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1 // 1 //
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4 // * 4 // * *
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24 // ******************************************* 21 // ********************************************************************
25 // 22 //
26 // G4Integrator inline methods implementation <<
27 // 23 //
28 // Author: V.Grichine, 04.09.1999 - First impl << 24 // $Id: G4Integrator.icc,v 1.12 2005/03/15 19:11:35 gcosmo Exp $
29 // G4SimpleIntegration class with H.P. << 25 // GEANT4 tag $Name: geant4-07-01-patch-01 $
30 // E.TCherniaev advises << 26 //
31 // ------------------------------------------- << 27 // Implementation of G4Integrator methods.
>> 28 //
>> 29 //
32 30
33 ////////////////////////////////////////////// 31 /////////////////////////////////////////////////////////////////////
34 // 32 //
35 // Sympson integration method 33 // Sympson integration method
36 // 34 //
37 ////////////////////////////////////////////// 35 /////////////////////////////////////////////////////////////////////
38 // 36 //
39 // Integration of class member functions T::f << 37 // Integration of class member functions T::f by Simpson method.
40 38
41 template <class T, class F> << 39 template <class T, class F>
42 G4double G4Integrator<T, F>::Simpson(T& typeT, << 40 G4double G4Integrator<T,F>::Simpson( T& typeT,
43 G4double << 41 F f,
44 { << 42 G4double xInitial,
45 G4int i; << 43 G4double xFinal,
46 G4double step = (xFinal - xInitial) / itera << 44 G4int iterationNumber )
47 G4double x = xInitial; << 45 {
48 G4double xPlus = xInitial + 0.5 * step; << 46 G4int i ;
49 G4double mean = ((typeT.*f)(xInitial) + (ty << 47 G4double step = (xFinal - xInitial)/iterationNumber ;
50 G4double sum = (typeT.*f)(xPlus); << 48 G4double x = xInitial ;
51 << 49 G4double xPlus = xInitial + 0.5*step ;
52 for(i = 1; i < iterationNumber; ++i) << 50 G4double mean = ( (typeT.*f)(xInitial) + (typeT.*f)(xFinal) )*0.5 ;
53 { << 51 G4double sum = (typeT.*f)(xPlus) ;
54 x += step; << 52
55 xPlus += step; << 53 for(i=1;i<iterationNumber;i++)
56 mean += (typeT.*f)(x); << 54 {
57 sum += (typeT.*f)(xPlus); << 55 x += step ;
58 } << 56 xPlus += step ;
59 mean += 2.0 * sum; << 57 mean += (typeT.*f)(x) ;
>> 58 sum += (typeT.*f)(xPlus) ;
>> 59 }
>> 60 mean += 2.0*sum ;
60 61
61 return mean * step / 3.0; << 62 return mean*step/3.0 ;
62 } 63 }
63 64
64 ////////////////////////////////////////////// 65 /////////////////////////////////////////////////////////////////////
65 // 66 //
66 // Integration of class member functions T::f 67 // Integration of class member functions T::f by Simpson method.
67 // Convenient to use with 'this' pointer 68 // Convenient to use with 'this' pointer
68 69
69 template <class T, class F> << 70 template <class T, class F>
70 G4double G4Integrator<T, F>::Simpson(T* ptrT, << 71 G4double G4Integrator<T,F>::Simpson( T* ptrT,
71 G4double << 72 F f,
72 { << 73 G4double xInitial,
73 G4int i; << 74 G4double xFinal,
74 G4double step = (xFinal - xInitial) / itera << 75 G4int iterationNumber )
75 G4double x = xInitial; << 76 {
76 G4double xPlus = xInitial + 0.5 * step; << 77 G4int i ;
77 G4double mean = ((ptrT->*f)(xInitial) + (pt << 78 G4double step = (xFinal - xInitial)/iterationNumber ;
78 G4double sum = (ptrT->*f)(xPlus); << 79 G4double x = xInitial ;
>> 80 G4double xPlus = xInitial + 0.5*step ;
>> 81 G4double mean = ( (ptrT->*f)(xInitial) + (ptrT->*f)(xFinal) )*0.5 ;
>> 82 G4double sum = (ptrT->*f)(xPlus) ;
>> 83
>> 84 for(i=1;i<iterationNumber;i++)
>> 85 {
>> 86 x += step ;
>> 87 xPlus += step ;
>> 88 mean += (ptrT->*f)(x) ;
>> 89 sum += (ptrT->*f)(xPlus) ;
>> 90 }
>> 91 mean += 2.0*sum ;
79 92
80 for(i = 1; i < iterationNumber; ++i) << 93 return mean*step/3.0 ;
81 { <<
82 x += step; <<
83 xPlus += step; <<
84 mean += (ptrT->*f)(x); <<
85 sum += (ptrT->*f)(xPlus); <<
86 } <<
87 mean += 2.0 * sum; <<
88 <<
89 return mean * step / 3.0; <<
90 } 94 }
91 95
92 ////////////////////////////////////////////// 96 /////////////////////////////////////////////////////////////////////
93 // 97 //
94 // Integration of class member functions T::f 98 // Integration of class member functions T::f by Simpson method.
95 // Convenient to use, when function f is defin 99 // Convenient to use, when function f is defined in global scope, i.e. in main()
96 // program 100 // program
97 101
98 template <class T, class F> << 102 template <class T, class F>
99 G4double G4Integrator<T, F>::Simpson(G4double << 103 G4double G4Integrator<T,F>::Simpson( G4double (*f)(G4double),
100 G4double << 104 G4double xInitial,
101 { << 105 G4double xFinal,
102 G4int i; << 106 G4int iterationNumber )
103 G4double step = (xFinal - xInitial) / itera << 107 {
104 G4double x = xInitial; << 108 G4int i ;
105 G4double xPlus = xInitial + 0.5 * step; << 109 G4double step = (xFinal - xInitial)/iterationNumber ;
106 G4double mean = ((*f)(xInitial) + (*f)(xFin << 110 G4double x = xInitial ;
107 G4double sum = (*f)(xPlus); << 111 G4double xPlus = xInitial + 0.5*step ;
108 << 112 G4double mean = ( (*f)(xInitial) + (*f)(xFinal) )*0.5 ;
109 for(i = 1; i < iterationNumber; ++i) << 113 G4double sum = (*f)(xPlus) ;
110 { << 114
111 x += step; << 115 for(i=1;i<iterationNumber;i++)
112 xPlus += step; << 116 {
113 mean += (*f)(x); << 117 x += step ;
114 sum += (*f)(xPlus); << 118 xPlus += step ;
115 } << 119 mean += (*f)(x) ;
116 mean += 2.0 * sum; << 120 sum += (*f)(xPlus) ;
>> 121 }
>> 122 mean += 2.0*sum ;
117 123
118 return mean * step / 3.0; << 124 return mean*step/3.0 ;
119 } 125 }
120 126
121 ////////////////////////////////////////////// 127 //////////////////////////////////////////////////////////////////////////
122 // 128 //
123 // Adaptive Gauss method 129 // Adaptive Gauss method
124 // 130 //
125 ////////////////////////////////////////////// 131 //////////////////////////////////////////////////////////////////////////
126 // 132 //
127 // 133 //
128 134
129 template <class T, class F> << 135 template <class T, class F>
130 G4double G4Integrator<T, F>::Gauss(T& typeT, F << 136 G4double G4Integrator<T,F>::Gauss( T& typeT, F f,
131 G4double xF << 137 G4double xInitial, G4double xFinal )
132 { << 138 {
133 static const G4double root = 1.0 / std::sqrt << 139 static G4double root = 1.0/std::sqrt(3.0) ;
134 << 140
135 G4double xMean = (xInitial + xFinal) / 2.0; << 141 G4double xMean = (xInitial + xFinal)/2.0 ;
136 G4double Step = (xFinal - xInitial) / 2.0; << 142 G4double Step = (xFinal - xInitial)/2.0 ;
137 G4double delta = Step * root; << 143 G4double delta = Step*root ;
138 G4double sum = ((typeT.*f)(xMean + delta) << 144 G4double sum = ((typeT.*f)(xMean + delta) +
139 << 145 (typeT.*f)(xMean - delta)) ;
140 return sum * Step; << 146
>> 147 return sum*Step ;
141 } 148 }
142 149
143 ////////////////////////////////////////////// 150 //////////////////////////////////////////////////////////////////////
144 // 151 //
145 // 152 //
146 153
147 template <class T, class F> << 154 template <class T, class F> G4double
148 G4double G4Integrator<T, F>::Gauss(T* ptrT, F << 155 G4Integrator<T,F>::Gauss( T* ptrT, F f, G4double a, G4double b )
149 { 156 {
150 return Gauss(*ptrT, f, a, b); << 157 return Gauss(*ptrT,f,a,b) ;
151 } 158 }
152 159
153 ////////////////////////////////////////////// 160 ///////////////////////////////////////////////////////////////////////
154 // 161 //
155 // 162 //
156 163
157 template <class T, class F> 164 template <class T, class F>
158 G4double G4Integrator<T, F>::Gauss(G4double (* << 165 G4double G4Integrator<T,F>::Gauss( G4double (*f)(G4double),
159 G4double xF << 166 G4double xInitial, G4double xFinal)
160 { 167 {
161 static const G4double root = 1.0 / std::sqrt << 168 static G4double root = 1.0/std::sqrt(3.0) ;
162 << 169
163 G4double xMean = (xInitial + xFinal) / 2.0; << 170 G4double xMean = (xInitial + xFinal)/2.0 ;
164 G4double Step = (xFinal - xInitial) / 2.0; << 171 G4double Step = (xFinal - xInitial)/2.0 ;
165 G4double delta = Step * root; << 172 G4double delta = Step*root ;
166 G4double sum = ((*f)(xMean + delta) + (*f) << 173 G4double sum = ( (*f)(xMean + delta) + (*f)(xMean - delta) ) ;
167 << 174
168 return sum * Step; << 175 return sum*Step ;
169 } 176 }
170 177
171 ////////////////////////////////////////////// 178 ///////////////////////////////////////////////////////////////////////////
172 // 179 //
173 // 180 //
174 181
175 template <class T, class F> << 182 template <class T, class F>
176 void G4Integrator<T, F>::AdaptGauss(T& typeT, << 183 void G4Integrator<T,F>::AdaptGauss( T& typeT, F f, G4double xInitial,
177 G4double x << 184 G4double xFinal, G4double fTolerance,
178 G4double& << 185 G4double& sum,
>> 186 G4int& depth )
>> 187 {
>> 188 if(depth > 100)
>> 189 {
>> 190 G4cout<<"G4Integrator<T,F>::AdaptGauss: WARNING !!!"<<G4endl ;
>> 191 G4cout<<"Function varies too rapidly to get stated accuracy in 100 steps "
>> 192 <<G4endl ;
>> 193
>> 194 return ;
>> 195 }
>> 196 G4double xMean = (xInitial + xFinal)/2.0 ;
>> 197 G4double leftHalf = Gauss(typeT,f,xInitial,xMean) ;
>> 198 G4double rightHalf = Gauss(typeT,f,xMean,xFinal) ;
>> 199 G4double full = Gauss(typeT,f,xInitial,xFinal) ;
>> 200 if(std::fabs(leftHalf+rightHalf-full) < fTolerance)
>> 201 {
>> 202 sum += full ;
>> 203 }
>> 204 else
>> 205 {
>> 206 depth++ ;
>> 207 AdaptGauss(typeT,f,xInitial,xMean,fTolerance,sum,depth) ;
>> 208 AdaptGauss(typeT,f,xMean,xFinal,fTolerance,sum,depth) ;
>> 209 }
>> 210 }
>> 211
>> 212 template <class T, class F>
>> 213 void G4Integrator<T,F>::AdaptGauss( T* ptrT, F f, G4double xInitial,
>> 214 G4double xFinal, G4double fTolerance,
>> 215 G4double& sum,
>> 216 G4int& depth )
179 { 217 {
180 if(depth > 100) << 218 AdaptGauss(*ptrT,f,xInitial,xFinal,fTolerance,sum,depth) ;
181 { <<
182 G4cout << "G4Integrator<T,F>::AdaptGauss: <<
183 G4cout << "Function varies too rapidly to <<
184 << G4endl; <<
185 <<
186 return; <<
187 } <<
188 G4double xMean = (xInitial + xFinal) / 2 <<
189 G4double leftHalf = Gauss(typeT, f, xInitia <<
190 G4double rightHalf = Gauss(typeT, f, xMean, <<
191 G4double full = Gauss(typeT, f, xInitia <<
192 if(std::fabs(leftHalf + rightHalf - full) < <<
193 { <<
194 sum += full; <<
195 } <<
196 else <<
197 { <<
198 ++depth; <<
199 AdaptGauss(typeT, f, xInitial, xMean, fTol <<
200 AdaptGauss(typeT, f, xMean, xFinal, fToler <<
201 } <<
202 } <<
203 <<
204 template <class T, class F> <<
205 void G4Integrator<T, F>::AdaptGauss(T* ptrT, F <<
206 G4double x <<
207 G4double& <<
208 { <<
209 AdaptGauss(*ptrT, f, xInitial, xFinal, fTole <<
210 } 219 }
211 220
212 ////////////////////////////////////////////// 221 /////////////////////////////////////////////////////////////////////////
213 // 222 //
214 // 223 //
215 template <class T, class F> 224 template <class T, class F>
216 void G4Integrator<T, F>::AdaptGauss(G4double ( << 225 void G4Integrator<T,F>::AdaptGauss( G4double (*f)(G4double),
217 G4double x << 226 G4double xInitial, G4double xFinal,
218 G4double& << 227 G4double fTolerance, G4double& sum,
219 { << 228 G4int& depth )
220 if(depth > 100) << 229 {
221 { << 230 if(depth > 100)
222 G4cout << "G4SimpleIntegration::AdaptGauss << 231 {
223 G4cout << "Function varies too rapidly to << 232 G4cout<<"G4SimpleIntegration::AdaptGauss: WARNING !!!"<<G4endl ;
224 << G4endl; << 233 G4cout<<"Function varies too rapidly to get stated accuracy in 100 steps "
225 << 234 <<G4endl ;
226 return; << 235
227 } << 236 return ;
228 G4double xMean = (xInitial + xFinal) / 2 << 237 }
229 G4double leftHalf = Gauss(f, xInitial, xMea << 238 G4double xMean = (xInitial + xFinal)/2.0 ;
230 G4double rightHalf = Gauss(f, xMean, xFinal) << 239 G4double leftHalf = Gauss(f,xInitial,xMean) ;
231 G4double full = Gauss(f, xInitial, xFin << 240 G4double rightHalf = Gauss(f,xMean,xFinal) ;
