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4c039060 | 1 | /************************************************************************** |
2 | * Copyright(c) 1998-1999, ALICE Experiment at CERN, All rights reserved. * | |
3 | * * | |
4 | * Author: The ALICE Off-line Project. * | |
5 | * Contributors are mentioned in the code where appropriate. * | |
6 | * * | |
7 | * Permission to use, copy, modify and distribute this software and its * | |
8 | * documentation strictly for non-commercial purposes is hereby granted * | |
9 | * without fee, provided that the above copyright notice appears in all * | |
10 | * copies and that both the copyright notice and this permission notice * | |
11 | * appear in the supporting documentation. The authors make no claims * | |
12 | * about the suitability of this software for any purpose. It is * | |
13 | * provided "as is" without express or implied warranty. * | |
14 | **************************************************************************/ | |
15 | ||
f531a546 | 16 | // $Id$ |
4c039060 | 17 | |
959fbac5 | 18 | /////////////////////////////////////////////////////////////////////////// |
19 | // Class AliMath | |
20 | // Various mathematical tools which may be very convenient while | |
21 | // performing physics analysis. | |
22 | // | |
23 | // Example : Probability of a Chi-squared value | |
24 | // ========= | |
25 | // | |
26 | // AliMath M; | |
27 | // Float_t chi2=20; // The chi-squared value | |
28 | // Int_t ndf=12; // The number of degrees of freedom | |
29 | // Float_t p=M.Prob(chi2,ndf); // The probability that at least a Chi-squared | |
30 | // // value of chi2 will be observed, even for a | |
31 | // // correct model | |
32 | // | |
33 | //--- Author: Nick van Eijndhoven 14-nov-1998 UU-SAP Utrecht | |
f531a546 | 34 | //- Modified: NvE $Date$ UU-SAP Utrecht |
959fbac5 | 35 | /////////////////////////////////////////////////////////////////////////// |
36 | ||
d88f97cc | 37 | #include "AliMath.h" |
c72198f1 | 38 | #include "Riostream.h" |
d88f97cc | 39 | |
40 | ClassImp(AliMath) // Class implementation to enable ROOT I/O | |
41 | ||
c72198f1 | 42 | AliMath::AliMath() : TObject() |
d88f97cc | 43 | { |
44 | // Default constructor | |
45 | } | |
46 | /////////////////////////////////////////////////////////////////////////// | |
47 | AliMath::~AliMath() | |
48 | { | |
49 | // Destructor | |
50 | } | |
51 | /////////////////////////////////////////////////////////////////////////// | |
c72198f1 | 52 | AliMath::AliMath(AliMath& m) : TObject(m) |
53 | { | |
54 | // Copy constructor | |
55 | } | |
56 | /////////////////////////////////////////////////////////////////////////// | |
29beb80d | 57 | Double_t AliMath::Gamma(Double_t z) |
d88f97cc | 58 | { |
59 | // Computation of gamma(z) for all z>0. | |
60 | // | |
61 | // The algorithm is based on the article by C.Lanczos [1] as denoted in | |
62 | // Numerical Recipes 2nd ed. on p. 207 (W.H.Press et al.). | |
63 | // | |
