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