232 if(std::fabs(leftHalf + rightHalf - full) < << 241 G4double full = Gauss(f,xInitial,xFinal) ;
233 { << 242 if(std::fabs(leftHalf+rightHalf-full) < fTolerance)
234 sum += full; << 243 {
235 } << 244 sum += full ;
236 else << 245 }
237 { << 246 else
238 ++depth; << 247 {
239 AdaptGauss(f, xInitial, xMean, fTolerance, << 248 depth++ ;
240 AdaptGauss(f, xMean, xFinal, fTolerance, s << 249 AdaptGauss(f,xInitial,xMean,fTolerance,sum,depth) ;
241 } << 250 AdaptGauss(f,xMean,xFinal,fTolerance,sum,depth) ;
>> 251 }
242 } 252 }
243 253
244 ////////////////////////////////////////////// 254 ////////////////////////////////////////////////////////////////////////
245 // 255 //
246 // Adaptive Gauss integration with accuracy 'e 256 // Adaptive Gauss integration with accuracy 'e'
247 // Convenient for using with class object type 257 // Convenient for using with class object typeT
248 << 258
249 template <class T, class F> << 259 template<class T, class F>
250 G4double G4Integrator<T, F>::AdaptiveGauss(T& << 260 G4double G4Integrator<T,F>::AdaptiveGauss( T& typeT, F f, G4double xInitial,
251 G4d << 261 G4double xFinal, G4double e )
252 { << 262 {
253 G4int depth = 0; << 263 G4int depth = 0 ;
254 G4double sum = 0.0; << 264 G4double sum = 0.0 ;
255 AdaptGauss(typeT, f, xInitial, xFinal, e, su << 265 AdaptGauss(typeT,f,xInitial,xFinal,e,sum,depth) ;
256 return sum; << 266 return sum ;
257 } 267 }
258 268
259 ////////////////////////////////////////////// 269 ////////////////////////////////////////////////////////////////////////
260 // 270 //
261 // Adaptive Gauss integration with accuracy 'e 271 // Adaptive Gauss integration with accuracy 'e'
262 // Convenient for using with 'this' pointer 272 // Convenient for using with 'this' pointer
263 << 273
264 template <class T, class F> << 274 template<class T, class F>
265 G4double G4Integrator<T, F>::AdaptiveGauss(T* << 275 G4double G4Integrator<T,F>::AdaptiveGauss( T* ptrT, F f, G4double xInitial,
266 G4d << 276 G4double xFinal, G4double e )
267 { 277 {
268 return AdaptiveGauss(*ptrT, f, xInitial, xFi << 278 return AdaptiveGauss(*ptrT,f,xInitial,xFinal,e) ;
269 } 279 }
270 280
271 ////////////////////////////////////////////// 281 ////////////////////////////////////////////////////////////////////////
272 // 282 //
273 // Adaptive Gauss integration with accuracy 'e 283 // Adaptive Gauss integration with accuracy 'e'
274 // Convenient for using with global scope func 284 // Convenient for using with global scope function f
275 << 285
276 template <class T, class F> 286 template <class T, class F>
277 G4double G4Integrator<T, F>::AdaptiveGauss(G4d << 287 G4double G4Integrator<T,F>::AdaptiveGauss( G4double (*f)(G4double),
278 G4d << 288 G4double xInitial, G4double xFinal, G4double e )
279 G4d << 289 {
280 { << 290 G4int depth = 0 ;
281 G4int depth = 0; << 291 G4double sum = 0.0 ;
282 G4double sum = 0.0; << 292 AdaptGauss(f,xInitial,xFinal,e,sum,depth) ;
283 AdaptGauss(f, xInitial, xFinal, e, sum, dept << 293 return sum ;
284 return sum; <<
285 } 294 }
286 295
287 ////////////////////////////////////////////// 296 ////////////////////////////////////////////////////////////////////////////
288 // Gauss integration methods involving ortogon 297 // Gauss integration methods involving ortogonal polynomials
289 ////////////////////////////////////////////// 298 ////////////////////////////////////////////////////////////////////////////
290 // 299 //
291 // Methods involving Legendre polynomials << 300 // Methods involving Legendre polynomials
292 // 301 //
293 ////////////////////////////////////////////// 302 /////////////////////////////////////////////////////////////////////////
294 // 303 //
295 // The value nLegendre set the accuracy requir 304 // The value nLegendre set the accuracy required, i.e the number of points
296 // where the function pFunction will be evalua 305 // where the function pFunction will be evaluated during integration.
297 // The function creates the arrays for absciss << 306 // The function creates the arrays for abscissas and weights that used
298 // in Gauss-Legendre quadrature method. << 307 // in Gauss-Legendre quadrature method.
299 // The values a and b are the limits of integr 308 // The values a and b are the limits of integration of the function f .
300 // nLegendre MUST BE EVEN !!! 309 // nLegendre MUST BE EVEN !!!
301 // Returns the integral of the function f betw << 310 // Returns the integral of the function f between a and b, by 2*fNumber point
302 // Gauss-Legendre integration: the function is 311 // Gauss-Legendre integration: the function is evaluated exactly
303 // 2*fNumber times at interior points in the r << 312 // 2*fNumber times at interior points in the range of integration.
304 // Since the weights and abscissas are, in thi << 313 // Since the weights and abscissas are, in this case, symmetric around
305 // the midpoint of the range of integration, t << 314 // the midpoint of the range of integration, there are actually only
306 // fNumber distinct values of each. 315 // fNumber distinct values of each.
307 // Convenient for using with some class object 316 // Convenient for using with some class object dataT
308 317
309 template <class T, class F> 318 template <class T, class F>
310 G4double G4Integrator<T, F>::Legendre(T& typeT << 319 G4double G4Integrator<T,F>::Legendre( T& typeT, F f, G4double a, G4double b,
311 G4int nL << 320 G4int nLegendre )
312 { 321 {
313 G4double nwt, nwt1, temp1, temp2, temp3, tem << 322 G4double newton, newton1, temp1, temp2, temp3, temp ;
314 G4double xDiff, xMean, dx, integral; << 323 G4double xDiff, xMean, dx, integral ;
315 324
316 const G4double tolerance = 1.6e-10; << 325 const G4double tolerance = 1.6e-10 ;
317 G4int i, j, k = nLegendre; << 326 G4int i, j, k = nLegendre ;
318 G4int fNumber = (nLegendre + 1) / 2; << 327 G4int fNumber = (nLegendre + 1)/2 ;
319 << 328
320 if(2 * fNumber != k) << 329 if(2*fNumber != k)
321 { << 330 {
322 G4Exception("G4Integrator<T,F>::Legendre(T << 331 G4Exception("G4Integrator<T,F>::Legendre(T&,F, ...)", "InvalidCall",
323 FatalException, "Invalid (odd) << 332 FatalException, "Invalid (odd) nLegendre in constructor.");
324 } << 333 }
325 << 334
326 G4double* fAbscissa = new G4double[fNumber]; << 335 G4double* fAbscissa = new G4double[fNumber] ;
327 G4double* fWeight = new G4double[fNumber]; << 336 G4double* fWeight = new G4double[fNumber] ;
328 << 337
329 for(i = 1; i <= fNumber; ++i) // Loop over << 338 for(i=1;i<=fNumber;i++) // Loop over the desired roots
330 { << 339 {
331 nwt = std::cos(CLHEP::pi * (i - 0.25) / << 340 newton = std::cos(pi*(i - 0.25)/(k + 0.5)) ; // Initial root approximation
332 (k + 0.5)); // Initial roo << 341
333 << 342 do // loop of Newton's method
334 do // loop of Newton's method << 343 {
335 { << 344 temp1 = 1.0 ;
336 temp1 = 1.0; << 345 temp2 = 0.0 ;
337 temp2 = 0.0; << 346 for(j=1;j<=k;j++)
338 for(j = 1; j <= k; ++j) << 347 {
339 { << 348 temp3 = temp2 ;
340 temp3 = temp2; << 349 temp2 = temp1 ;
341 temp2 = temp1; << 350 temp1 = ((2.0*j - 1.0)*newton*temp2 - (j - 1.0)*temp3)/j ;
342 temp1 = ((2.0 * j - 1.0) * nwt * temp2 << 351 }
343 } << 352 temp = k*(newton*temp1 - temp2)/(newton*newton - 1.0) ;
344 temp = k * (nwt * temp1 - temp2) / (nwt << 353 newton1 = newton ;
345 nwt1 = nwt; << 354 newton = newton1 - temp1/temp ; // Newton's method
346 nwt = nwt1 - temp1 / temp; // Newton's << 355 }
347 } while(std::fabs(nwt - nwt1) > tolerance) << 356 while(std::fabs(newton - newton1) > tolerance) ;
348 << 357
349 fAbscissa[fNumber - i] = nwt; << 358 fAbscissa[fNumber-i] = newton ;
350 fWeight[fNumber - i] = 2.0 / ((1.0 - nwt << 359 fWeight[fNumber-i] = 2.0/((1.0 - newton*newton)*temp*temp) ;
351 } << 360 }
352 << 361
353 // << 362 //
354 // Now we ready to get integral << 363 // Now we ready to get integral
355 // << 364 //
356 << 365
357 xMean = 0.5 * (a + b); << 366 xMean = 0.5*(a + b) ;
358 xDiff = 0.5 * (b - a); << 367 xDiff = 0.5*(b - a) ;
359 integral = 0.0; << 368 integral = 0.0 ;
360 for(i = 0; i < fNumber; ++i) << 369 for(i=0;i<fNumber;i++)
361 { << 370 {
362 dx = xDiff * fAbscissa[i]; << 371 dx = xDiff*fAbscissa[i] ;
363 integral += fWeight[i] * ((typeT.*f)(xMean << 372 integral += fWeight[i]*( (typeT.*f)(xMean + dx) +
364 } << 373 (typeT.*f)(xMean - dx) ) ;
365 delete[] fAbscissa; << 374 }
366 delete[] fWeight; << 375 delete[] fAbscissa;
367 return integral *= xDiff; << 376 delete[] fWeight;
368 } << 377 return integral *= xDiff ;
>> 378 }
369 379
370 ////////////////////////////////////////////// 380 ///////////////////////////////////////////////////////////////////////
371 // 381 //
372 // Convenient for using with the pointer 'this 382 // Convenient for using with the pointer 'this'
373 383
374 template <class T, class F> 384 template <class T, class F>
375 G4double G4Integrator<T, F>::Legendre(T* ptrT, << 385 G4double G4Integrator<T,F>::Legendre( T* ptrT, F f, G4double a,
376 G4int nL << 386 G4double b, G4int nLegendre )
377 { 387 {
378 return Legendre(*ptrT, f, a, b, nLegendre); << 388 return Legendre(*ptrT,f,a,b,nLegendre) ;
379 } 389 }
380 390
381 ////////////////////////////////////////////// 391 ///////////////////////////////////////////////////////////////////////
382 // 392 //
383 // Convenient for using with global scope func 393 // Convenient for using with global scope function f
384 394
385 template <class T, class F> 395 template <class T, class F>
386 G4double G4Integrator<T, F>::Legendre(G4double << 396 G4double G4Integrator<T,F>::Legendre( G4double (*f)(G4double),
387 G4double << 397 G4double a, G4double b, G4int nLegendre)
388 { 398 {
389 G4double nwt, nwt1, temp1, temp2, temp3, tem << 399 G4double newton, newton1, temp1, temp2, temp3, temp ;
390 G4double xDiff, xMean, dx, integral; << 400 G4double xDiff, xMean, dx, integral ;
391 401
392 const G4double tolerance = 1.6e-10; << 402 const G4double tolerance = 1.6e-10 ;
393 G4int i, j, k = nLegendre; << 403 G4int i, j, k = nLegendre ;
394 G4int fNumber = (nLegendre + 1) / 2; << 404 G4int fNumber = (nLegendre + 1)/2 ;
395 << 405
396 if(2 * fNumber != k) << 406 if(2*fNumber != k)
397 { << 407 {
398 G4Exception("G4Integrator<T,F>::Legendre(. << 408 G4Exception("G4Integrator<T,F>::Legendre(...)", "InvalidCall",
399 FatalException, "Invalid (odd) << 409 FatalException, "Invalid (odd) nLegendre in constructor.");
400 } << 410 }
401 << 411
402 G4double* fAbscissa = new G4double[fNumber]; << 412 G4double* fAbscissa = new G4double[fNumber] ;
403 G4double* fWeight = new G4double[fNumber]; << 413 G4double* fWeight = new G4double[fNumber] ;
404 << 414
405 for(i = 1; i <= fNumber; i++) // Loop over << 415 for(i=1;i<=fNumber;i++) // Loop over the desired roots
406 { << 416 {
407 nwt = std::cos(CLHEP::pi * (i - 0.25) / << 417 newton = std::cos(pi*(i - 0.25)/(k + 0.5)) ; // Initial root approximation
408 (k + 0.5)); // Initial roo << 418
409 << 419 do // loop of Newton's method
410 do // loop of Newton's method << 420 {
411 { << 421 temp1 = 1.0 ;
412 temp1 = 1.0; << 422 temp2 = 0.0 ;
413 temp2 = 0.0; << 423 for(j=1;j<=k;j++)
414 for(j = 1; j <= k; ++j) << 424 {
415 { << 425 temp3 = temp2 ;
416 temp3 = temp2; << 426 temp2 = temp1 ;
417 temp2 = temp1; << 427 temp1 = ((2.0*j - 1.0)*newton*temp2 - (j - 1.0)*temp3)/j ;
418 temp1 = ((2.0 * j - 1.0) * nwt * temp2 << 428 }
419 } << 429 temp = k*(newton*temp1 - temp2)/(newton*newton - 1.0) ;
420 temp = k * (nwt * temp1 - temp2) / (nwt << 430 newton1 = newton ;
421 nwt1 = nwt; << 431 newton = newton1 - temp1/temp ; // Newton's method
422 nwt = nwt1 - temp1 / temp; // Newton's << 432 }
423 } while(std::fabs(nwt - nwt1) > tolerance) << 433 while(std::fabs(newton - newton1) > tolerance) ;
424 << 434
425 fAbscissa[fNumber - i] = nwt; << 435 fAbscissa[fNumber-i] = newton ;