64 | // [1] C.Lanczos, SIAM Journal of Numerical Analysis B1 (1964), 86. | |
65 | // | |
66 | //--- Nve 14-nov-1998 UU-SAP Utrecht | |
67 | ||
68 | if (z<=0.) | |
69 | { | |
70 | cout << "*Gamma(z)* Wrong argument z = " << z << endl; | |
71 | return 0; | |
72 | } | |
73 | ||
29beb80d | 74 | Double_t v=LnGamma(z); |
d88f97cc | 75 | return exp(v); |
76 | } | |
77 | /////////////////////////////////////////////////////////////////////////// | |
29beb80d | 78 | Double_t AliMath::Gamma(Double_t a,Double_t x) |
d88f97cc | 79 | { |
80 | // Computation of the incomplete gamma function P(a,x) | |
81 | // | |
82 | // The algorithm is based on the formulas and code as denoted in | |
83 | // Numerical Recipes 2nd ed. on p. 210-212 (W.H.Press et al.). | |
84 | // | |
85 | //--- Nve 14-nov-1998 UU-SAP Utrecht | |
86 | ||
87 | if (a<=0.) | |
88 | { | |
89 | cout << "*Gamma(a,x)* Invalid argument a = " << a << endl; | |
90 | return 0; | |
91 | } | |
92 | ||
93 | if (x<=0.) | |
94 | { | |
95 | if (x<0) cout << "*Gamma(a,x)* Invalid argument x = " << x << endl; | |
96 | return 0; | |
97 | } | |
98 | ||
99 | if (x<(a+1.)) | |
100 | { | |
101 | return GamSer(a,x); | |
102 | } | |
103 | else | |
104 | { | |
105 | return GamCf(a,x); | |
106 | } | |
107 | } | |
108 | /////////////////////////////////////////////////////////////////////////// | |
29beb80d | 109 | Double_t AliMath::LnGamma(Double_t z) |
d88f97cc | 110 | { |
111 | // Computation of ln[gamma(z)] for all z>0. | |
112 | // | |
113 | // The algorithm is based on the article by C.Lanczos [1] as denoted in | |
114 | // Numerical Recipes 2nd ed. on p. 207 (W.H.Press et al.). | |
115 | // | |
116 | // [1] C.Lanczos, SIAM Journal of Numerical Analysis B1 (1964), 86. | |
117 | // | |
118 | // The accuracy of the result is better than 2e-10. | |
119 | // | |
120 | //--- Nve 14-nov-1998 UU-SAP Utrecht | |
121 | ||
122 | if (z<=0.) | |
123 | { | |
124 | cout << "*LnGamma(z)* Wrong argument z = " << z << endl; | |
125 | return 0; | |
126 | } | |
127 | ||
128 | // Coefficients for the series expansion | |
129 | Double_t c[7]; | |
130 | c[0]= 2.5066282746310005; | |
131 | c[1]= 76.18009172947146; | |
132 | c[2]=-86.50532032941677; | |
133 | c[3]= 24.01409824083091; | |
134 | c[4]= -1.231739572450155; | |
135 | c[5]= 0.1208650973866179e-2; | |
136 | c[6]= -0.5395239384953e-5; | |
137 | ||
138 | Double_t x=z; | |
139 | Double_t y=x; | |
140 | Double_t tmp=x+5.5; | |
141 | tmp=(x+0.5)*log(tmp)-tmp; | |
142 | Double_t ser=1.000000000190015; | |
143 | for (Int_t i=1; i<7; i++) | |
144 | { | |
145 | y+=1.; | |
146 | ser+=c[i]/y; | |
147 | } | |
29beb80d | 148 | Double_t v=tmp+log(c[0]*ser/x); |
d88f97cc | 149 | return v; |
150 | } | |
151 | /////////////////////////////////////////////////////////////////////////// | |
29beb80d | 152 | Double_t AliMath::GamSer(Double_t a,Double_t x) |
d88f97cc | 153 | { |
154 | // Computation of the incomplete gamma function P(a,x) | |
155 | // via its series representation. | |
156 | // | |
