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35 // 35 //
36 // %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 36 // %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
37 // 37 //
38 // MODULE: G4Bessel.cc 38 // MODULE: G4Bessel.cc
39 // 39 //
40 // Version: B.1 40 // Version: B.1
41 // Date: 15/04/04 41 // Date: 15/04/04
42 // Author: P R Truscott 42 // Author: P R Truscott
43 // Organisation: QinetiQ Ltd, UK 43 // Organisation: QinetiQ Ltd, UK
44 // Customer: ESA/ESTEC, NOORDWIJK 44 // Customer: ESA/ESTEC, NOORDWIJK
45 // Contract: 17191/03/NL/LvH 45 // Contract: 17191/03/NL/LvH
46 // 46 //
47 // %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 47 // %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
48 // 48 //
49 // CHANGE HISTORY 49 // CHANGE HISTORY
50 // -------------- 50 // --------------
51 // 51 //
52 // 18 Noevmber 2003, P R Truscott, QinetiQ Ltd 52 // 18 Noevmber 2003, P R Truscott, QinetiQ Ltd, UK
53 // Created. 53 // Created.
54 // 54 //
55 // 15 March 2004, P R Truscott, QinetiQ Ltd, U 55 // 15 March 2004, P R Truscott, QinetiQ Ltd, UK
56 // Beta release 56 // Beta release
57 // 57 //
58 // 06 August 2015, A. Ribon, CERN <<
59 // Migrated to G4Exp, G4Log and G4Pow. <<
60 // <<
61 // %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 58 // %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
62 ////////////////////////////////////////////// 59 ////////////////////////////////////////////////////////////////////////////////
63 // 60 //
64 #include "G4Bessel.hh" 61 #include "G4Bessel.hh"
65 #include "G4PhysicalConstants.hh" 62 #include "G4PhysicalConstants.hh"
66 <<
67 #include "G4Exp.hh" <<
68 #include "G4Log.hh" <<
69 #include "G4Pow.hh" <<
70 ////////////////////////////////////////////// 63 ////////////////////////////////////////////////////////////////////////////////
71 // 64 //
72 G4Bessel::G4Bessel () 65 G4Bessel::G4Bessel ()
73 {;} 66 {;}
74 ////////////////////////////////////////////// 67 ////////////////////////////////////////////////////////////////////////////////
75 // 68 //
76 G4Bessel::~G4Bessel () 69 G4Bessel::~G4Bessel ()
77 {;} 70 {;}
78 ////////////////////////////////////////////// 71 ////////////////////////////////////////////////////////////////////////////////
79 // 72 //
80 G4double G4Bessel::I0 (G4double x) 73 G4double G4Bessel::I0 (G4double x)
81 { 74 {
82 const G4double P1 = 1.0, 75 const G4double P1 = 1.0,
83 P2 = 3.5156229, 76 P2 = 3.5156229,
84 P3 = 3.0899424, 77 P3 = 3.0899424,
85 P4 = 1.2067492, 78 P4 = 1.2067492,
86 P5 = 0.2659732, 79 P5 = 0.2659732,
87 P6 = 0.0360768, 80 P6 = 0.0360768,
88 P7 = 0.0045813; 81 P7 = 0.0045813;
89 const G4double Q1 = 0.39894228, 82 const G4double Q1 = 0.39894228,
90 Q2 = 0.01328592, 83 Q2 = 0.01328592,
91 Q3 = 0.00225319, 84 Q3 = 0.00225319,
92 Q4 =-0.00157565, 85 Q4 =-0.00157565,
93 Q5 = 0.00916281, 86 Q5 = 0.00916281,
94 Q6 =-0.02057706, 87 Q6 =-0.02057706,
95 Q7 = 0.02635537, 88 Q7 = 0.02635537,
96 Q8 =-0.01647633, 89 Q8 =-0.01647633,