426 fWeight[fNumber - i] = 2.0 / ((1.0 - nwt << 436 fWeight[fNumber-i] = 2.0/((1.0 - newton*newton)*temp*temp) ;
427 } << 437 }
428 << 438
429 // << 439 //
430 // Now we ready to get integral << 440 // Now we ready to get integral
431 // << 441 //
432 << 442
433 xMean = 0.5 * (a + b); << 443 xMean = 0.5*(a + b) ;
434 xDiff = 0.5 * (b - a); << 444 xDiff = 0.5*(b - a) ;
435 integral = 0.0; << 445 integral = 0.0 ;
436 for(i = 0; i < fNumber; ++i) << 446 for(i=0;i<fNumber;i++)
437 { << 447 {
438 dx = xDiff * fAbscissa[i]; << 448 dx = xDiff*fAbscissa[i] ;
439 integral += fWeight[i] * ((*f)(xMean + dx) << 449 integral += fWeight[i]*( (*f)(xMean + dx) + (*f)(xMean - dx) ) ;
440 } << 450 }
441 delete[] fAbscissa; << 451 delete[] fAbscissa;
442 delete[] fWeight; << 452 delete[] fWeight;
443 453
444 return integral *= xDiff; << 454 return integral *= xDiff ;
445 } << 455 }
446 456
447 ////////////////////////////////////////////// 457 ////////////////////////////////////////////////////////////////////////////
448 // 458 //
449 // Returns the integral of the function to be 459 // Returns the integral of the function to be pointed by T::f between a and b,
450 // by ten point Gauss-Legendre integration: th 460 // by ten point Gauss-Legendre integration: the function is evaluated exactly
451 // ten times at interior points in the range o 461 // ten times at interior points in the range of integration. Since the weights
452 // and abscissas are, in this case, symmetric << 462 // and abscissas are, in this case, symmetric around the midpoint of the
453 // range of integration, there are actually on 463 // range of integration, there are actually only five distinct values of each
454 // Convenient for using with class object type 464 // Convenient for using with class object typeT
455 465
456 template <class T, class F> << 466 template <class T, class F>
457 G4double G4Integrator<T, F>::Legendre10(T& typ << 467 G4double G4Integrator<T,F>::Legendre10( T& typeT, F f,G4double a, G4double b)
458 { 468 {
459 G4int i; << 469 G4int i ;
460 G4double xDiff, xMean, dx, integral; << 470 G4double xDiff, xMean, dx, integral ;
461 << 471
462 // From Abramowitz M., Stegan I.A. 1964 , Ha << 472 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 916
463 << 473
464 static const G4double abscissa[] = { 0.14887 << 474 static G4double abscissa[] = { 0.148874338981631, 0.433395394129247,
465 0.67940 << 475 0.679409568299024, 0.865063366688985,
466 0.97390 << 476 0.973906528517172 } ;
467 << 477
468 static const G4double weight[] = { 0.2955242 << 478 static G4double weight[] = { 0.295524224714753, 0.269266719309996,
469 0.2190863 << 479 0.219086362515982, 0.149451349150581,
470 0.0666713 << 480 0.066671344308688 } ;
471 xMean = 0.5 * (a + << 481 xMean = 0.5*(a + b) ;
472 xDiff = 0.5 * (b - << 482 xDiff = 0.5*(b - a) ;
473 integral = 0.0; << 483 integral = 0.0 ;
474 for(i = 0; i < 5; ++i) << 484 for(i=0;i<5;i++)
475 { << 485 {
476 dx = xDiff * abscissa[i]; << 486 dx = xDiff*abscissa[i] ;
477 integral += weight[i] * ((typeT.*f)(xMean << 487 integral += weight[i]*( (typeT.*f)(xMean + dx) + (typeT.*f)(xMean - dx)) ;
478 } << 488 }
479 return integral *= xDiff; << 489 return integral *= xDiff ;
480 } 490 }
481 491
482 ////////////////////////////////////////////// 492 ///////////////////////////////////////////////////////////////////////////
483 // 493 //
484 // Convenient for using with the pointer 'this 494 // Convenient for using with the pointer 'this'
485 495
486 template <class T, class F> << 496 template <class T, class F>
487 G4double G4Integrator<T, F>::Legendre10(T* ptr << 497 G4double G4Integrator<T,F>::Legendre10( T* ptrT, F f,G4double a, G4double b)
488 { 498 {
489 return Legendre10(*ptrT, f, a, b); << 499 return Legendre10(*ptrT,f,a,b) ;
490 } << 500 }
491 501
492 ////////////////////////////////////////////// 502 //////////////////////////////////////////////////////////////////////////
493 // 503 //
494 // Convenient for using with global scope func 504 // Convenient for using with global scope functions
495 505
496 template <class T, class F> 506 template <class T, class F>
497 G4double G4Integrator<T, F>::Legendre10(G4doub << 507 G4double G4Integrator<T,F>::Legendre10( G4double (*f)(G4double),
498 G4doub << 508 G4double a, G4double b )
499 { 509 {
500 G4int i; << 510 G4int i ;
501 G4double xDiff, xMean, dx, integral; << 511 G4double xDiff, xMean, dx, integral ;
502 << 512
503 // From Abramowitz M., Stegan I.A. 1964 , Ha << 513 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 916
504 << 514
505 static const G4double abscissa[] = { 0.14887 << 515 static G4double abscissa[] = { 0.148874338981631, 0.433395394129247,
506 0.67940 << 516 0.679409568299024, 0.865063366688985,
507 0.97390 << 517 0.973906528517172 } ;
508 << 518
509 static const G4double weight[] = { 0.2955242 << 519 static G4double weight[] = { 0.295524224714753, 0.269266719309996,
510 0.2190863 << 520 0.219086362515982, 0.149451349150581,
511 0.0666713 << 521 0.066671344308688 } ;
512 xMean = 0.5 * (a + << 522 xMean = 0.5*(a + b) ;
513 xDiff = 0.5 * (b - << 523 xDiff = 0.5*(b - a) ;
514 integral = 0.0; << 524 integral = 0.0 ;
515 for(i = 0; i < 5; ++i) << 525 for(i=0;i<5;i++)
516 { << 526 {
517 dx = xDiff * abscissa[i]; << 527 dx = xDiff*abscissa[i] ;
518 integral += weight[i] * ((*f)(xMean + dx) << 528 integral += weight[i]*( (*f)(xMean + dx) + (*f)(xMean - dx)) ;
519 } << 529 }
520 return integral *= xDiff; << 530 return integral *= xDiff ;
521 } 531 }
522 532
523 ////////////////////////////////////////////// 533 ///////////////////////////////////////////////////////////////////////
524 // 534 //
525 // Returns the integral of the function to be 535 // Returns the integral of the function to be pointed by T::f between a and b,
526 // by 96 point Gauss-Legendre integration: the 536 // by 96 point Gauss-Legendre integration: the function is evaluated exactly
527 // ten Times at interior points in the range o 537 // ten Times at interior points in the range of integration. Since the weights
528 // and abscissas are, in this case, symmetric << 538 // and abscissas are, in this case, symmetric around the midpoint of the
529 // range of integration, there are actually on 539 // range of integration, there are actually only five distinct values of each
530 // Convenient for using with some class object 540 // Convenient for using with some class object typeT
531 541
532 template <class T, class F> << 542 template <class T, class F>
533 G4double G4Integrator<T, F>::Legendre96(T& typ << 543 G4double G4Integrator<T,F>::Legendre96( T& typeT, F f,G4double a, G4double b)
534 { 544 {
535 G4int i; << 545 G4int i ;
536 G4double xDiff, xMean, dx, integral; << 546 G4double xDiff, xMean, dx, integral ;
537 << 547
538 // From Abramowitz M., Stegan I.A. 1964 , Ha << 548 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 919
539 << 549
540 static const G4double abscissa[] = { << 550 static G4double
541 0.016276744849602969579, 0.048812985136049 << 551 abscissa[] = {
542 0.081297495464425558994, 0.113695850110665 << 552 0.016276744849602969579, 0.048812985136049731112,
543 0.145973714654896941989, 0.178096882367618 << 553 0.081297495464425558994, 0.113695850110665920911,
544 << 554 0.145973714654896941989, 0.178096882367618602759, // 6
545 0.210031310460567203603, 0.241743156163840 << 555
546 0.273198812591049141487, 0.304364944354496 << 556 0.210031310460567203603, 0.241743156163840012328,
547 0.335208522892625422616, 0.365696861472313 << 557 0.273198812591049141487, 0.304364944354496353024,
548 << 558 0.335208522892625422616, 0.365696861472313635031, // 12
549 0.395797649828908603285, 0.425478988407300 << 559
550 0.454709422167743008636, 0.483457973920596 << 560 0.395797649828908603285, 0.425478988407300545365,
551 0.511694177154667673586, 0.539388108324357 << 561 0.454709422167743008636, 0.483457973920596359768,
552 << 562 0.511694177154667673586, 0.539388108324357436227, // 18
553 0.566510418561397168404, 0.593032364777572 << 563
554 0.618925840125468570386, 0.644163403784967 << 564 0.566510418561397168404, 0.593032364777572080684,
555 0.668718310043916153953, 0.692564536642171 << 565 0.618925840125468570386, 0.644163403784967106798,
556 << 566 0.668718310043916153953, 0.692564536642171561344, // 24
557 0.715676812348967626225, 0.738030643744400 << 567
558 0.759602341176647498703, 0.780369043867433 << 568 0.715676812348967626225, 0.738030643744400132851,
559 0.800308744139140817229, 0.819400310737931 << 569 0.759602341176647498703, 0.780369043867433217604,
560 << 570 0.800308744139140817229, 0.819400310737931675539, // 30
561 0.837623511228187121494, 0.854959033434601 << 571
562 0.871388505909296502874, 0.886894517402420 << 572 0.837623511228187121494, 0.854959033434601455463,
563 0.901460635315852341319, 0.915071423120898 << 573 0.871388505909296502874, 0.886894517402420416057,
564 << 574 0.901460635315852341319, 0.915071423120898074206, // 36
565 0.927712456722308690965, 0.939370339752755 << 575
566 0.950032717784437635756, 0.959688291448742 << 576 0.927712456722308690965, 0.939370339752755216932,
567 0.968326828463264212174, 0.975939174585136 << 577 0.950032717784437635756, 0.959688291448742539300,
568 << 578 0.968326828463264212174, 0.975939174585136466453, // 42
569 0.982517263563014677447, 0.988054126329623 << 579
570 0.992543900323762624572, 0.995981842987209 << 580 0.982517263563014677447, 0.988054126329623799481,
571 0.998364375863181677724, 0.999689503883230 << 581 0.992543900323762624572, 0.995981842987209290650,
572 }; << 582 0.998364375863181677724, 0.999689503883230766828 // 48
573 << 583 } ;
574 static const G4double weight[] = { << 584
575 0.032550614492363166242, 0.032516118713868 << 585 static G4double
576 0.032447163714064269364, 0.032343822568575 << 586 weight[] = {
577 0.032206204794030250669, 0.032034456231992 << 587 0.032550614492363166242, 0.032516118713868835987,
578 << 588 0.032447163714064269364, 0.032343822568575928429,
579 0.031828758894411006535, 0.031589330770727 << 589 0.032206204794030250669, 0.032034456231992663218, // 6
580 0.031316425596862355813, 0.031010332586313 << 590
581 0.030671376123669149014, 0.030299915420827 << 591 0.031828758894411006535, 0.031589330770727168558,
582 << 592 0.031316425596862355813, 0.031010332586313837423,
583 0.029896344136328385984, 0.029461089958167 << 593 0.030671376123669149014, 0.030299915420827593794, // 12
584 0.028994614150555236543, 0.028497411065085 << 594
585 0.027970007616848334440, 0.027412962726029 << 595 0.029896344136328385984, 0.029461089958167905970,
586 << 596 0.028994614150555236543, 0.028497411065085385646,
587 0.026826866725591762198, 0.026212340735672 << 597 0.027970007616848334440, 0.027412962726029242823, // 18
588 0.025570036005349361499, 0.024900633222483 << 598
589 0.024204841792364691282, 0.023483399085926 << 599 0.026826866725591762198, 0.026212340735672413913,
590 << 600 0.025570036005349361499, 0.024900633222483610288,
591 0.022737069658329374001, 0.021966644438744 << 601 0.024204841792364691282, 0.023483399085926219842, // 24
592 0.021172939892191298988, 0.020356797154333 << 602
593 0.019519081140145022410, 0.018660679627411 << 603 0.022737069658329374001, 0.021966644438744349195,
594 << 604 0.021172939892191298988, 0.020356797154333324595,
595 0.017782502316045260838, 0.016885479864245 << 605 0.019519081140145022410, 0.018660679627411467385, // 30
596 0.015970562902562291381, 0.015038721026994 << 606
597 0.014090941772314860916, 0.013128229566961 << 607 0.017782502316045260838, 0.016885479864245172450,
598 << 608 0.015970562902562291381, 0.015038721026994938006,
599 0.012151604671088319635, 0.011162102099838 << 609 0.014090941772314860916, 0.013128229566961572637, // 36
600 0.010160770535008415758, 0.009148671230783 << 610