157 | // The algorithm is based on the formulas and code as denoted in | |
158 | // Numerical Recipes 2nd ed. on p. 210-212 (W.H.Press et al.). | |
159 | // | |
160 | //--- Nve 14-nov-1998 UU-SAP Utrecht | |
161 | ||
162 | Int_t itmax=100; // Maximum number of iterations | |
29beb80d | 163 | Double_t eps=3.e-7; // Relative accuracy |
d88f97cc | 164 | |
165 | if (a<=0.) | |
166 | { | |
167 | cout << "*GamSer(a,x)* Invalid argument a = " << a << endl; | |
168 | return 0; | |
169 | } | |
170 | ||
171 | if (x<=0.) | |
172 | { | |
173 | if (x<0) cout << "*GamSer(a,x)* Invalid argument x = " << x << endl; | |
174 | return 0; | |
175 | } | |
176 | ||
29beb80d | 177 | Double_t gln=LnGamma(a); |
178 | Double_t ap=a; | |
179 | Double_t sum=1./a; | |
180 | Double_t del=sum; | |
d88f97cc | 181 | for (Int_t n=1; n<=itmax; n++) |
182 | { | |
183 | ap+=1.; | |
184 | del=del*x/ap; | |
185 | sum+=del; | |
186 | if (fabs(del)<fabs(sum*eps)) break; | |
187 | if (n==itmax) cout << "*GamSer(a,x)* a too large or itmax too small" << endl; | |
188 | } | |
29beb80d | 189 | Double_t v=sum*exp(-x+a*log(x)-gln); |
d88f97cc | 190 | return v; |
191 | } | |
192 | /////////////////////////////////////////////////////////////////////////// | |
29beb80d | 193 | Double_t AliMath::GamCf(Double_t a,Double_t x) |
d88f97cc | 194 | { |
195 | // Computation of the incomplete gamma function P(a,x) | |
196 | // via its continued fraction representation. | |
197 | // | |
198 | // The algorithm is based on the formulas and code as denoted in | |
199 | // Numerical Recipes 2nd ed. on p. 210-212 (W.H.Press et al.). | |
200 | // | |
201 | //--- Nve 14-nov-1998 UU-SAP Utrecht | |
202 | ||
203 | Int_t itmax=100; // Maximum number of iterations | |
29beb80d | 204 | Double_t eps=3.e-7; // Relative accuracy |
205 | Double_t fpmin=1.e-30; // Smallest Double_t value allowed here | |
d88f97cc | 206 | |
207 | if (a<=0.) | |
208 | { | |
209 | cout << "*GamCf(a,x)* Invalid argument a = " << a << endl; | |
210 | return 0; | |
211 | } | |
212 | ||
213 | if (x<=0.) | |
214 | { | |
215 | if (x<0) cout << "*GamCf(a,x)* Invalid argument x = " << x << endl; | |
216 | return 0; | |
217 | } | |
218 | ||
29beb80d | 219 | Double_t gln=LnGamma(a); |
220 | Double_t b=x+1.-a; | |
221 | Double_t c=1./fpmin; | |
222 | Double_t d=1./b; | |
223 | Double_t h=d; | |
224 | Double_t an,del; | |
d88f97cc | 225 | for (Int_t i=1; i<=itmax; i++) |
226 | { | |
29beb80d | 227 | an=double(-i)*(double(i)-a); |
d88f97cc | 228 | b+=2.; |
229 | d=an*d+b; | |
230 | if (fabs(d)<fpmin) d=fpmin; | |
231 | c=b+an/c; | |
232 | if (fabs(c)<fpmin) c=fpmin; | |
233 | d=1./d; | |
234 | del=d*c; | |
235 | h=h*del; | |
236 | if (fabs(del-1.)<eps) break; | |
237 | if (i==itmax) cout << "*GamCf(a,x)* a too large or itmax too small" << endl; | |
238 | } | |
29beb80d | 239 | Double_t v=exp(-x+a*log(x)-gln)*h; |
d88f97cc | 240 | return (1.-v); |
241 | } | |
242 | /////////////////////////////////////////////////////////////////////////// | |