97 Q9 = 0.00392377; 90 Q9 = 0.00392377;
98 91
99 G4double I = 0.0; 92 G4double I = 0.0;
100 if (std::abs(x) < 3.75) << 93 if (std::fabs(x) < 3.75)
101 { 94 {
102 G4double y = G4Pow::GetInstance()->powN(x/ << 95 G4double y = std::pow(x/3.75, 2.0);
103 I = P1+y*(P2+y*(P3+y*(P4+y*(P5+y*(P6+y*P7) 96 I = P1+y*(P2+y*(P3+y*(P4+y*(P5+y*(P6+y*P7)))));
104 } 97 }
105 else 98 else
106 { 99 {
107 G4double ax = std::abs(x); << 100 G4double ax = std::fabs(x);
108 G4double y = 3.75/ax; 101 G4double y = 3.75/ax;
109 I = G4Exp(ax) / std::sqrt(ax) * << 102 I = std::exp(ax) / std::sqrt(ax) *
110 (Q1+y*(Q2+y*(Q3+y*(Q4+y*(Q5+y*(Q6+y*(Q7+ 103 (Q1+y*(Q2+y*(Q3+y*(Q4+y*(Q5+y*(Q6+y*(Q7+y*(Q8+y*Q9))))))));
111 } 104 }
112 return I; 105 return I;
113 } 106 }
114 ////////////////////////////////////////////// 107 ////////////////////////////////////////////////////////////////////////////////
115 // 108 //
116 G4double G4Bessel::K0 (G4double x) 109 G4double G4Bessel::K0 (G4double x)
117 { 110 {
118 const G4double P1 =-0.57721566, 111 const G4double P1 =-0.57721566,
119 P2 = 0.42278420, 112 P2 = 0.42278420,
120 P3 = 0.23069756, 113 P3 = 0.23069756,
121 P4 = 0.03488590, 114 P4 = 0.03488590,
122 P5 = 0.00262698, 115 P5 = 0.00262698,
123 P6 = 0.00010750, 116 P6 = 0.00010750,
124 P7 = 0.00000740; 117 P7 = 0.00000740;
125 const G4double Q1 = 1.25331414, 118 const G4double Q1 = 1.25331414,
126 Q2 =-0.07832358, 119 Q2 =-0.07832358,
127 Q3 = 0.02189568, 120 Q3 = 0.02189568,
128 Q4 =-0.01062446, 121 Q4 =-0.01062446,
129 Q5 = 0.00587872, 122 Q5 = 0.00587872,
130 Q6 =-0.00251540, 123 Q6 =-0.00251540,
131 Q7 = 0.00053208; 124 Q7 = 0.00053208;
132 125
133 G4double K = 0.0; 126 G4double K = 0.0;
134 if (x <= 2.0) 127 if (x <= 2.0)
135 { 128 {
136 G4double y = x * x / 4.0; 129 G4double y = x * x / 4.0;
137 K = (-G4Log(x/2.0)) * I0(x) + << 130 K = (-std::log(x/2.0)) * I0(x) +
138 P1+y*(P2+y*(P3+y*(P4+y*(P5+y*(P6+y*P7))) 131 P1+y*(P2+y*(P3+y*(P4+y*(P5+y*(P6+y*P7)))));
139 } 132 }
140 else 133 else
141 { 134 {
142 G4double y = 2.0 / x; 135 G4double y = 2.0 / x;
143 K = G4Exp(-x) / std::sqrt(x) * << 136 K = std::exp(-x) / std::sqrt(x) *
144 (Q1+y*(Q2+y*(Q3+y*(Q4+y*(Q5+y*(Q6+y*Q7)) 137 (Q1+y*(Q2+y*(Q3+y*(Q4+y*(Q5+y*(Q6+y*Q7))))));
145 } 138 }
146 return K; 139 return K;
147 } 140 }
148 ////////////////////////////////////////////// 141 ////////////////////////////////////////////////////////////////////////////////
149 // 142 //
150 G4double G4Bessel::I1 (G4double x) 143 G4double G4Bessel::I1 (G4double x)
151 { 144 {
152 const G4double P1 = 0.5, 145 const G4double P1 = 0.5,
153 P2 = 0.87890594, 146 P2 = 0.87890594,
154 P3 = 0.51498869, 147 P3 = 0.51498869,
155 P4 = 0.15084934, 148 P4 = 0.15084934,
156 P5 = 0.02658733, 149 P5 = 0.02658733,
157 P6 = 0.00301532, 150 P6 = 0.00301532,
158 P7 = 0.00032411; 151 P7 = 0.00032411;
159 const G4double Q1 = 0.39894228, 152 const G4double Q1 = 0.39894228,
160 Q2 =-0.03988024, 153 Q2 =-0.03988024,
161 Q3 =-0.00362018, 154 Q3 =-0.00362018,