601 0.008126876925698759217, 0.007096470791153 << 611 0.012151604671088319635, 0.011162102099838498591,
602 << 612 0.010160770535008415758, 0.009148671230783386633,
603 0.006058545504235961683, 0.005014202742927 << 613 0.008126876925698759217, 0.007096470791153865269, // 42
604 0.003964554338444686674, 0.002910731817934 << 614
605 0.001853960788946921732, 0.000796792065552 << 615 0.006058545504235961683, 0.005014202742927517693,
606 }; << 616 0.003964554338444686674, 0.002910731817934946408,
607 xMean = 0.5 * (a + b); << 617 0.001853960788946921732, 0.000796792065552012429 // 48
608 xDiff = 0.5 * (b - a); << 618 } ;
609 integral = 0.0; << 619 xMean = 0.5*(a + b) ;
610 for(i = 0; i < 48; ++i) << 620 xDiff = 0.5*(b - a) ;
611 { << 621 integral = 0.0 ;
612 dx = xDiff * abscissa[i]; << 622 for(i=0;i<48;i++)
613 integral += weight[i] * ((typeT.*f)(xMean << 623 {
614 } << 624 dx = xDiff*abscissa[i] ;
615 return integral *= xDiff; << 625 integral += weight[i]*((typeT.*f)(xMean + dx) + (typeT.*f)(xMean - dx)) ;
>> 626 }
>> 627 return integral *= xDiff ;
616 } 628 }
617 629
618 ////////////////////////////////////////////// 630 ///////////////////////////////////////////////////////////////////////
619 // 631 //
620 // Convenient for using with the pointer 'this 632 // Convenient for using with the pointer 'this'
621 633
622 template <class T, class F> << 634 template <class T, class F>
623 G4double G4Integrator<T, F>::Legendre96(T* ptr << 635 G4double G4Integrator<T,F>::Legendre96( T* ptrT, F f,G4double a, G4double b)
624 { 636 {
625 return Legendre96(*ptrT, f, a, b); << 637 return Legendre96(*ptrT,f,a,b) ;
626 } << 638 }
627 639
628 ////////////////////////////////////////////// 640 ///////////////////////////////////////////////////////////////////////
629 // 641 //
630 // Convenient for using with global scope func << 642 // Convenient for using with global scope function f
631 643
632 template <class T, class F> 644 template <class T, class F>
633 G4double G4Integrator<T, F>::Legendre96(G4doub << 645 G4double G4Integrator<T,F>::Legendre96( G4double (*f)(G4double),
634 G4doub << 646 G4double a, G4double b )
635 { 647 {
636 G4int i; << 648 G4int i ;
637 G4double xDiff, xMean, dx, integral; << 649 G4double xDiff, xMean, dx, integral ;
638 << 650
639 // From Abramowitz M., Stegan I.A. 1964 , Ha << 651 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 919
640 << 652
641 static const G4double abscissa[] = { << 653 static G4double
642 0.016276744849602969579, 0.048812985136049 << 654 abscissa[] = {
643 0.081297495464425558994, 0.113695850110665 << 655 0.016276744849602969579, 0.048812985136049731112,
644 0.145973714654896941989, 0.178096882367618 << 656 0.081297495464425558994, 0.113695850110665920911,
645 << 657 0.145973714654896941989, 0.178096882367618602759, // 6
646 0.210031310460567203603, 0.241743156163840 << 658
647 0.273198812591049141487, 0.304364944354496 << 659 0.210031310460567203603, 0.241743156163840012328,
648 0.335208522892625422616, 0.365696861472313 << 660 0.273198812591049141487, 0.304364944354496353024,
649 << 661 0.335208522892625422616, 0.365696861472313635031, // 12
650 0.395797649828908603285, 0.425478988407300 << 662
651 0.454709422167743008636, 0.483457973920596 << 663 0.395797649828908603285, 0.425478988407300545365,
652 0.511694177154667673586, 0.539388108324357 << 664 0.454709422167743008636, 0.483457973920596359768,
653 << 665 0.511694177154667673586, 0.539388108324357436227, // 18
654 0.566510418561397168404, 0.593032364777572 << 666
655 0.618925840125468570386, 0.644163403784967 << 667 0.566510418561397168404, 0.593032364777572080684,
656 0.668718310043916153953, 0.692564536642171 << 668 0.618925840125468570386, 0.644163403784967106798,
657 << 669 0.668718310043916153953, 0.692564536642171561344, // 24
658 0.715676812348967626225, 0.738030643744400 << 670
659 0.759602341176647498703, 0.780369043867433 << 671 0.715676812348967626225, 0.738030643744400132851,
660 0.800308744139140817229, 0.819400310737931 << 672 0.759602341176647498703, 0.780369043867433217604,
661 << 673 0.800308744139140817229, 0.819400310737931675539, // 30
662 0.837623511228187121494, 0.854959033434601 << 674
663 0.871388505909296502874, 0.886894517402420 << 675 0.837623511228187121494, 0.854959033434601455463,
664 0.901460635315852341319, 0.915071423120898 << 676 0.871388505909296502874, 0.886894517402420416057,
665 << 677 0.901460635315852341319, 0.915071423120898074206, // 36
666 0.927712456722308690965, 0.939370339752755 << 678
667 0.950032717784437635756, 0.959688291448742 << 679 0.927712456722308690965, 0.939370339752755216932,
668 0.968326828463264212174, 0.975939174585136 << 680 0.950032717784437635756, 0.959688291448742539300,
669 << 681 0.968326828463264212174, 0.975939174585136466453, // 42
670 0.982517263563014677447, 0.988054126329623 << 682
671 0.992543900323762624572, 0.995981842987209 << 683 0.982517263563014677447, 0.988054126329623799481,
672 0.998364375863181677724, 0.999689503883230 << 684 0.992543900323762624572, 0.995981842987209290650,
673 }; << 685 0.998364375863181677724, 0.999689503883230766828 // 48
674 << 686 } ;
675 static const G4double weight[] = { << 687
676 0.032550614492363166242, 0.032516118713868 << 688 static G4double
677 0.032447163714064269364, 0.032343822568575 << 689 weight[] = {
678 0.032206204794030250669, 0.032034456231992 << 690 0.032550614492363166242, 0.032516118713868835987,
679 << 691 0.032447163714064269364, 0.032343822568575928429,
680 0.031828758894411006535, 0.031589330770727 << 692 0.032206204794030250669, 0.032034456231992663218, // 6
681 0.031316425596862355813, 0.031010332586313 << 693
682 0.030671376123669149014, 0.030299915420827 << 694 0.031828758894411006535, 0.031589330770727168558,
683 << 695 0.031316425596862355813, 0.031010332586313837423,
684 0.029896344136328385984, 0.029461089958167 << 696 0.030671376123669149014, 0.030299915420827593794, // 12
685 0.028994614150555236543, 0.028497411065085 << 697
686 0.027970007616848334440, 0.027412962726029 << 698 0.029896344136328385984, 0.029461089958167905970,
687 << 699 0.028994614150555236543, 0.028497411065085385646,
688 0.026826866725591762198, 0.026212340735672 << 700 0.027970007616848334440, 0.027412962726029242823, // 18
689 0.025570036005349361499, 0.024900633222483 << 701
690 0.024204841792364691282, 0.023483399085926 << 702 0.026826866725591762198, 0.026212340735672413913,
691 << 703 0.025570036005349361499, 0.024900633222483610288,
692 0.022737069658329374001, 0.021966644438744 << 704 0.024204841792364691282, 0.023483399085926219842, // 24
693 0.021172939892191298988, 0.020356797154333 << 705
694 0.019519081140145022410, 0.018660679627411 << 706 0.022737069658329374001, 0.021966644438744349195,
695 << 707 0.021172939892191298988, 0.020356797154333324595,
696 0.017782502316045260838, 0.016885479864245 << 708 0.019519081140145022410, 0.018660679627411467385, // 30
697 0.015970562902562291381, 0.015038721026994 << 709
698 0.014090941772314860916, 0.013128229566961 << 710 0.017782502316045260838, 0.016885479864245172450,
699 << 711 0.015970562902562291381, 0.015038721026994938006,
700 0.012151604671088319635, 0.011162102099838 << 712 0.014090941772314860916, 0.013128229566961572637, // 36
701 0.010160770535008415758, 0.009148671230783 << 713
702 0.008126876925698759217, 0.007096470791153 << 714 0.012151604671088319635, 0.011162102099838498591,
703 << 715 0.010160770535008415758, 0.009148671230783386633,
704 0.006058545504235961683, 0.005014202742927 << 716 0.008126876925698759217, 0.007096470791153865269, // 42
705 0.003964554338444686674, 0.002910731817934 << 717
706 0.001853960788946921732, 0.000796792065552 << 718 0.006058545504235961683, 0.005014202742927517693,
707 }; << 719 0.003964554338444686674, 0.002910731817934946408,
708 xMean = 0.5 * (a + b); << 720 0.001853960788946921732, 0.000796792065552012429 // 48
709 xDiff = 0.5 * (b - a); << 721 } ;
710 integral = 0.0; << 722 xMean = 0.5*(a + b) ;
711 for(i = 0; i < 48; ++i) << 723 xDiff = 0.5*(b - a) ;
712 { << 724 integral = 0.0 ;
713 dx = xDiff * abscissa[i]; << 725 for(i=0;i<48;i++)
714 integral += weight[i] * ((*f)(xMean + dx) << 726 {
715 } << 727 dx = xDiff*abscissa[i] ;
716 return integral *= xDiff; << 728 integral += weight[i]*((*f)(xMean + dx) + (*f)(xMean - dx)) ;
>> 729 }
>> 730 return integral *= xDiff ;
717 } 731 }
718 732
719 ////////////////////////////////////////////// 733 //////////////////////////////////////////////////////////////////////////////
720 // 734 //
721 // Methods involving Chebyshev polynomials << 735 // Methods involving Chebyshev polynomials
722 // 736 //
723 ////////////////////////////////////////////// 737 ///////////////////////////////////////////////////////////////////////////
724 // 738 //
725 // Integrates function pointed by T::f from a << 739 // Integrates function pointed by T::f from a to b by Gauss-Chebyshev
726 // quadrature method. 740 // quadrature method.
727 // Convenient for using with class object type 741 // Convenient for using with class object typeT
728 742
729 template <class T, class F> 743 template <class T, class F>
730 G4double G4Integrator<T, F>::Chebyshev(T& type << 744 G4double G4Integrator<T,F>::Chebyshev( T& typeT, F f, G4double a,
731 G4int n << 745 G4double b, G4int nChebyshev )
732 { 746 {
733 G4int i; << 747 G4int i ;
734 G4double xDiff, xMean, dx, integral = 0.0; << 748 G4double xDiff, xMean, dx, integral = 0.0 ;
735 << 749
736 G4int fNumber = nChebyshev; // Try to << 750 G4int fNumber = nChebyshev ; // Try to reduce fNumber twice ??
737 G4double cof = CLHEP::pi / fNumber; << 751 G4double cof = pi/fNumber ;
738 G4double* fAbscissa = new G4double[fNumber]; << 752 G4double* fAbscissa = new G4double[fNumber] ;
739 G4double* fWeight = new G4double[fNumber]; << 753 G4double* fWeight = new G4double[fNumber] ;
740 for(i = 0; i < fNumber; ++i) << 754 for(i=0;i<fNumber;i++)
741 { << 755 {
742 fAbscissa[i] = std::cos(cof * (i + 0.5)); << 756 fAbscissa[i] = std::cos(cof*(i + 0.5)) ;
743 fWeight[i] = cof * std::sqrt(1 - fAbscis << 757 fWeight[i] = cof*std::sqrt(1 - fAbscissa[i]*fAbscissa[i]) ;
744 } << 758 }
745 << 759
746 // << 760 //
747 // Now we ready to estimate the integral << 761 // Now we ready to estimate the integral
748 // << 762 //
749 << 763
750 xMean = 0.5 * (a + b); << 764 xMean = 0.5*(a + b) ;
751 xDiff = 0.5 * (b - a); << 765 xDiff = 0.5*(b - a) ;
752 for(i = 0; i < fNumber; ++i) << 766 for(i=0;i<fNumber;i++)
753 { << 767 {
754 dx = xDiff * fAbscissa[i]; << 768 dx = xDiff*fAbscissa[i] ;
755 integral += fWeight[i] * (typeT.*f)(xMean << 769 integral += fWeight[i]*(typeT.*f)(xMean + dx) ;
756 } << 770 }
757 delete[] fAbscissa; << 771 delete[] fAbscissa;
758 delete[] fWeight; << 772 delete[] fWeight;
759 return integral *= xDiff; << 773 return integral *= xDiff ;
760 } 774 }
761 775
762 ////////////////////////////////////////////// 776 ///////////////////////////////////////////////////////////////////////
763 // 777 //
764 // Convenient for using with 'this' pointer 778 // Convenient for using with 'this' pointer
765 779
766 template <class T, class F> 780 template <class T, class F>
767 G4double G4Integrator<T, F>::Chebyshev(T* ptrT << 781 G4double G4Integrator<T,F>::Chebyshev( T* ptrT, F f, G4double a,
768 G4int n << 782 G4double b, G4int n )
769 { 783 {
770 return Chebyshev(*ptrT, f, a, b, n); << 784 return Chebyshev(*ptrT,f,a,b,n) ;
771 } << 785 }
772 786
773 ////////////////////////////////////////////// 787 ////////////////////////////////////////////////////////////////////////
774 // 788 //
775 // For use with global scope functions f << 789 // For use with global scope functions f
776 790
777 template <class T, class F> 791 template <class T, class F>
778 G4double G4Integrator<T, F>::Chebyshev(G4doubl << 792 G4double G4Integrator<T,F>::Chebyshev( G4double (*f)(G4double),