29beb80d | 243 | Double_t AliMath::Erf(Double_t x) |
d88f97cc | 244 | { |
245 | // Computation of the error function erf(x). | |
246 | // | |
247 | //--- NvE 14-nov-1998 UU-SAP Utrecht | |
248 | ||
249 | return (1.-Erfc(x)); | |
250 | } | |
251 | /////////////////////////////////////////////////////////////////////////// | |
29beb80d | 252 | Double_t AliMath::Erfc(Double_t x) |
d88f97cc | 253 | { |
254 | // Computation of the complementary error function erfc(x). | |
255 | // | |
256 | // The algorithm is based on a Chebyshev fit as denoted in | |
257 | // Numerical Recipes 2nd ed. on p. 214 (W.H.Press et al.). | |
258 | // | |
259 | // The fractional error is always less than 1.2e-7. | |
260 | // | |
261 | //--- Nve 14-nov-1998 UU-SAP Utrecht | |
262 | ||
263 | // The parameters of the Chebyshev fit | |
387a745b | 264 | const Double_t ka1=-1.26551223, ka2=1.00002368, |
265 | ka3= 0.37409196, ka4=0.09678418, | |
266 | ka5=-0.18628806, ka6=0.27886807, | |
267 | ka7=-1.13520398, ka8=1.48851587, | |
268 | ka9=-0.82215223, ka10=0.17087277; | |
d88f97cc | 269 | |
29beb80d | 270 | Double_t v=1.; // The return value |
d88f97cc | 271 | |
29beb80d | 272 | Double_t z=fabs(x); |
d88f97cc | 273 | |
274 | if (z <= 0.) return v; // erfc(0)=1 | |
275 | ||
29beb80d | 276 | Double_t t=1./(1.+0.5*z); |
d88f97cc | 277 | |
278 | v=t*exp((-z*z) | |
387a745b | 279 | +ka1+t*(ka2+t*(ka3+t*(ka4+t*(ka5+t*(ka6+t*(ka7+t*(ka8+t*(ka9+t*ka10))))))))); |
d88f97cc | 280 | |
281 | if (x < 0.) v=2.-v; // erfc(-x)=2-erfc(x) | |
282 | ||
283 | return v; | |
284 | } | |
285 | /////////////////////////////////////////////////////////////////////////// | |
176f88c0 | 286 | Double_t AliMath::Prob(Double_t chi2,Int_t ndf,Int_t mode) |
d88f97cc | 287 | { |
288 | // Computation of the probability for a certain Chi-squared (chi2) | |
289 | // and number of degrees of freedom (ndf). | |
290 | // | |
176f88c0 | 291 | // According to the value of the parameter "mode" various algorithms |
292 | // can be selected. | |
293 | // | |
294 | // mode = 0 : Calculations are based on the incomplete gamma function P(a,x), | |
295 | // where a=ndf/2 and x=chi2/2. | |
296 | // | |
297 | // 1 : Same as for mode=0. However, in case ndf=1 an exact expression | |
298 | // based on the error function Erf() is used. | |
299 | // | |
300 | // 2 : Same as for mode=0. However, in case ndf>30 a Gaussian approximation | |
301 | // is used instead of the gamma function. | |
302 | // | |
303 | // When invoked as Prob(chi2,ndf) the default mode=1 is used. | |
d88f97cc | 304 | // |
305 | // P(a,x) represents the probability that the observed Chi-squared | |
306 | // for a correct model should be less than the value chi2. | |
307 | // | |
308 | // The returned probability corresponds to 1-P(a,x), | |
309 | // which denotes the probability that an observed Chi-squared exceeds | |
310 | // the value chi2 by chance, even for a correct model. | |
311 | // | |
312 | //--- NvE 14-nov-1998 UU-SAP Utrecht | |
313 | ||