162 Q4 = 0.00163801, 155 Q4 = 0.00163801,
163 Q5 =-0.01031555, 156 Q5 =-0.01031555,
164 Q6 = 0.02282967, 157 Q6 = 0.02282967,
165 Q7 =-0.02895312, 158 Q7 =-0.02895312,
166 Q8 = 0.01787654, 159 Q8 = 0.01787654,
167 Q9 =-0.00420059; 160 Q9 =-0.00420059;
168 161
169 G4double I = 0.0; 162 G4double I = 0.0;
170 if (std::abs(x) < 3.75) << 163 if (std::fabs(x) < 3.75)
171 { 164 {
172 G4double ax = std::abs(x); << 165 G4double ax = std::fabs(x);
173 G4double y = G4Pow::GetInstance()->powN(x/ << 166 G4double y = std::pow(x/3.75, 2.0);
174 I = ax*(P1+y*(P2+y*(P3+y*(P4+y*(P5+y*(P6+y 167 I = ax*(P1+y*(P2+y*(P3+y*(P4+y*(P5+y*(P6+y*P7))))));
175 } 168 }
176 else 169 else
177 { 170 {
178 G4double ax = std::abs(x); << 171 G4double ax = std::fabs(x);
179 G4double y = 3.75/ax; 172 G4double y = 3.75/ax;
180 I = G4Exp(ax) / std::sqrt(ax) * << 173 I = std::exp(ax) / std::sqrt(ax) *
181 (Q1+y*(Q2+y*(Q3+y*(Q4+y*(Q5+y*(Q6+y*(Q7+ 174 (Q1+y*(Q2+y*(Q3+y*(Q4+y*(Q5+y*(Q6+y*(Q7+y*(Q8+y*Q9))))))));
182 } 175 }
183 if (x < 0.0) I = -I; 176 if (x < 0.0) I = -I;
184 return I; 177 return I;
185 } 178 }
186 ////////////////////////////////////////////// 179 ////////////////////////////////////////////////////////////////////////////////
187 // 180 //
188 G4double G4Bessel::K1 (G4double x) 181 G4double G4Bessel::K1 (G4double x)
189 { 182 {
190 const G4double P1 = 1.0, 183 const G4double P1 = 1.0,
191 P2 = 0.15443144, 184 P2 = 0.15443144,
192 P3 =-0.67278579, 185 P3 =-0.67278579,
193 P4 =-0.18156897, 186 P4 =-0.18156897,
194 P5 =-0.01919402, 187 P5 =-0.01919402,
195 P6 =-0.00110404, 188 P6 =-0.00110404,
196 P7 =-0.00004686; 189 P7 =-0.00004686;
197 const G4double Q1 = 1.25331414, 190 const G4double Q1 = 1.25331414,
198 Q2 = 0.23498619, 191 Q2 = 0.23498619,
199 Q3 =-0.03655620, 192 Q3 =-0.03655620,
200 Q4 = 0.01504268, 193 Q4 = 0.01504268,
201 Q5 =-0.00780353, 194 Q5 =-0.00780353,
202 Q6 = 0.00325614, 195 Q6 = 0.00325614,
203 Q7 =-0.00068245; 196 Q7 =-0.00068245;
204 197
205 G4double K = 0.0; 198 G4double K = 0.0;
206 if (x <= 2.0) 199 if (x <= 2.0)
207 { 200 {
208 G4double y = x * x / 4.0; 201 G4double y = x * x / 4.0;
209 K = G4Log(x/2.0)*I1(x) + 1.0/x * << 202 K = std::log(x/2.0)*I1(x) + 1.0/x *
210 (P1+y*(P2+y*(P3+y*(P4+y*(P5+y*(P6+y*P7)) 203 (P1+y*(P2+y*(P3+y*(P4+y*(P5+y*(P6+y*P7))))));
211 } 204 }
212 else 205 else
213 { 206 {
214 G4double y = 2.0 / x; 207 G4double y = 2.0 / x;
215 K = G4Exp(-x) / std::sqrt(x) * << 208 K = std::exp(-x) / std::sqrt(x) *
216 (Q1+y*(Q2+y*(Q3+y*(Q4+y*(Q5+y*(Q6+y*Q7)) 209 (Q1+y*(Q2+y*(Q3+y*(Q4+y*(Q5+y*(Q6+y*Q7))))));
217 } 210 }
218 return K; 211 return K;
219 } 212 }
220 ////////////////////////////////////////////// 213 ////////////////////////////////////////////////////////////////////////////////
221 // 214 //
222 G4double G4Bessel::pI0 (G4double x) 215 G4double G4Bessel::pI0 (G4double x)
223 { 216 {
224 const G4double A0 = 0.1250000000000E+00, 217 const G4double A0 = 0.1250000000000E+00,
225 A1 = 7.0312500000000E-02, 218 A1 = 7.0312500000000E-02,