779 G4doubl << 793 G4double a, G4double b, G4int nChebyshev )
780 { 794 {
781 G4int i; << 795 G4int i ;
782 G4double xDiff, xMean, dx, integral = 0.0; << 796 G4double xDiff, xMean, dx, integral = 0.0 ;
783 << 797
784 G4int fNumber = nChebyshev; // Try to << 798 G4int fNumber = nChebyshev ; // Try to reduce fNumber twice ??
785 G4double cof = CLHEP::pi / fNumber; << 799 G4double cof = pi/fNumber ;
786 G4double* fAbscissa = new G4double[fNumber]; << 800 G4double* fAbscissa = new G4double[fNumber] ;
787 G4double* fWeight = new G4double[fNumber]; << 801 G4double* fWeight = new G4double[fNumber] ;
788 for(i = 0; i < fNumber; ++i) << 802 for(i=0;i<fNumber;i++)
789 { << 803 {
790 fAbscissa[i] = std::cos(cof * (i + 0.5)); << 804 fAbscissa[i] = std::cos(cof*(i + 0.5)) ;
791 fWeight[i] = cof * std::sqrt(1 - fAbscis << 805 fWeight[i] = cof*std::sqrt(1 - fAbscissa[i]*fAbscissa[i]) ;
792 } << 806 }
793 << 807
794 // << 808 //
795 // Now we ready to estimate the integral << 809 // Now we ready to estimate the integral
796 // << 810 //
797 << 811
798 xMean = 0.5 * (a + b); << 812 xMean = 0.5*(a + b) ;
799 xDiff = 0.5 * (b - a); << 813 xDiff = 0.5*(b - a) ;
800 for(i = 0; i < fNumber; ++i) << 814 for(i=0;i<fNumber;i++)
801 { << 815 {
802 dx = xDiff * fAbscissa[i]; << 816 dx = xDiff*fAbscissa[i] ;
803 integral += fWeight[i] * (*f)(xMean + dx); << 817 integral += fWeight[i]*(*f)(xMean + dx) ;
804 } << 818 }
805 delete[] fAbscissa; << 819 delete[] fAbscissa;
806 delete[] fWeight; << 820 delete[] fWeight;
807 return integral *= xDiff; << 821 return integral *= xDiff ;
808 } 822 }
809 823
810 ////////////////////////////////////////////// 824 //////////////////////////////////////////////////////////////////////
811 // 825 //
812 // Method involving Laguerre polynomials 826 // Method involving Laguerre polynomials
813 // 827 //
814 ////////////////////////////////////////////// 828 //////////////////////////////////////////////////////////////////////
815 // 829 //
816 // Integral from zero to infinity of std::pow( << 830 // Integral from zero to infinity of std::pow(x,alpha)*std::exp(-x)*f(x).
817 // The value of nLaguerre sets the accuracy. 831 // The value of nLaguerre sets the accuracy.
818 // The function creates arrays fAbscissa[0,.., << 832 // The function creates arrays fAbscissa[0,..,nLaguerre-1] and
819 // fWeight[0,..,nLaguerre-1] . << 833 // fWeight[0,..,nLaguerre-1] .
820 // Convenient for using with class object 'typ 834 // Convenient for using with class object 'typeT' and (typeT.*f) function
821 // (T::f) 835 // (T::f)
822 836
823 template <class T, class F> 837 template <class T, class F>
824 G4double G4Integrator<T, F>::Laguerre(T& typeT << 838 G4double G4Integrator<T,F>::Laguerre( T& typeT, F f, G4double alpha,
825 G4int nL << 839 G4int nLaguerre )
826 { 840 {
827 const G4double tolerance = 1.0e-10; << 841 const G4double tolerance = 1.0e-10 ;
828 const G4int maxNumber = 12; << 842 const G4int maxNumber = 12 ;
829 G4int i, j, k; << 843 G4int i, j, k ;
830 G4double nwt = 0., nwt1, temp1, temp2, << 844 G4double newton=0., newton1, temp1, temp2, temp3, temp, cofi ;
831 G4double integral = 0.0; << 845 G4double integral = 0.0 ;
832 << 846
833 G4int fNumber = nLaguerre; << 847 G4int fNumber = nLaguerre ;
834 G4double* fAbscissa = new G4double[fNumber]; << 848 G4double* fAbscissa = new G4double[fNumber] ;
835 G4double* fWeight = new G4double[fNumber]; << 849 G4double* fWeight = new G4double[fNumber] ;
>> 850
>> 851 for(i=1;i<=fNumber;i++) // Loop over the desired roots
>> 852 {
>> 853 if(i == 1)
>> 854 {
>> 855 newton = (1.0 + alpha)*(3.0 + 0.92*alpha)
>> 856 / (1.0 + 2.4*fNumber + 1.8*alpha) ;
>> 857 }
>> 858 else if(i == 2)
>> 859 {
>> 860 newton += (15.0 + 6.25*alpha)/(1.0 + 0.9*alpha + 2.5*fNumber) ;
>> 861 }
>> 862 else
>> 863 {
>> 864 cofi = i - 2 ;
>> 865 newton += ((1.0+2.55*cofi)/(1.9*cofi)
>> 866 + 1.26*cofi*alpha/(1.0+3.5*cofi))
>> 867 * (newton - fAbscissa[i-3])/(1.0 + 0.3*alpha) ;
>> 868 }
>> 869 for(k=1;k<=maxNumber;k++)
>> 870 {
>> 871 temp1 = 1.0 ;
>> 872 temp2 = 0.0 ;
836 873
837 for(i = 1; i <= fNumber; ++i) // Loop over << 874 for(j=1;j<=fNumber;j++)
838 { << 875 {
839 if(i == 1) << 876 temp3 = temp2 ;
840 { << 877 temp2 = temp1 ;
841 nwt = (1.0 + alpha) * (3.0 + 0.92 * alph << 878 temp1 = ((2*j - 1 + alpha - newton)*temp2 - (j - 1 + alpha)*temp3)/j ;
842 (1.0 + 2.4 * fNumber + 1.8 * alpha << 879 }
843 } << 880 temp = (fNumber*temp1 - (fNumber +alpha)*temp2)/newton ;
844 else if(i == 2) << 881 newton1 = newton ;
845 { << 882 newton = newton1 - temp1/temp ;
846 nwt += (15.0 + 6.25 * alpha) / (1.0 + 0. << 883
847 } << 884 if(std::fabs(newton - newton1) <= tolerance)
848 else << 885 {
849 { << 886 break ;
850 cofi = i - 2; << 887 }
851 nwt += ((1.0 + 2.55 * cofi) / (1.9 * cof << 888 }
852 1.26 * cofi * alpha / (1.0 + 3.5 << 889 if(k > maxNumber)
853 (nwt - fAbscissa[i - 3]) / (1.0 + << 890 {
854 } << 891 G4Exception("G4Integrator<T,F>::Laguerre(T,F, ...)", "Error",
855 for(k = 1; k <= maxNumber; ++k) << 892 FatalException, "Too many (>12) iterations.");
856 { << 893 }
857 temp1 = 1.0; << 894
858 temp2 = 0.0; << 895 fAbscissa[i-1] = newton ;
859 << 896 fWeight[i-1] = -std::exp(GammaLogarithm(alpha + fNumber) -
860 for(j = 1; j <= fNumber; ++j) << 897 GammaLogarithm((G4double)fNumber))/(temp*fNumber*temp2) ;
861 { << 898 }
862 temp3 = temp2; << 899
863 temp2 = temp1; << 900 //
864 temp1 = << 901 // Integral evaluation
865 ((2 * j - 1 + alpha - nwt) * temp2 - << 902 //
866 } << 903
867 temp = (fNumber * temp1 - (fNumber + alp << 904 for(i=0;i<fNumber;i++)
868 nwt1 = nwt; << 905 {
869 nwt = nwt1 - temp1 / temp; << 906 integral += fWeight[i]*(typeT.*f)(fAbscissa[i]) ;
870 << 907 }
871 if(std::fabs(nwt - nwt1) <= tolerance) << 908 delete[] fAbscissa;
872 { << 909 delete[] fWeight;
873 break; << 910 return integral ;
874 } << 911 }
875 } <<
876 if(k > maxNumber) <<
877 { <<
878 G4Exception("G4Integrator<T,F>::Laguerre <<
879 FatalException, "Too many (> <<
880 } <<
881 <<
882 fAbscissa[i - 1] = nwt; <<
883 fWeight[i - 1] = -std::exp(GammaLogarith <<
884 GammaLogarithm( <<
885 (temp * fNumber * temp2); <<
886 } <<
887 912
888 // <<
889 // Integral evaluation <<
890 // <<
891 913
892 for(i = 0; i < fNumber; ++i) <<
893 { <<
894 integral += fWeight[i] * (typeT.*f)(fAbsci <<
895 } <<
896 delete[] fAbscissa; <<
897 delete[] fWeight; <<
898 return integral; <<
899 } <<
900 914
901 ////////////////////////////////////////////// 915 //////////////////////////////////////////////////////////////////////
902 // 916 //
903 // 917 //
904 918
905 template <class T, class F> << 919 template <class T, class F> G4double
906 G4double G4Integrator<T, F>::Laguerre(T* ptrT, << 920 G4Integrator<T,F>::Laguerre( T* ptrT, F f, G4double alpha, G4int nLaguerre )
907 G4int nL <<
908 { 921 {
909 return Laguerre(*ptrT, f, alpha, nLaguerre); << 922 return Laguerre(*ptrT,f,alpha,nLaguerre) ;
910 } 923 }
911 924
912 ////////////////////////////////////////////// 925 ////////////////////////////////////////////////////////////////////////
913 // 926 //
914 // For use with global scope functions f << 927 // For use with global scope functions f
915 928
916 template <class T, class F> << 929 template <class T, class F> G4double
917 G4double G4Integrator<T, F>::Laguerre(G4double << 930 G4Integrator<T,F>::Laguerre( G4double (*f)(G4double),
918 G4int nL << 931 G4double alpha, G4int nLaguerre )
919 { << 932 {
920 const G4double tolerance = 1.0e-10; << 933 const G4double tolerance = 1.0e-10 ;
921 const G4int maxNumber = 12; << 934 const G4int maxNumber = 12 ;
922 G4int i, j, k; << 935 G4int i, j, k ;
923 G4double nwt = 0., nwt1, temp1, temp2, << 936 G4double newton=0., newton1, temp1, temp2, temp3, temp, cofi ;
924 G4double integral = 0.0; << 937 G4double integral = 0.0 ;
925 << 938
926 G4int fNumber = nLaguerre; << 939 G4int fNumber = nLaguerre ;
927 G4double* fAbscissa = new G4double[fNumber]; << 940 G4double* fAbscissa = new G4double[fNumber] ;
928 G4double* fWeight = new G4double[fNumber]; << 941 G4double* fWeight = new G4double[fNumber] ;
929 << 942
930 for(i = 1; i <= fNumber; ++i) // Loop over << 943 for(i=1;i<=fNumber;i++) // Loop over the desired roots
931 { << 944 {
932 if(i == 1) << 945 if(i == 1)
933 { << 946 {
934 nwt = (1.0 + alpha) * (3.0 + 0.92 * alph << 947 newton = (1.0 + alpha)*(3.0 + 0.92*alpha)
935 (1.0 + 2.4 * fNumber + 1.8 * alpha << 948 / (1.0 + 2.4*fNumber + 1.8*alpha) ;
936 } << 949 }
937 else if(i == 2) << 950 else if(i == 2)
938 { << 951 {
939 nwt += (15.0 + 6.25 * alpha) / (1.0 + 0. << 952 newton += (15.0 + 6.25*alpha)/(1.0 + 0.9*alpha + 2.5*fNumber) ;
940 } << 953 }
941 else << 954 else
942 { << 955 {
943 cofi = i - 2; << 956 cofi = i - 2 ;
944 nwt += ((1.0 + 2.55 * cofi) / (1.9 * cof << 957 newton += ((1.0+2.55*cofi)/(1.9*cofi)
945 1.26 * cofi * alpha / (1.0 + 3.5 << 958 + 1.26*cofi*alpha/(1.0+3.5*cofi))
946 (nwt - fAbscissa[i - 3]) / (1.0 + << 959 * (newton - fAbscissa[i-3])/(1.0 + 0.3*alpha) ;
947 } << 960 }
948 for(k = 1; k <= maxNumber; ++k) << 961 for(k=1;k<=maxNumber;k++)
949 { << 962 {
950 temp1 = 1.0; << 963 temp1 = 1.0 ;
951 temp2 = 0.0; << 964 temp2 = 0.0 ;
952 <<
953 for(j = 1; j <= fNumber; ++j) <<
954 { <<
955 temp3 = temp2; <<
956 temp2 = temp1; <<
957 temp1 = <<
958 ((2 * j - 1 + alpha - nwt) * temp2 - <<
959 } <<
960 temp = (fNumber * temp1 - (fNumber + alp <<
961 nwt1 = nwt; <<
962 nwt = nwt1 - temp1 / temp; <<
963 <<
964 if(std::fabs(nwt - nwt1) <= tolerance) <<
965 { <<
966 break; <<
967 } <<
968 } <<
969 if(k > maxNumber) <<
970 { <<
971 G4Exception("G4Integrator<T,F>::Laguerre <<
972 "Too many (>12) iterations." <<
973 } <<
974 <<
975 fAbscissa[i - 1] = nwt; <<
976 fWeight[i - 1] = -std::exp(GammaLogarith <<
977 GammaLogarithm( <<
978 (temp * fNumber * temp2); <<
979 } <<
980 <<
981 // <<
982 // Integral evaluation <<
983 // <<
984 965
985 for(i = 0; i < fNumber; i++) << 966 for(j=1;j<=fNumber;j++)
986 { << 967 {
987 integral += fWeight[i] * (*f)(fAbscissa[i] << 968 temp3 = temp2 ;
988 } << 969 temp2 = temp1 ;
989 delete[] fAbscissa; << 970 temp1 = ((2*j - 1 + alpha - newton)*temp2 - (j - 1 + alpha)*temp3)/j ;
990 delete[] fWeight; << 971 }
991 return integral; << 972 temp = (fNumber*temp1 - (fNumber +alpha)*temp2)/newton ;
>> 973 newton1 = newton ;
>> 974 newton = newton1 - temp1/temp ;
>> 975
>> 976 if(std::fabs(newton - newton1) <= tolerance)
>> 977 {
>> 978 break ;
>> 979 }
>> 980 }
>> 981 if(k > maxNumber)
>> 982 {
>> 983 G4Exception("G4Integrator<T,F>::Laguerre( ...)", "Error",
>> 984 FatalException, "Too many (>12) iterations.");
>> 985 }
>> 986
>> 987 fAbscissa[i-1] = newton ;
>> 988 fWeight[i-1] = -std::exp(GammaLogarithm(alpha + fNumber) -
>> 989 GammaLogarithm((G4double)fNumber))/(temp*fNumber*temp2) ;
>> 990 }
>> 991
>> 992 //
>> 993 // Integral evaluation
>> 994 //
>> 995
>> 996 for(i=0;i<fNumber;i++)
>> 997 {
>> 998 integral += fWeight[i]*(*f)(fAbscissa[i]) ;
>> 999 }
>> 1000 delete[] fAbscissa;
>> 1001 delete[] fWeight;
>> 1002 return integral ;
992 } 1003 }
993 1004
994 ////////////////////////////////////////////// 1005 ///////////////////////////////////////////////////////////////////////
995 // 1006 //
996 // Auxiliary function which returns the value 1007 // Auxiliary function which returns the value of std::log(gamma-function(x))
997 // Returns the value ln(Gamma(xx) for xx > 0. << 1008 // Returns the value ln(Gamma(xx) for xx > 0. Full accuracy is obtained for
998 // xx > 1. For 0 < xx < 1. the reflection form 1009 // xx > 1. For 0 < xx < 1. the reflection formula (6.1.4) can be used first.