314 | if (ndf <= 0) return 0; // Set CL to zero in case ndf<=0 | |
315 | ||
316 | if (chi2 <= 0.) | |
317 | { | |
318 | if (chi2 < 0.) | |
319 | { | |
320 | return 0; | |
321 | } | |
322 | else | |
323 | { | |
324 | return 1; | |
325 | } | |
326 | } | |
176f88c0 | 327 | |
328 | Double_t v=-1.; | |
329 | ||
330 | switch (mode) | |
331 | { | |
332 | case 1: // Exact expression for ndf=1 as alternative for the gamma function | |
333 | if (ndf==1) v=1.-Erf(sqrt(chi2)/sqrt(2.)); | |
334 | break; | |
335 | ||
336 | case 2: // Gaussian approximation for large ndf (i.e. ndf>30) as alternative for the gamma function | |
337 | if (ndf>30) | |
338 | { | |
339 | Double_t q=sqrt(2.*chi2)-sqrt(double(2*ndf-1)); | |
340 | if (q>0.) v=0.5*(1.-Erf(q/sqrt(2.))); | |
341 | } | |
342 | break; | |
343 | } | |
d88f97cc | 344 | |
176f88c0 | 345 | if (v<0.) |
346 | { | |
347 | // Evaluate the incomplete gamma function | |
348 | Double_t a=double(ndf)/2.; | |
349 | Double_t x=chi2/2.; | |
350 | v=1.-Gamma(a,x); | |
351 | } | |
352 | ||
353 | return v; | |
d88f97cc | 354 | } |
355 | /////////////////////////////////////////////////////////////////////////// | |
29beb80d | 356 | Double_t AliMath::BesselI0(Double_t x) |
357 | { | |
358 | // Computation of the modified Bessel function I_0(x) for any real x. | |
359 | // | |
360 | // The algorithm is based on the article by Abramowitz and Stegun [1] | |
361 | // as denoted in Numerical Recipes 2nd ed. on p. 230 (W.H.Press et al.). | |
362 | // | |
363 | // [1] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions, | |
364 | // Applied Mathematics Series vol. 55 (1964), Washington. | |
365 | // | |
366 | //--- NvE 12-mar-2000 UU-SAP Utrecht | |
367 | ||
368 | // Parameters of the polynomial approximation | |
387a745b | 369 | const Double_t kp1=1.0, kp2=3.5156229, kp3=3.0899424, |
370 | kp4=1.2067492, kp5=0.2659732, kp6=3.60768e-2, kp7=4.5813e-3; | |
29beb80d | 371 | |
387a745b | 372 | const Double_t kq1= 0.39894228, kq2= 1.328592e-2, kq3= 2.25319e-3, |
373 | kq4=-1.57565e-3, kq5= 9.16281e-3, kq6=-2.057706e-2, | |
374 | kq7= 2.635537e-2, kq8=-1.647633e-2, kq9= 3.92377e-3; | |
29beb80d | 375 | |
376 | Double_t ax=fabs(x); | |
377 | ||
378 | Double_t y=0,result=0; | |
379 | ||
380 | if (ax < 3.75) | |
381 | { | |
382 | y=pow(x/3.75,2); | |
387a745b | 383 | result=kp1+y*(kp2+y*(kp3+y*(kp4+y*(kp5+y*(kp6+y*kp7))))); |
29beb80d | 384 | } |
385 | else | |
386 | { | |
387 | y=3.75/ax; | |
387a745b | 388 | result=(exp(ax)/sqrt(ax)) |
389 | *(kq1+y*(kq2+y*(kq3+y*(kq4+y*(kq5+y*(kq6+y*(kq7+y*(kq8+y*kq9)))))))); | |
29beb80d | 390 | } |
391 | ||
392 | return result; | |
393 | } | |
394 | /////////////////////////////////////////////////////////////////////////// | |
395 | Double_t AliMath::BesselK0(Double_t x) | |
396 | { | |
397 | // Computation of the modified Bessel function K_0(x) for positive real x. | |
398 | // | |
399 | // The algorithm is based on the article by Abramowitz and Stegun [1] | |
400 | // as denoted in Numerical Recipes 2nd ed. on p. 230 (W.H.Press et al.). | |
