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34 // ********************************************************************
35 //
36 // %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
37 //
38 // MODULE: G4Bessel.cc
39 //
40 // Version: B.1
41 // Date: 15/04/04
42 // Author: P R Truscott
43 // Organisation: QinetiQ Ltd, UK
44 // Customer: ESA/ESTEC, NOORDWIJK
45 // Contract: 17191/03/NL/LvH
46 //
47 // %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
48 //
49 // CHANGE HISTORY
50 // --------------
51 //
52 // 18 Noevmber 2003, P R Truscott, QinetiQ Ltd, UK
53 // Created.
54 //
55 // 15 March 2004, P R Truscott, QinetiQ Ltd, UK
56 // Beta release
57 //
58 // 06 August 2015, A. Ribon, CERN
59 // Migrated to G4Exp, G4Log and G4Pow.
60 //
61 // %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
62 ////////////////////////////////////////////////////////////////////////////////
63 //
64 #include "G4Bessel.hh"
65 #include "G4PhysicalConstants.hh"
66
67 #include "G4Exp.hh"
68 #include "G4Log.hh"
69 #include "G4Pow.hh"
70 ////////////////////////////////////////////////////////////////////////////////
71 //
72 G4Bessel::G4Bessel ()
73 {;}
74 ////////////////////////////////////////////////////////////////////////////////
75 //
76 G4Bessel::~G4Bessel ()
77 {;}
78 ////////////////////////////////////////////////////////////////////////////////
79 //
80 G4double G4Bessel::I0 (G4double x)
81 {
82 const G4double P1 = 1.0,
83 P2 = 3.5156229,
84 P3 = 3.0899424,
85 P4 = 1.2067492,
86 P5 = 0.2659732,
87 P6 = 0.0360768,
88 P7 = 0.0045813;
89 const G4double Q1 = 0.39894228,
90 Q2 = 0.01328592,
91 Q3 = 0.00225319,
92 Q4 =-0.00157565,
93 Q5 = 0.00916281,
94 Q6 =-0.02057706,
95 Q7 = 0.02635537,
96 Q8 =-0.01647633,
97 Q9 = 0.00392377;
98
99 G4double I = 0.0;
100 if (std::abs(x) < 3.75)
101 {
102 G4double y = G4Pow::GetInstance()->powN(x/3.75, 2);
103 I = P1+y*(P2+y*(P3+y*(P4+y*(P5+y*(P6+y*P7)))));
104 }
105 else
106 {
107 G4double ax = std::abs(x);
108 G4double y = 3.75/ax;
109 I = G4Exp(ax) / std::sqrt(ax) *
110 (Q1+y*(Q2+y*(Q3+y*(Q4+y*(Q5+y*(Q6+y*(Q7+y*(Q8+y*Q9))))))));
111 }
112 return I;
113 }
114 ////////////////////////////////////////////////////////////////////////////////
115 //
116 G4double G4Bessel::K0 (G4double x)
117 {
118 const G4double P1 =-0.57721566,
119 P2 = 0.42278420,
120 P3 = 0.23069756,
121 P4 = 0.03488590,
122 P5 = 0.00262698,
123 P6 = 0.00010750,
124 P7 = 0.00000740;
125 const G4double Q1 = 1.25331414,
126 Q2 =-0.07832358,
127 Q3 = 0.02189568,
128 Q4 =-0.01062446,
129 Q5 = 0.00587872,
130 Q6 =-0.00251540,
131 Q7 = 0.00053208;
132
133 G4double K = 0.0;
134 if (x <= 2.0)
135 {
136 G4double y = x * x / 4.0;
137 K = (-G4Log(x/2.0)) * I0(x) +
138 P1+y*(P2+y*(P3+y*(P4+y*(P5+y*(P6+y*P7)))));
139 }
140 else
141 {
142 G4double y = 2.0 / x;
143 K = G4Exp(-x) / std::sqrt(x) *
144 (Q1+y*(Q2+y*(Q3+y*(Q4+y*(Q5+y*(Q6+y*Q7))))));
145 }
146 return K;
147 }
148 ////////////////////////////////////////////////////////////////////////////////
149 //
150 G4double G4Bessel::I1 (G4double x)
151 {
152 const G4double P1 = 0.5,
153 P2 = 0.87890594,
154 P3 = 0.51498869,
155 P4 = 0.15084934,
156 P5 = 0.02658733,
157 P6 = 0.00301532,
158 P7 = 0.00032411;
159 const G4double Q1 = 0.39894228,
160 Q2 =-0.03988024,
161 Q3 =-0.00362018,
162 Q4 = 0.00163801,
163 Q5 =-0.01031555,
164 Q6 = 0.02282967,
165 Q7 =-0.02895312,
166 Q8 = 0.01787654,
167 Q9 =-0.00420059;
168
169 G4double I = 0.0;
170 if (std::abs(x) < 3.75)
171 {
172 G4double ax = std::abs(x);
173 G4double y = G4Pow::GetInstance()->powN(x/3.75, 2);
174 I = ax*(P1+y*(P2+y*(P3+y*(P4+y*(P5+y*(P6+y*P7))))));
175 }
176 else
177 {
178 G4double ax = std::abs(x);
179 G4double y = 3.75/ax;
180 I = G4Exp(ax) / std::sqrt(ax) *
181 (Q1+y*(Q2+y*(Q3+y*(Q4+y*(Q5+y*(Q6+y*(Q7+y*(Q8+y*Q9))))))));
182 }
183 if (x < 0.0) I = -I;
184 return I;
185 }
186 ////////////////////////////////////////////////////////////////////////////////
187 //
