } << 1339 {
1400 else if(i == nJacobi - 1) << 1340 root1=(1.0+0.235002*beta)/(0.766001+0.118998*beta) ;
1401 { << 1341 root2=1.0/(1.0+0.639002*(nJacobi-4.0)/(1.0+0.71001*(nJacobi-4.0))) ;
1402 root1 = (1.0 + 0.235002 * beta) / (0.76 << 1342 root3=1.0/(1.0+20.0*alpha/((7.5+alpha)*nJacobi*nJacobi)) ;
1403 root2 = 1.0 / (1.0 + 0.639002 * (nJacob << 1343 root += (root-fAbscissa[nJacobi-4])*root1*root2*root3 ;
1404 (1.0 + 0.71001 * << 1344 }
1405 root3 = 1.0 / (1.0 + 20.0 * alpha / ((7 << 1345 else if (i == nJacobi)
1406 root += (root - fAbscissa[nJacobi - 4]) << 1346 {
1407 } << 1347 root1 = (1.0+0.37002*beta)/(1.67001+0.27998*beta) ;
1408 else if(i == nJacobi) << 1348 root2 = 1.0/(1.0+0.22*(nJacobi-8.0)/nJacobi) ;
1409 { << 1349 root3 = 1.0/(1.0+8.0*alpha/((6.28002+alpha)*nJacobi*nJacobi)) ;
1410 root1 = (1.0 + 0.37002 * beta) / (1.670 << 1350 root += (root-fAbscissa[nJacobi-3])*root1*root2*root3 ;
1411 root2 = 1.0 / (1.0 + 0.22 * (nJacobi - << 1351 }
1412 root3 = << 1352 else
1413 1.0 / (1.0 + 8.0 * alpha / ((6.28002 << 1353 {
1414 root += (root - fAbscissa[nJacobi - 3]) << 1354 root = 3.0*fAbscissa[i-2]-3.0*fAbscissa[i-3]+fAbscissa[i-4] ;
1415 } << 1355 }
1416 else << 1356 alphaBeta = alpha + beta ;
1417 { << 1357 for (k=1;k<=maxNumber;k++)
1418 root = 3.0 * fAbscissa[i - 2] - 3.0 * f << 1358 {
1419 } << 1359 temp = 2.0 + alphaBeta ;
1420 alphaBeta = alpha + beta; << 1360 newton1 = (alpha-beta+temp*root)/2.0 ;
1421 for(k = 1; k <= maxNumber; ++k) << 1361 newton2 = 1.0 ;
1422 { << 1362 for (j=2;j<=nJacobi;j++)
1423 temp = 2.0 + alphaBeta; << 1363 {
1424 nwt1 = (alpha - beta + temp * root) / 2 << 1364 newton3 = newton2 ;
1425 nwt2 = 1.0; << 1365 newton2 = newton1 ;
1426 for(j = 2; j <= nJacobi; ++j) << 1366 temp = 2*j+alphaBeta ;
1427 { << 1367 a = 2*j*(j+alphaBeta)*(temp-2.0) ;
1428 nwt3 = nwt2; << 1368 b = (temp-1.0)*(alpha*alpha-beta*beta+temp*(temp-2.0)*root) ;
1429 nwt2 = nwt1; << 1369 c = 2.0*(j-1+alpha)*(j-1+beta)*temp ;
1430 temp = 2 * j + alphaBeta; << 1370 newton1 = (b*newton2-c*newton3)/a ;
1431 a = 2 * j * (j + alphaBeta) * (tem << 1371 }
1432 b = (temp - 1.0) * << 1372 newton = (nJacobi*(alpha - beta - temp*root)*newton1 +
1433 (alpha * alpha - beta * beta + te << 1373 2.0*(nJacobi + alpha)*(nJacobi + beta)*newton2)/
1434 c = 2.0 * (j - 1 + alpha) * (j - 1 << 1374 (temp*(1.0 - root*root)) ;
1435 nwt1 = (b * nwt2 - c * nwt3) / a; << 1375 rootTemp = root ;
1436 } << 1376 root = rootTemp - newton1/newton ;
1437 nwt = (nJacobi * (alpha - beta - temp * << 1377 if (fabs(root-rootTemp) <= tolerance)
1438 2.0 * (nJacobi + alpha) * (nJaco << 1378 {
1439 (temp * (1.0 - root * root)); << 1379 break ;
1440 rootTemp = root; << 1380 }
1441 root = rootTemp - nwt1 / nwt; << 1381 }
1442 if(std::fabs(root - rootTemp) <= tolera << 1382 if (k > maxNumber)
1443 { << 1383 {
1444 break; << 1384 G4Exception("Too many iterations (>12) in G4Integrator<T,F>::Jacobi") ;
1445 } << 1385 }
1446 } << 1386 fAbscissa[i-1] = root ;
1447 if(k > maxNumber) << 1387 fWeight[i-1] = exp(GammaLogarithm((G4double)(alpha+nJacobi)) +
1448 { << 1388 GammaLogarithm((G4double)(beta+nJacobi)) -
1449 G4Exception("G4Integrator<T,F>::Jacobi( << 1389 GammaLogarithm((G4double)(nJacobi+1.0)) -
1450 "Too many (>12) iterations. << 1390 GammaLogarithm((G4double)(nJacobi + alphaBeta + 1.0)))
1451 } << 1391 *temp*pow(2.0,alphaBeta)/(newton*newton2) ;
1452 fAbscissa[i - 1] = root; << 1392 }
1453 fWeight[i - 1] = << 1393 //
1454 std::exp(GammaLogarithm((G4double)(alph << 1394 // Calculation of the integral
1455 GammaLogarithm((G4double)(beta << 1395 //
1456 GammaLogarithm((G4double)(nJac << 1396 G4double integral = 0.0 ;
1457 GammaLogarithm((G4double)(nJac << 1397 for(i=0;i<fNumber;i++)
1458 temp * std::pow(2.0, alphaBeta) / (nwt << 1398 {
1459 } << 1399 integral += fWeight[i]*(*f)(fAbscissa[i]) ;
>> 1400 }
>> 1401 return integral ;
>> 1402 }
1460 1403
1461 // <<
1462 // Calculation of the integral <<
1463 // <<
1464 1404
1465 G4double integral = 0.0; <<
1466 for(i = 0; i < fNumber; ++i) <<
1467 { <<
1468 integral += fWeight[i] * (*f)(fAbscissa[i <<
1469 } <<
1470 delete[] fAbscissa; <<
1471 delete[] fWeight; <<
1472 return integral; <<
1473 } <<
1474 1405
1475 // 1406 //
1476 // 1407 //
1477 ///////////////////////////////////////////// 1408 ///////////////////////////////////////////////////////////////////
>> 1409
>> 1410
>> 1411
>> 1412
1478 1413