226 A2 = 7.3242187500000E-02, 219 A2 = 7.3242187500000E-02,
227 A3 = 1.1215209960938E-01, 220 A3 = 1.1215209960938E-01,
228 A4 = 2.2710800170898E-01, 221 A4 = 2.2710800170898E-01,
229 A5 = 5.7250142097473E-01, 222 A5 = 5.7250142097473E-01,
230 A6 = 1.7277275025845E+00, 223 A6 = 1.7277275025845E+00,
231 A7 = 6.0740420012735E+00, 224 A7 = 6.0740420012735E+00,
232 A8 = 2.4380529699556E+01, 225 A8 = 2.4380529699556E+01,
233 A9 = 1.1001714026925E+02, 226 A9 = 1.1001714026925E+02,
234 A10 = 5.5133589612202E+02, 227 A10 = 5.5133589612202E+02,
235 A11 = 3.0380905109224E+03; 228 A11 = 3.0380905109224E+03;
236 229
237 G4double I = 0.0; 230 G4double I = 0.0;
238 if (x == 0.0) 231 if (x == 0.0)
239 { 232 {
240 I = 1.0; 233 I = 1.0;
241 } 234 }
242 else if (x < 18.0) 235 else if (x < 18.0)
243 { 236 {
244 I = 1.0; 237 I = 1.0;
245 G4double y = x * x; 238 G4double y = x * x;
246 G4double q = 1.0; 239 G4double q = 1.0;
247 for (G4int i=1; i<101; i++) 240 for (G4int i=1; i<101; i++)
248 { 241 {
249 q *= 0.25 * y / i / i; 242 q *= 0.25 * y / i / i;
250 I += q; 243 I += q;
251 if (std::abs(q/I) < 1.0E-15) break; 244 if (std::abs(q/I) < 1.0E-15) break;
252 } 245 }
253 } 246 }
254 else 247 else
255 { 248 {
256 G4double y = 1.0 / x; 249 G4double y = 1.0 / x;
257 I = G4Exp(x) / std::sqrt(twopi*x) * << 250 I = std::exp(x) / std::sqrt(twopi*x) *
258 (1.0 + y*(A0+y*(A1+y*(A2+y*(A3+y*(A4+y*( 251 (1.0 + y*(A0+y*(A1+y*(A2+y*(A3+y*(A4+y*(A5+y*(A6+y*(A7+y*(A8+y*(A9+y*(A10+y*A11))))))))))));
259 } 252 }
260 253
261 return I; 254 return I;
262 } 255 }
263 ////////////////////////////////////////////// 256 ////////////////////////////////////////////////////////////////////////////////
264 // 257 //
265 G4double G4Bessel::pI1 (G4double x) 258 G4double G4Bessel::pI1 (G4double x)
266 { 259 {
267 const G4double A0 = -0.3750000000000E+00, 260 const G4double A0 = -0.3750000000000E+00,
268 A1 = -1.1718750000000E-01, 261 A1 = -1.1718750000000E-01,
269 A2 = -1.0253906250000E-01, 262 A2 = -1.0253906250000E-01,
270 A3 = -1.4419555664063E-01, 263 A3 = -1.4419555664063E-01,
271 A4 = -2.775764465332E-01, 264 A4 = -2.775764465332E-01,
272 A5 = -6.7659258842468E-01, 265 A5 = -6.7659258842468E-01,
273 A6 = -1.9935317337513E+00, 266 A6 = -1.9935317337513E+00,
274 A7 = -6.8839142681099E+00, 267 A7 = -6.8839142681099E+00,
275 A8 = -2.7248827311269E+01, 268 A8 = -2.7248827311269E+01,
276 A9 = -1.2159789187654E+02, 269 A9 = -1.2159789187654E+02,
277 A10 = -6.0384407670507E+02, 270 A10 = -6.0384407670507E+02,
278 A11 = -3.3022722944809E+03; 271 A11 = -3.3022722944809E+03;
279 272
280 G4double I = 0.0; 273 G4double I = 0.0;
281 if (x == 0.0) 274 if (x == 0.0)
282 { 275 {
283 I = 0.0; 276 I = 0.0;
284 } 277 }
285 else if (x < 18.0) 278 else if (x < 18.0)
286 { 279 {
287 I = 1.0; 280 I = 1.0;
288 G4double y = x * x; 281 G4double y = x * x;
289 G4double q = 1.0; 282 G4double q = 1.0;
290 for (G4int i=1; i<101; i++) 283 for (G4int i=1; i<101; i++)
291 { 284 {
292 q *= 0.25 * y / i / (i+1.0); 285 q *= 0.25 * y / i / (i+1.0);