999 // (Adapted from Numerical Recipes in C) 1010 // (Adapted from Numerical Recipes in C)
1000 // 1011 //
1001 1012
1002 template <class T, class F> 1013 template <class T, class F>
1003 G4double G4Integrator<T, F>::GammaLogarithm(G << 1014 G4double G4Integrator<T,F>::GammaLogarithm(G4double xx)
1004 { 1015 {
1005 static const G4double cof[6] = { 76.1800917 << 1016 static G4double cof[6] = { 76.18009172947146, -86.50532032941677,
1006 24.0140982 << 1017 24.01409824083091, -1.231739572450155,
1007 0.12086509 << 1018 0.1208650973866179e-2, -0.5395239384953e-5 } ;
1008 G4int j; << 1019 register G4int j;
1009 G4double x = xx - 1.0; << 1020 G4double x = xx - 1.0 ;
1010 G4double tmp = x + 5.5; << 1021 G4double tmp = x + 5.5 ;
1011 tmp -= (x + 0.5) * std::log(tmp); << 1022 tmp -= (x + 0.5) * std::log(tmp) ;
1012 G4double ser = 1.000000000190015; << 1023 G4double ser = 1.000000000190015 ;
1013 1024
1014 for(j = 0; j <= 5; ++j) << 1025 for ( j = 0; j <= 5; j++ )
1015 { 1026 {
1016 x += 1.0; << 1027 x += 1.0 ;
1017 ser += cof[j] / x; << 1028 ser += cof[j]/x ;
1018 } 1029 }
1019 return -tmp + std::log(2.5066282746310005 * << 1030 return -tmp + std::log(2.5066282746310005*ser) ;
1020 } 1031 }
1021 1032
1022 ///////////////////////////////////////////// 1033 ///////////////////////////////////////////////////////////////////////
1023 // 1034 //
1024 // Method involving Hermite polynomials 1035 // Method involving Hermite polynomials
1025 // 1036 //
1026 ///////////////////////////////////////////// 1037 ///////////////////////////////////////////////////////////////////////
1027 // 1038 //
1028 // 1039 //
1029 // Gauss-Hermite method for integration of st << 1040 // Gauss-Hermite method for integration of std::exp(-x*x)*f(x)
1030 // from minus infinity to plus infinity . << 1041 // from minus infinity to plus infinity .
1031 // 1042 //
1032 1043
1033 template <class T, class F> << 1044 template <class T, class F>
1034 G4double G4Integrator<T, F>::Hermite(T& typeT << 1045 G4double G4Integrator<T,F>::Hermite( T& typeT, F f, G4int nHermite )
1035 { 1046 {
1036 const G4double tolerance = 1.0e-12; << 1047 const G4double tolerance = 1.0e-12 ;
1037 const G4int maxNumber = 12; << 1048 const G4int maxNumber = 12 ;
1038 << 1049
1039 G4int i, j, k; << 1050 G4int i, j, k ;
1040 G4double integral = 0.0; << 1051 G4double integral = 0.0 ;
1041 G4double nwt = 0., nwt1, temp1, temp2, << 1052 G4double newton=0., newton1, temp1, temp2, temp3, temp ;
1042 1053
1043 G4double piInMinusQ = << 1054 G4double piInMinusQ = std::pow(pi,-0.25) ; // 1.0/std::sqrt(std::sqrt(pi)) ??
1044 std::pow(CLHEP::pi, -0.25); // 1.0/std:: <<
1045 1055
1046 G4int fNumber = (nHermite + 1) / 2; << 1056 G4int fNumber = (nHermite +1)/2 ;
1047 G4double* fAbscissa = new G4double[fNumber] << 1057 G4double* fAbscissa = new G4double[fNumber] ;
1048 G4double* fWeight = new G4double[fNumber] << 1058 G4double* fWeight = new G4double[fNumber] ;
1049 1059
1050 for(i = 1; i <= fNumber; ++i) << 1060 for(i=1;i<=fNumber;i++)
1051 { << 1061 {
1052 if(i == 1) << 1062 if(i == 1)
1053 { << 1063 {
1054 nwt = std::sqrt((G4double)(2 * nHermite << 1064 newton = std::sqrt((G4double)(2*nHermite + 1)) -
1055 1.85575001 * std::pow((G4double)( << 1065 1.85575001*std::pow((G4double)(2*nHermite + 1),-0.16666999) ;
1056 } << 1066 }
1057 else if(i == 2) << 1067 else if(i == 2)
1058 { << 1068 {
1059 nwt -= 1.14001 * std::pow((G4double) nH << 1069 newton -= 1.14001*std::pow((G4double)nHermite,0.425999)/newton ;
1060 } << 1070 }
1061 else if(i == 3) << 1071 else if(i == 3)
1062 { << 1072 {
1063 nwt = 1.86002 * nwt - 0.86002 * fAbscis << 1073 newton = 1.86002*newton - 0.86002*fAbscissa[0] ;
1064 } << 1074 }
1065 else if(i == 4) << 1075 else if(i == 4)
1066 { << 1076 {
1067 nwt = 1.91001 * nwt - 0.91001 * fAbscis << 1077 newton = 1.91001*newton - 0.91001*fAbscissa[1] ;
1068 } << 1078 }
1069 else << 1079 else
1070 { << 1080 {
1071 nwt = 2.0 * nwt - fAbscissa[i - 3]; << 1081 newton = 2.0*newton - fAbscissa[i - 3] ;
1072 } << 1082 }
1073 for(k = 1; k <= maxNumber; ++k) << 1083 for(k=1;k<=maxNumber;k++)
1074 { << 1084 {
1075 temp1 = piInMinusQ; << 1085 temp1 = piInMinusQ ;
1076 temp2 = 0.0; << 1086 temp2 = 0.0 ;
1077 <<
1078 for(j = 1; j <= nHermite; ++j) <<
1079 { <<
1080 temp3 = temp2; <<
1081 temp2 = temp1; <<
1082 temp1 = nwt * std::sqrt(2.0 / j) * te <<
1083 std::sqrt(((G4double)(j - 1)) <<
1084 } <<
1085 temp = std::sqrt((G4double) 2 * nHermit <<
1086 nwt1 = nwt; <<
1087 nwt = nwt1 - temp1 / temp; <<
1088 <<
1089 if(std::fabs(nwt - nwt1) <= tolerance) <<
1090 { <<
1091 break; <<
1092 } <<
1093 } <<
1094 if(k > maxNumber) <<
1095 { <<
1096 G4Exception("G4Integrator<T,F>::Hermite <<
1097 FatalException, "Too many ( <<
1098 } <<
1099 fAbscissa[i - 1] = nwt; <<
1100 fWeight[i - 1] = 2.0 / (temp * temp); <<
1101 } <<
1102 <<
1103 // <<
1104 // Integral calculation <<
1105 // <<
1106 1087
1107 for(i = 0; i < fNumber; ++i) << 1088 for(j=1;j<=nHermite;j++)
1108 { << 1089 {
1109 integral += << 1090 temp3 = temp2 ;
1110 fWeight[i] * ((typeT.*f)(fAbscissa[i]) << 1091 temp2 = temp1 ;
1111 } << 1092 temp1 = newton*std::sqrt(2.0/j)*temp2 -
1112 delete[] fAbscissa; << 1093 std::sqrt(((G4double)(j - 1))/j)*temp3 ;
1113 delete[] fWeight; << 1094 }
1114 return integral; << 1095 temp = std::sqrt((G4double)2*nHermite)*temp2 ;
>> 1096 newton1 = newton ;
>> 1097 newton = newton1 - temp1/temp ;
>> 1098
>> 1099 if(std::fabs(newton - newton1) <= tolerance)
>> 1100 {
>> 1101 break ;
>> 1102 }
>> 1103 }
>> 1104 if(k > maxNumber)
>> 1105 {
>> 1106 G4Exception("G4Integrator<T,F>::Hermite(T,F, ...)", "Error",
>> 1107 FatalException, "Too many (>12) iterations.");
>> 1108 }
>> 1109 fAbscissa[i-1] = newton ;
>> 1110 fWeight[i-1] = 2.0/(temp*temp) ;
>> 1111 }
>> 1112
>> 1113 //
>> 1114 // Integral calculation
>> 1115 //
>> 1116
>> 1117 for(i=0;i<fNumber;i++)
>> 1118 {
>> 1119 integral += fWeight[i]*( (typeT.*f)(fAbscissa[i]) +
>> 1120 (typeT.*f)(-fAbscissa[i]) ) ;
>> 1121 }
>> 1122 delete[] fAbscissa;
>> 1123 delete[] fWeight;
>> 1124 return integral ;
1115 } 1125 }
1116 1126
>> 1127
1117 ///////////////////////////////////////////// 1128 ////////////////////////////////////////////////////////////////////////
1118 // 1129 //
1119 // For use with 'this' pointer 1130 // For use with 'this' pointer
1120 1131
1121 template <class T, class F> << 1132 template <class T, class F>
1122 G4double G4Integrator<T, F>::Hermite(T* ptrT, << 1133 G4double G4Integrator<T,F>::Hermite( T* ptrT, F f, G4int n )
1123 { 1134 {
1124 return Hermite(*ptrT, f, n); << 1135 return Hermite(*ptrT,f,n) ;
1125 } << 1136 }
1126 1137
1127 ///////////////////////////////////////////// 1138 ////////////////////////////////////////////////////////////////////////
1128 // 1139 //
1129 // For use with global scope f 1140 // For use with global scope f
1130 1141
1131 template <class T, class F> 1142 template <class T, class F>
1132 G4double G4Integrator<T, F>::Hermite(G4double << 1143 G4double G4Integrator<T,F>::Hermite( G4double (*f)(G4double), G4int nHermite)
1133 { 1144 {
1134 const G4double tolerance = 1.0e-12; << 1145 const G4double tolerance = 1.0e-12 ;
1135 const G4int maxNumber = 12; << 1146 const G4int maxNumber = 12 ;