188 G4double G4Bessel::K1 (G4double x)
189 {
190 const G4double P1 = 1.0,
191 P2 = 0.15443144,
192 P3 =-0.67278579,
193 P4 =-0.18156897,
194 P5 =-0.01919402,
195 P6 =-0.00110404,
196 P7 =-0.00004686;
197 const G4double Q1 = 1.25331414,
198 Q2 = 0.23498619,
199 Q3 =-0.03655620,
200 Q4 = 0.01504268,
201 Q5 =-0.00780353,
202 Q6 = 0.00325614,
203 Q7 =-0.00068245;
204
205 G4double K = 0.0;
206 if (x <= 2.0)
207 {
208 G4double y = x * x / 4.0;
209 K = G4Log(x/2.0)*I1(x) + 1.0/x *
210 (P1+y*(P2+y*(P3+y*(P4+y*(P5+y*(P6+y*P7))))));
211 }
212 else
213 {
214 G4double y = 2.0 / x;
215 K = G4Exp(-x) / std::sqrt(x) *
216 (Q1+y*(Q2+y*(Q3+y*(Q4+y*(Q5+y*(Q6+y*Q7))))));
217 }
218 return K;
219 }
220 ////////////////////////////////////////////////////////////////////////////////
221 //
222 G4double G4Bessel::pI0 (G4double x)
223 {
224 const G4double A0 = 0.1250000000000E+00,
225 A1 = 7.0312500000000E-02,
226 A2 = 7.3242187500000E-02,
227 A3 = 1.1215209960938E-01,
228 A4 = 2.2710800170898E-01,
229 A5 = 5.7250142097473E-01,
230 A6 = 1.7277275025845E+00,
231 A7 = 6.0740420012735E+00,
232 A8 = 2.4380529699556E+01,
233 A9 = 1.1001714026925E+02,
234 A10 = 5.5133589612202E+02,
235 A11 = 3.0380905109224E+03;
236
237 G4double I = 0.0;
238 if (x == 0.0)
239 {
240 I = 1.0;
241 }
242 else if (x < 18.0)
243 {
244 I = 1.0;
245 G4double y = x * x;
246 G4double q = 1.0;
247 for (G4int i=1; i<101; i++)
248 {
249 q *= 0.25 * y / i / i;
250 I += q;
251 if (std::abs(q/I) < 1.0E-15) break;
252 }
253 }
254 else
255 {
256 G4double y = 1.0 / x;
257 I = G4Exp(x) / std::sqrt(twopi*x) *
258 (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 }
260
261 return I;
262 }
263 ////////////////////////////////////////////////////////////////////////////////
264 //
265 G4double G4Bessel::pI1 (G4double x)
266 {
267 const G4double A0 = -0.3750000000000E+00,
268 A1 = -1.1718750000000E-01,
269 A2 = -1.0253906250000E-01,
270 A3 = -1.4419555664063E-01,
271 A4 = -2.775764465332E-01,
272 A5 = -6.7659258842468E-01,
273 A6 = -1.9935317337513E+00,
274 A7 = -6.8839142681099E+00,
275 A8 = -2.7248827311269E+01,
276 A9 = -1.2159789187654E+02,
277 A10 = -6.0384407670507E+02,
278 A11 = -3.3022722944809E+03;
279
280 G4double I = 0.0;
281 if (x == 0.0)
282 {
283 I = 0.0;
284 }
285 else if (x < 18.0)
286 {
287 I = 1.0;
288 G4double y = x * x;
289 G4double q = 1.0;
290 for (G4int i=1; i<101; i++)
291 {
292 q *= 0.25 * y / i / (i+1.0);
293 I += q;
294 if (std::abs(q/I) < 1.0E-15) break;
295 }
296 I *= 0.5 * x;
297
298 }
299 else
300 {
301 G4double y = 1.0 / x;
302 I = G4Exp(x) / std::sqrt(twopi*x) *
303 (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 }
305
306 return I;
307 }
308 ////////////////////////////////////////////////////////////////////////////////
309 //
310 G4double G4Bessel::pK0 (G4double x)
311 {
312 const G4double A0 = 0.1250000000000E+00,
313 A1 = 0.2109375000000E+00,
314 A2 = 1.0986328125000E+00,
315 A3 = 1.1775970458984E+01,
316 A4 = 2.1461706161499E+02,
317 A5 = 5.9511522710323E+03,
318 A6 = 2.3347645606175E+05,
319 A7 = 1.2312234987631E+07;
320
321 G4double K = 0.0;
322 if (x == 0.0)
323 {
324 K = 1.0E+307;
325 }
326 else if (x < 9.0)
327 {
328 G4double y = x * x;
329 G4double C = -G4Log(x/2.0) - 0.5772156649015329;
330 G4double q = 1.0;
331 G4double t = 0.0;
332 for (G4int i=1; i<51; i++)
333 {
334 q *= 0.25 * y / i / i;
335 t += 1.0 / i ;
336 K += q * (t+C);
337 }
338 K += C;
339 }
340 else
341 {
342 G4double y = 1.0 / x / x;
343 K = 0.5 / x / pI0(x) *
344 (1.0 + y*(A0+y*(A1+y*(A2+y*(A3+y*(A4+y*(A5+y*(A6+y*A7))))))));
345 }
346
347 return K;
348 }
349 ////////////////////////////////////////////////////////////////////////////////
350 //
351 G4double G4Bessel::pK1 (G4double x)
352 {
353 G4double K = 0.0;
354 if (x == 0.0)
355 K = 1.0E+307;
356 else
357 K = (1.0/x - pI1(x)*pK0(x)) / pI0(x);
358 return K;
359 }
360 ////////////////////////////////////////////////////////////////////////////////
361 //
362