293 I += q; 286 I += q;
294 if (std::abs(q/I) < 1.0E-15) break; 287 if (std::abs(q/I) < 1.0E-15) break;
295 } 288 }
296 I *= 0.5 * x; 289 I *= 0.5 * x;
297 290
298 } 291 }
299 else 292 else
300 { 293 {
301 G4double y = 1.0 / x; 294 G4double y = 1.0 / x;
302 I = G4Exp(x) / std::sqrt(twopi*x) * << 295 I = std::exp(x) / std::sqrt(twopi*x) *
303 (1.0 + y*(A0+y*(A1+y*(A2+y*(A3+y*(A4+y*( 296 (1.0 + y*(A0+y*(A1+y*(A2+y*(A3+y*(A4+y*(A5+y*(A6+y*(A7+y*(A8+y*(A9+y*(A10+y*A11))))))))))));
304 } 297 }
305 298
306 return I; 299 return I;
307 } 300 }
308 ////////////////////////////////////////////// 301 ////////////////////////////////////////////////////////////////////////////////
309 // 302 //
310 G4double G4Bessel::pK0 (G4double x) 303 G4double G4Bessel::pK0 (G4double x)
311 { 304 {
312 const G4double A0 = 0.1250000000000E+00, 305 const G4double A0 = 0.1250000000000E+00,
313 A1 = 0.2109375000000E+00, 306 A1 = 0.2109375000000E+00,
314 A2 = 1.0986328125000E+00, 307 A2 = 1.0986328125000E+00,
315 A3 = 1.1775970458984E+01, 308 A3 = 1.1775970458984E+01,
316 A4 = 2.1461706161499E+02, 309 A4 = 2.1461706161499E+02,
317 A5 = 5.9511522710323E+03, 310 A5 = 5.9511522710323E+03,
318 A6 = 2.3347645606175E+05, 311 A6 = 2.3347645606175E+05,
319 A7 = 1.2312234987631E+07; 312 A7 = 1.2312234987631E+07;
320 313
321 G4double K = 0.0; 314 G4double K = 0.0;
322 if (x == 0.0) 315 if (x == 0.0)
323 { 316 {
324 K = 1.0E+307; 317 K = 1.0E+307;
325 } 318 }
326 else if (x < 9.0) 319 else if (x < 9.0)
327 { 320 {
328 G4double y = x * x; 321 G4double y = x * x;
329 G4double C = -G4Log(x/2.0) - 0.57721566490 << 322 G4double C = -std::log(x/2.0) - 0.5772156649015329;
330 G4double q = 1.0; 323 G4double q = 1.0;
331 G4double t = 0.0; 324 G4double t = 0.0;
332 for (G4int i=1; i<51; i++) 325 for (G4int i=1; i<51; i++)
333 { 326 {
334 q *= 0.25 * y / i / i; 327 q *= 0.25 * y / i / i;
335 t += 1.0 / i ; 328 t += 1.0 / i ;
336 K += q * (t+C); 329 K += q * (t+C);
337 } 330 }
338 K += C; 331 K += C;
339 } 332 }
340 else 333 else
341 { 334 {
342 G4double y = 1.0 / x / x; 335 G4double y = 1.0 / x / x;
343 K = 0.5 / x / pI0(x) * 336 K = 0.5 / x / pI0(x) *
344 (1.0 + y*(A0+y*(A1+y*(A2+y*(A3+y*(A4+y*( 337 (1.0 + y*(A0+y*(A1+y*(A2+y*(A3+y*(A4+y*(A5+y*(A6+y*A7))))))));
345 } 338 }
346 339
347 return K; 340 return K;
348 } 341 }
349 ////////////////////////////////////////////// 342 ////////////////////////////////////////////////////////////////////////////////
350 // 343 //
351 G4double G4Bessel::pK1 (G4double x) 344 G4double G4Bessel::pK1 (G4double x)
352 { 345 {
353 G4double K = 0.0; 346 G4double K = 0.0;
354 if (x == 0.0) 347 if (x == 0.0)
355 K = 1.0E+307; 348 K = 1.0E+307;
356 else 349 else
357 K = (1.0/x - pI1(x)*pK0(x)) / pI0(x); 350 K = (1.0/x - pI1(x)*pK0(x)) / pI0(x);
358 return K; 351 return K;
359 } 352 }
360 ////////////////////////////////////////////// 353 ////////////////////////////////////////////////////////////////////////////////
361 // 354 //
362 355