1136 << 1147
1137 G4int i, j, k; << 1148 G4int i, j, k ;
1138 G4double integral = 0.0; << 1149 G4double integral = 0.0 ;
1139 G4double nwt = 0., nwt1, temp1, temp2, << 1150 G4double newton=0., newton1, temp1, temp2, temp3, temp ;
1140 << 1151
1141 G4double piInMinusQ = << 1152 G4double piInMinusQ = std::pow(pi,-0.25) ; // 1.0/std::sqrt(std::sqrt(pi)) ??
1142 std::pow(CLHEP::pi, -0.25); // 1.0/std:: << 1153
1143 << 1154 G4int fNumber = (nHermite +1)/2 ;
1144 G4int fNumber = (nHermite + 1) / 2; << 1155 G4double* fAbscissa = new G4double[fNumber] ;
1145 G4double* fAbscissa = new G4double[fNumber] << 1156 G4double* fWeight = new G4double[fNumber] ;
1146 G4double* fWeight = new G4double[fNumber] << 1157
1147 << 1158 for(i=1;i<=fNumber;i++)
1148 for(i = 1; i <= fNumber; ++i) << 1159 {
1149 { << 1160 if(i == 1)
1150 if(i == 1) << 1161 {
1151 { << 1162 newton = std::sqrt((G4double)(2*nHermite + 1)) -
1152 nwt = std::sqrt((G4double)(2 * nHermite << 1163 1.85575001*std::pow((G4double)(2*nHermite + 1),-0.16666999) ;
1153 1.85575001 * std::pow((G4double)( << 1164 }
1154 } << 1165 else if(i == 2)
1155 else if(i == 2) << 1166 {
1156 { << 1167 newton -= 1.14001*std::pow((G4double)nHermite,0.425999)/newton ;
1157 nwt -= 1.14001 * std::pow((G4double) nH << 1168 }
1158 } << 1169 else if(i == 3)
1159 else if(i == 3) << 1170 {
1160 { << 1171 newton = 1.86002*newton - 0.86002*fAbscissa[0] ;
1161 nwt = 1.86002 * nwt - 0.86002 * fAbscis << 1172 }
1162 } << 1173 else if(i == 4)
1163 else if(i == 4) << 1174 {
1164 { << 1175 newton = 1.91001*newton - 0.91001*fAbscissa[1] ;
1165 nwt = 1.91001 * nwt - 0.91001 * fAbscis << 1176 }
1166 } << 1177 else
1167 else << 1178 {
1168 { << 1179 newton = 2.0*newton - fAbscissa[i - 3] ;
1169 nwt = 2.0 * nwt - fAbscissa[i - 3]; << 1180 }
1170 } << 1181 for(k=1;k<=maxNumber;k++)
1171 for(k = 1; k <= maxNumber; ++k) << 1182 {
1172 { << 1183 temp1 = piInMinusQ ;
1173 temp1 = piInMinusQ; << 1184 temp2 = 0.0 ;
1174 temp2 = 0.0; <<
1175 <<
1176 for(j = 1; j <= nHermite; ++j) <<
1177 { <<
1178 temp3 = temp2; <<
1179 temp2 = temp1; <<
1180 temp1 = nwt * std::sqrt(2.0 / j) * te <<
1181 std::sqrt(((G4double)(j - 1)) <<
1182 } <<
1183 temp = std::sqrt((G4double) 2 * nHermit <<
1184 nwt1 = nwt; <<
1185 nwt = nwt1 - temp1 / temp; <<
1186 <<
1187 if(std::fabs(nwt - nwt1) <= tolerance) <<
1188 { <<
1189 break; <<
1190 } <<
1191 } <<
1192 if(k > maxNumber) <<
1193 { <<
1194 G4Exception("G4Integrator<T,F>::Hermite <<
1195 "Too many (>12) iterations. <<
1196 } <<
1197 fAbscissa[i - 1] = nwt; <<
1198 fWeight[i - 1] = 2.0 / (temp * temp); <<
1199 } <<
1200 <<
1201 // <<
1202 // Integral calculation <<
1203 // <<
1204 1185
1205 for(i = 0; i < fNumber; ++i) << 1186 for(j=1;j<=nHermite;j++)
1206 { << 1187 {
1207 integral += fWeight[i] * ((*f)(fAbscissa[ << 1188 temp3 = temp2 ;
1208 } << 1189 temp2 = temp1 ;
1209 delete[] fAbscissa; << 1190 temp1 = newton*std::sqrt(2.0/j)*temp2 -
1210 delete[] fWeight; << 1191 std::sqrt(((G4double)(j - 1))/j)*temp3 ;
1211 return integral; << 1192 }
>> 1193 temp = std::sqrt((G4double)2*nHermite)*temp2 ;
>> 1194 newton1 = newton ;
>> 1195 newton = newton1 - temp1/temp ;
>> 1196
>> 1197 if(std::fabs(newton - newton1) <= tolerance)
>> 1198 {
>> 1199 break ;
>> 1200 }
>> 1201 }
>> 1202 if(k > maxNumber)
>> 1203 {
>> 1204 G4Exception("G4Integrator<T,F>::Hermite(...)", "Error",
>> 1205 FatalException, "Too many (>12) iterations.");
>> 1206 }
>> 1207 fAbscissa[i-1] = newton ;
>> 1208 fWeight[i-1] = 2.0/(temp*temp) ;
>> 1209 }
>> 1210
>> 1211 //
>> 1212 // Integral calculation
>> 1213 //
>> 1214
>> 1215 for(i=0;i<fNumber;i++)
>> 1216 {
>> 1217 integral += fWeight[i]*( (*f)(fAbscissa[i]) + (*f)(-fAbscissa[i]) ) ;
>> 1218 }
>> 1219 delete[] fAbscissa;
>> 1220 delete[] fWeight;
>> 1221 return integral ;
1212 } 1222 }
1213 1223
1214 ///////////////////////////////////////////// 1224 ////////////////////////////////////////////////////////////////////////////
1215 // 1225 //
1216 // Method involving Jacobi polynomials 1226 // Method involving Jacobi polynomials
1217 // 1227 //
1218 ///////////////////////////////////////////// 1228 ////////////////////////////////////////////////////////////////////////////
1219 // 1229 //
1220 // Gauss-Jacobi method for integration of ((1 1230 // Gauss-Jacobi method for integration of ((1-x)^alpha)*((1+x)^beta)*f(x)
1221 // from minus unit to plus unit . 1231 // from minus unit to plus unit .
1222 // 1232 //
1223 1233
1224 template <class T, class F> << 1234 template <class T, class F>
1225 G4double G4Integrator<T, F>::Jacobi(T& typeT, << 1235 G4double G4Integrator<T,F>::Jacobi( T& typeT, F f, G4double alpha,
1226 G4double << 1236 G4double beta, G4int nJacobi)
1227 { << 1237 {
1228 const G4double tolerance = 1.0e-12; << 1238 const G4double tolerance = 1.0e-12 ;
1229 const G4double maxNumber = 12; << 1239 const G4double maxNumber = 12 ;
1230 G4int i, k, j; << 1240 G4int i, k, j ;
1231 G4double alphaBeta, alphaReduced, betaReduc << 1241 G4double alphaBeta, alphaReduced, betaReduced, root1=0., root2=0., root3=0. ;
1232 << 1242 G4double a, b, c, newton1, newton2, newton3, newton, temp, root=0., rootTemp ;
1233 G4double a, b, c, nwt1, nwt2, nwt3, nwt, te << 1243
1234 << 1244 G4int fNumber = nJacobi ;
1235 G4int fNumber = nJacobi; << 1245 G4double* fAbscissa = new G4double[fNumber] ;
1236 G4double* fAbscissa = new G4double[fNumber] << 1246 G4double* fWeight = new G4double[fNumber] ;
1237 G4double* fWeight = new G4double[fNumber] << 1247
1238 << 1248 for (i=1;i<=nJacobi;i++)
1239 for(i = 1; i <= nJacobi; ++i) << 1249 {
1240 { << 1250 if (i == 1)
1241 if(i == 1) << 1251 {
1242 { << 1252 alphaReduced = alpha/nJacobi ;
1243 alphaReduced = alpha / nJacobi; << 1253 betaReduced = beta/nJacobi ;
1244 betaReduced = beta / nJacobi; << 1254 root1 = (1.0+alpha)*(2.78002/(4.0+nJacobi*nJacobi)+
1245 root1 = (1.0 + alpha) * (2.78002 << 1255 0.767999*alphaReduced/nJacobi) ;
1246 0.767999 * alp << 1256 root2 = 1.0+1.48*alphaReduced+0.96002*betaReduced +
1247 root2 = 1.0 + 1.48 * alphaReduce << 1257 0.451998*alphaReduced*alphaReduced +
1248 0.451998 * alphaReduced * alpha << 1258 0.83001*alphaReduced*betaReduced ;
1249 0.83001 * alphaReduced * betaRe << 1259 root = 1.0-root1/root2 ;
1250 root = 1.0 - root1 / root2; << 1260 }
1251 } << 1261 else if (i == 2)
1252 else if(i == 2) << 1262 {
1253 { << 1263 root1=(4.1002+alpha)/((1.0+alpha)*(1.0+0.155998*alpha)) ;
1254 root1 = (4.1002 + alpha) / ((1.0 + alph << 1264 root2=1.0+0.06*(nJacobi-8.0)*(1.0+0.12*alpha)/nJacobi ;
1255 root2 = 1.0 + 0.06 * (nJacobi - 8.0) * << 1265 root3=1.0+0.012002*beta*(1.0+0.24997*std::fabs(alpha))/nJacobi ;
1256 root3 = << 1266 root -= (1.0-root)*root1*root2*root3 ;
1257 1.0 + 0.012002 * beta * (1.0 + 0.2499 << 1267 }
1258 root -= (1.0 - root) * root1 * root2 * << 1268 else if (i == 3)
1259 } << 1269 {
1260 else if(i == 3) << 1270 root1=(1.67001+0.27998*alpha)/(1.0+0.37002*alpha) ;
1261 { << 1271 root2=1.0+0.22*(nJacobi-8.0)/nJacobi ;
1262 root1 = (1.67001 + 0.27998 * alpha) / ( << 1272 root3=1.0+8.0*beta/((6.28001+beta)*nJacobi*nJacobi) ;
1263 root2 = 1.0 + 0.22 * (nJacobi - 8.0) / << 1273 root -= (fAbscissa[0]-root)*root1*root2*root3 ;
1264 root3 = 1.0 + 8.0 * beta / ((6.28001 + << 1274 }
1265 root -= (fAbscissa[0] - root) * root1 * << 1275 else if (i == nJacobi-1)
1266 } << 1276 {
1267 else if(i == nJacobi - 1) << 1277 root1=(1.0+0.235002*beta)/(0.766001+0.118998*beta) ;
1268 { << 1278 root2=1.0/(1.0+0.639002*(nJacobi-4.0)/(1.0+0.71001*(nJacobi-4.0))) ;
1269 root1 = (1.0 + 0.235002 * beta) / (0.76 << 1279 root3=1.0/(1.0+20.0*alpha/((7.5+alpha)*nJacobi*nJacobi)) ;
1270 root2 = 1.0 / (1.0 + 0.639002 * (nJacob << 1280 root += (root-fAbscissa[nJacobi-4])*root1*root2*root3 ;
1271 (1.0 + 0.71001 * << 1281 }
1272 root3 = 1.0 / (1.0 + 20.0 * alpha / ((7 << 1282 else if (i == nJacobi)
1273 root += (root - fAbscissa[nJacobi - 4]) << 1283 {
1274 } << 1284 root1 = (1.0+0.37002*beta)/(1.67001+0.27998*beta) ;
1275 else if(i == nJacobi) << 1285 root2 = 1.0/(1.0+0.22*(nJacobi-8.0)/nJacobi) ;
1276 { << 1286 root3 = 1.0/(1.0+8.0*alpha/((6.28002+alpha)*nJacobi*nJacobi)) ;
1277 root1 = (1.0 + 0.37002 * beta) / (1.670 << 1287 root += (root-fAbscissa[nJacobi-3])*root1*root2*root3 ;
1278 root2 = 1.0 / (1.0 + 0.22 * (nJacobi - << 1288 }
1279 root3 = << 1289 else
1280 1.0 / (1.0 + 8.0 * alpha / ((6.28002 << 1290 {
1281 root += (root - fAbscissa[nJacobi - 3]) << 1291 root = 3.0*fAbscissa[i-2]-3.0*fAbscissa[i-3]+fAbscissa[i-4] ;
1282 } << 1292 }
1283 else << 1293 alphaBeta = alpha + beta ;
1284 { << 1294 for (k=1;k<=maxNumber;k++)
1285 root = 3.0 * fAbscissa[i - 2] - 3.0 * f << 1295 {
1286 } << 1296 temp = 2.0 + alphaBeta ;
1287 alphaBeta = alpha + beta; << 1297 newton1 = (alpha-beta+temp*root)/2.0 ;
1288 for(k = 1; k <= maxNumber; ++k) << 1298 newton2 = 1.0 ;
1289 { << 1299 for (j=2;j<=nJacobi;j++)
1290 temp = 2.0 + alphaBeta; << 1300 {
1291 nwt1 = (alpha - beta + temp * root) / 2 << 1301 newton3 = newton2 ;
1292 nwt2 = 1.0; << 1302 newton2 = newton1 ;
1293 for(j = 2; j <= nJacobi; ++j) << 1303 temp = 2*j+alphaBeta ;
1294 { << 1304 a = 2*j*(j+alphaBeta)*(temp-2.0) ;
1295 nwt3 = nwt2; << 1305 b = (temp-1.0)*(alpha*alpha-beta*beta+temp*(temp-2.0)*root) ;
1296 nwt2 = nwt1; << 1306 c = 2.0*(j-1+alpha)*(j-1+beta)*temp ;
1297 temp = 2 * j + alphaBeta; << 1307 newton1 = (b*newton2-c*newton3)/a ;
1298 a = 2 * j * (j + alphaBeta) * (tem << 1308 }
1299 b = (temp - 1.0) * << 1309 newton = (nJacobi*(alpha - beta - temp*root)*newton1 +
1300 (alpha * alpha - beta * beta + te << 1310 2.0*(nJacobi + alpha)*(nJacobi + beta)*newton2)/
1301 c = 2.0 * (j - 1 + alpha) * (j - 1 << 1311 (temp*(1.0 - root*root)) ;
1302 nwt1 = (b * nwt2 - c * nwt3) / a; << 1312 rootTemp = root ;
1303 } << 1313 root = rootTemp - newton1/newton ;
1304 nwt = (nJacobi * (alpha - beta - temp * << 1314 if (std::fabs(root-rootTemp) <= tolerance)
1305 2.0 * (nJacobi + alpha) * (nJaco << 1315 {
1306 (temp * (1.0 - root * root)); << 1316 break ;
1307 rootTemp = root; << 1317 }
1308 root = rootTemp - nwt1 / nwt; << 1318 }
1309 if(std::fabs(root - rootTemp) <= tolera << 1319 if (k > maxNumber)
1310 { << 1320 {
1311 break; << 1321 G4Exception("G4Integrator<T,F>::Jacobi(T,F, ...)", "Error",
1312 } << 1322 FatalException, "Too many (>12) iterations.");
1313 } << 1323 }
1314 if(k > maxNumber) << 1324 fAbscissa[i-1] = root ;
1315 { << 1325 fWeight[i-1] = std::exp(GammaLogarithm((G4double)(alpha+nJacobi)) +
1316 G4Exception("G4Integrator<T,F>::Jacobi( << 1326 GammaLogarithm((G4double)(beta+nJacobi)) -
1317 FatalException, "Too many ( << 1327 GammaLogarithm((G4double)(nJacobi+1.0)) -
1318 } << 1328 GammaLogarithm((G4double)(nJacobi + alphaBeta + 1.0)))
1319 fAbscissa[i - 1] = root; << 1329 *temp*std::pow(2.0,alphaBeta)/(newton*newton2) ;
1320 fWeight[i - 1] = << 1330 }
1321 std::exp(GammaLogarithm((G4double)(alph << 1331
1322 GammaLogarithm((G4double)(beta << 1332 //
1323 GammaLogarithm((G4double)(nJac << 1333 // Calculation of the integral
1324 GammaLogarithm((G4double)(nJac << 1334 //
1325 temp * std::pow(2.0, alphaBeta) / (nwt << 1335
1326 } << 1336 G4double integral = 0.0 ;
1327 << 1337 for(i=0;i<fNumber;i++)
1328 // << 1338 {
1329 // Calculation of the integral << 1339 integral += fWeight[i]*(typeT.*f)(fAbscissa[i]) ;
1330 // << 1340 }
1331 << 1341 delete[] fAbscissa;
1332 G4double integral = 0.0; << 1342 delete[] fWeight;
1333 for(i = 0; i < fNumber; ++i) << 1343 return integral ;
1334 { <<
1335 integral += fWeight[i] * (typeT.*f)(fAbsc <<
1336 } <<
1337 delete[] fAbscissa; <<
1338 delete[] fWeight; <<
1339 return integral; <<
1340 } 1344 }
1341 1345
>> 1346
1342 ///////////////////////////////////////////// 1347 /////////////////////////////////////////////////////////////////////////
1343 // 1348 //
1344 // For use with 'this' pointer 1349 // For use with 'this' pointer
1345 1350
1346 template <class T, class F> << 1351 template <class T, class F>
1347 G4double G4Integrator<T, F>::Jacobi(T* ptrT, << 1352 G4double G4Integrator<T,F>::Jacobi( T* ptrT, F f, G4double alpha,
1348 G4int n) << 1353 G4double beta, G4int n)
1349 { 1354 {
1350 return Jacobi(*ptrT, f, alpha, beta, n); << 1355 return Jacobi(*ptrT,f,alpha,beta,n) ;
1351 } << 1356 }
1352 1357
1353 ///////////////////////////////////////////// 1358 /////////////////////////////////////////////////////////////////////////
1354 // 1359 //
1355 // For use with global scope f << 1360 // For use with global scope f
1356 1361
1357 template <class T, class F> 1362 template <class T, class F>
1358 G4double G4Integrator<T, F>::Jacobi(G4double << 1363 G4double G4Integrator<T,F>::Jacobi( G4double (*f)(G4double), G4double alpha,
1359 G4double << 1364 G4double beta, G4int nJacobi)
1360 { 1365 {
1361 const G4double tolerance = 1.0e-12; << 1366 const G4double tolerance = 1.0e-12 ;
1362 const G4double maxNumber = 12; << 1367 const G4double maxNumber = 12 ;
1363 G4int i, k, j; << 1368 G4int i, k, j ;
1364 G4double alphaBeta, alphaReduced, betaReduc << 1369 G4double alphaBeta, alphaReduced, betaReduced, root1=0., root2=0., root3=0. ;
1365 << 1370 G4double a, b, c, newton1, newton2, newton3, newton, temp, root=0., rootTemp ;
1366 G4double a, b, c, nwt1, nwt2, nwt3, nwt, te << 1371
1367 << 1372 G4int fNumber = nJacobi ;
1368 G4int fNumber = nJacobi; << 1373 G4double* fAbscissa = new G4double[fNumber] ;
1369 G4double* fAbscissa = new G4double[fNumber] << 1374 G4double* fWeight = new G4double[fNumber] ;
1370 G4double* fWeight = new G4double[fNumber] << 1375
1371 << 1376 for (i=1;i<=nJacobi;i++)
1372 for(i = 1; i <= nJacobi; ++i) << 1377 {
1373 { << 1378 if (i == 1)
1374 if(i == 1) << 1379 {
1375 { << 1380 alphaReduced = alpha/nJacobi ;
1376 alphaReduced = alpha / nJacobi; << 1381 betaReduced = beta/nJacobi ;
1377 betaReduced = beta / nJacobi; << 1382 root1 = (1.0+alpha)*(2.78002/(4.0+nJacobi*nJacobi)+
1378 root1 = (1.0 + alpha) * (2.78002 << 1383 0.767999*alphaReduced/nJacobi) ;
1379 0.767999 * alp << 1384 root2 = 1.0+1.48*alphaReduced+0.96002*betaReduced +
1380 root2 = 1.0 + 1.48 * alphaReduce << 1385 0.451998*alphaReduced*alphaReduced +
1381 0.451998 * alphaReduced * alpha << 1386 0.83001*alphaReduced*betaReduced ;
1382 0.83001 * alphaReduced * betaRe << 1387 root = 1.0-root1/root2 ;
1383 root = 1.0 - root1 / root2; << 1388 }
1384 } << 1389 else if (i == 2)
1385 else if(i == 2) << 1390 {
1386 { << 1391 root1=(4.1002+alpha)/((1.0+alpha)*(1.0+0.155998*alpha)) ;
1387 root1 = (4.1002 + alpha) / ((1.0 + alph << 1392 root2=1.0+0.06*(nJacobi-8.0)*(1.0+0.12*alpha)/nJacobi ;
1388 root2 = 1.0 + 0.06 * (nJacobi - 8.0) * << 1393 root3=1.0+0.012002*beta*(1.0+0.24997*std::fabs(alpha))/nJacobi ;
1389 root3 = << 1394 root -= (1.0-root)*root1*root2*root3 ;
1390 1.0 + 0.012002 * beta * (1.0 + 0.2499 << 1395 }
1391 root -= (1.0 - root) * root1 * root2 * << 1396 else if (i == 3)
1392 } << 1397 {
1393 else if(i == 3) << 1398 root1=(1.67001+0.27998*alpha)/(1.0+0.37002*alpha) ;
1394 { << 1399 root2=1.0+0.22*(nJacobi-8.0)/nJacobi ;
1395 root1 = (1.67001 + 0.27998 * alpha) / ( << 1400 root3=1.0+8.0*beta/((6.28001+beta)*nJacobi*nJacobi) ;
1396 root2 = 1.0 + 0.22 * (nJacobi - 8.0) / << 1401 root -= (fAbscissa[0]-root)*root1*root2*root3 ;
1397 root3 = 1.0 + 8.0 * beta / ((6.28001 + << 1402 }
1398 root -= (fAbscissa[0] - root) * root1 * << 1403 else if (i == nJacobi-1)
1399 } << 1404 {
1400 else if(i == nJacobi - 1) << 1405 root1=(1.0+0.235002*beta)/(0.766001+0.118998*beta) ;
1401 { << 1406 root2=1.0/(1.0+0.639002*(nJacobi-4.0)/(1.0+0.71001*(nJacobi-4.0))) ;
1402 root1 = (1.0 + 0.235002 * beta) / (0.76 << 1407 root3=1.0/(1.0+20.0*alpha/((7.5+alpha)*nJacobi*nJacobi)) ;
1403 root2 = 1.0 / (1.0 + 0.639002 * (nJacob << 1408 root += (root-fAbscissa[nJacobi-4])*root1*root2*root3 ;
1404 (1.0 + 0.71001 * << 1409 }
1405 root3 = 1.0 / (1.0 + 20.0 * alpha / ((7 << 1410 else if (i == nJacobi)
1406 root += (root - fAbscissa[nJacobi - 4]) << 1411 {
1407 } << 1412 root1 = (1.0+0.37002*beta)/(1.67001+0.27998*beta) ;
1408 else if(i == nJacobi) << 1413 root2 = 1.0/(1.0+0.22*(nJacobi-8.0)/nJacobi) ;
1409 { << 1414 root3 = 1.0/(1.0+8.0*alpha/((6.28002+alpha)*nJacobi*nJacobi)) ;
1410 root1 = (1.0 + 0.37002 * beta) / (1.670 << 1415 root += (root-fAbscissa[nJacobi-3])*root1*root2*root3 ;
1411 root2 = 1.0 / (1.0 + 0.22 * (nJacobi - << 1416 }
1412 root3 = << 1417 else
1413 1.0 / (1.0 + 8.0 * alpha / ((6.28002 << 1418 {
1414 root += (root - fAbscissa[nJacobi - 3]) << 1419 root = 3.0*fAbscissa[i-2]-3.0*fAbscissa[i-3]+fAbscissa[i-4] ;
1415 } << 1420 }
1416 else << 1421 alphaBeta = alpha + beta ;
1417 { << 1422 for (k=1;k<=maxNumber;k++)
1418 root = 3.0 * fAbscissa[i - 2] - 3.0 * f << 1423 {
1419 } << 1424 temp = 2.0 + alphaBeta ;
1420 alphaBeta = alpha + beta; << 1425 newton1 = (alpha-beta+temp*root)/2.0 ;
1421 for(k = 1; k <= maxNumber; ++k) << 1426 newton2 = 1.0 ;
1422 { << 1427 for (j=2;j<=nJacobi;j++)
1423 temp = 2.0 + alphaBeta; << 1428 {
1424 nwt1 = (alpha - beta + temp * root) / 2 << 1429 newton3 = newton2 ;
1425 nwt2 = 1.0; << 1430 newton2 = newton1 ;
1426 for(j = 2; j <= nJacobi; ++j) << 1431 temp = 2*j+alphaBeta ;
1427 { << 1432 a = 2*j*(j+alphaBeta)*(temp-2.0) ;
1428 nwt3 = nwt2; << 1433 b = (temp-1.0)*(alpha*alpha-beta*beta+temp*(temp-2.0)*root) ;
1429 nwt2 = nwt1; << 1434 c = 2.0*(j-1+alpha)*(j-1+beta)*temp ;
1430 temp = 2 * j + alphaBeta; << 1435 newton1 = (b*newton2-c*newton3)/a ;
1431 a = 2 * j * (j + alphaBeta) * (tem << 1436 }
1432 b = (temp - 1.0) * << 1437 newton = (nJacobi*(alpha - beta - temp*root)*newton1 +
1433 (alpha * alpha - beta * beta + te << 1438 2.0*(nJacobi + alpha)*(nJacobi + beta)*newton2) /
1434 c = 2.0 * (j - 1 + alpha) * (j - 1 << 1439 (temp*(1.0 - root*root)) ;
1435 nwt1 = (b * nwt2 - c * nwt3) / a; << 1440 rootTemp = root ;
1436 } << 1441 root = rootTemp - newton1/newton ;
1437 nwt = (nJacobi * (alpha - beta - temp * << 1442 if (std::fabs(root-rootTemp) <= tolerance)
1438 2.0 * (nJacobi + alpha) * (nJaco << 1443 {
1439 (temp * (1.0 - root * root)); << 1444 break ;
1440 rootTemp = root; << 1445 }
1441 root = rootTemp - nwt1 / nwt; << 1446 }
1442 if(std::fabs(root - rootTemp) <= tolera << 1447 if (k > maxNumber)
1443 { << 1448 {
1444 break; << 1449 G4Exception("G4Integrator<T,F>::Jacobi(...)", "Error",
1445 } << 1450 FatalException, "Too many (>12) iterations.");
1446 } << 1451 }
1447 if(k > maxNumber) << 1452 fAbscissa[i-1] = root ;
1448 { << 1453 fWeight[i-1] =
1449 G4Exception("G4Integrator<T,F>::Jacobi( << 1454 std::exp(GammaLogarithm((G4double)(alpha+nJacobi)) +
1450 "Too many (>12) iterations. << 1455 GammaLogarithm((G4double)(beta+nJacobi)) -
1451 } << 1456 GammaLogarithm((G4double)(nJacobi+1.0)) -
1452 fAbscissa[i - 1] = root; << 1457 GammaLogarithm((G4double)(nJacobi + alphaBeta + 1.0)))
1453 fWeight[i - 1] = << 1458 *temp*std::pow(2.0,alphaBeta)/(newton*newton2);
1454 std::exp(GammaLogarithm((G4double)(alph << 1459 }
1455 GammaLogarithm((G4double)(beta << 1460
1456 GammaLogarithm((G4double)(nJac << 1461 //
1457 GammaLogarithm((G4double)(nJac << 1462 // Calculation of the integral
1458 temp * std::pow(2.0, alphaBeta) / (nwt << 1463 //
1459 } << 1464
1460 << 1465 G4double integral = 0.0 ;
1461 // << 1466 for(i=0;i<fNumber;i++)
1462 // Calculation of the integral << 1467 {
1463 // << 1468 integral += fWeight[i]*(*f)(fAbscissa[i]) ;
1464 << 1469 }
1465 G4double integral = 0.0; << 1470 delete[] fAbscissa;
1466 for(i = 0; i < fNumber; ++i) << 1471 delete[] fWeight;
1467 { << 1472 return integral ;
1468 integral += fWeight[i] * (*f)(fAbscissa[i <<
1469 } <<
1470 delete[] fAbscissa; <<
1471 delete[] fWeight; <<
1472 return integral; <<
1473 } 1473 }
1474 1474
1475 // 1475 //
1476 // 1476 //
1477 ///////////////////////////////////////////// 1477 ///////////////////////////////////////////////////////////////////
1478 1478