401 | // | |
402 | // [1] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions, | |
403 | // Applied Mathematics Series vol. 55 (1964), Washington. | |
404 | // | |
405 | //--- NvE 12-mar-2000 UU-SAP Utrecht | |
406 | ||
407 | // Parameters of the polynomial approximation | |
387a745b | 408 | const Double_t kp1=-0.57721566, kp2=0.42278420, kp3=0.23069756, |
409 | kp4= 3.488590e-2, kp5=2.62698e-3, kp6=1.0750e-4, kp7=7.4e-5; | |
29beb80d | 410 | |
387a745b | 411 | const Double_t kq1= 1.25331414, kq2=-7.832358e-2, kq3= 2.189568e-2, |
412 | kq4=-1.062446e-2, kq5= 5.87872e-3, kq6=-2.51540e-3, kq7=5.3208e-4; | |
29beb80d | 413 | |
414 | if (x <= 0) | |
415 | { | |
416 | cout << " *BesselK0* Invalid argument x = " << x << endl; | |
417 | return 0; | |
418 | } | |
419 | ||
420 | Double_t y=0,result=0; | |
421 | ||
422 | if (x <= 2) | |
423 | { | |
424 | y=x*x/4.; | |
387a745b | 425 | result=(-log(x/2.)*BesselI0(x)) |
426 | +(kp1+y*(kp2+y*(kp3+y*(kp4+y*(kp5+y*(kp6+y*kp7)))))); | |
29beb80d | 427 | } |
428 | else | |
429 | { | |
430 | y=2./x; | |
387a745b | 431 | result=(exp(-x)/sqrt(x)) |
432 | *(kq1+y*(kq2+y*(kq3+y*(kq4+y*(kq5+y*(kq6+y*kq7)))))); | |
29beb80d | 433 | } |
434 | ||
435 | return result; | |
436 | } | |
437 | /////////////////////////////////////////////////////////////////////////// | |
438 | Double_t AliMath::BesselI1(Double_t x) | |
439 | { | |
440 | // Computation of the modified Bessel function I_1(x) for any real x. | |
441 | // | |
442 | // The algorithm is based on the article by Abramowitz and Stegun [1] | |
443 | // as denoted in Numerical Recipes 2nd ed. on p. 230 (W.H.Press et al.). | |
444 | // | |
445 | // [1] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions, | |
446 | // Applied Mathematics Series vol. 55 (1964), Washington. | |
447 | // | |
448 | //--- NvE 12-mar-2000 UU-SAP Utrecht | |
449 | ||
450 | // Parameters of the polynomial approximation | |
387a745b | 451 | const Double_t kp1=0.5, kp2=0.87890594, kp3=0.51498869, |
452 | kp4=0.15084934, kp5=2.658733e-2, kp6=3.01532e-3, kp7=3.2411e-4; | |
29beb80d | 453 | |
387a745b | 454 | const Double_t kq1= 0.39894228, kq2=-3.988024e-2, kq3=-3.62018e-3, |
455 | kq4= 1.63801e-3, kq5=-1.031555e-2, kq6= 2.282967e-2, | |
456 | kq7=-2.895312e-2, kq8= 1.787654e-2, kq9=-4.20059e-3; | |
29beb80d | 457 | |
458 | Double_t ax=fabs(x); | |
459 | ||
460 | Double_t y=0,result=0; | |
461 | ||
462 | if (ax < 3.75) | |
463 | { | |
464 | y=pow(x/3.75,2); | |
387a745b | 465 | result=x*(kp1+y*(kp2+y*(kp3+y*(kp4+y*(kp5+y*(kp6+y*kp7)))))); |
29beb80d | 466 | } |
467 | else | |
468 | { | |
469 | y=3.75/ax; | |
387a745b | 470 | result=(exp(ax)/sqrt(ax)) |
471 | *(kq1+y*(kq2+y*(kq3+y*(kq4+y*(kq5+y*(kq6+y*(kq7+y*(kq8+y*kq9)))))))); | |
29beb80d | 472 | if (x < 0) result=-result; |
473 | } | |
474 | ||
475 | return result; | |
476 | } | |
477 | /////////////////////////////////////////////////////////////////////////// | |
478 | Double_t AliMath::BesselK1(Double_t x) | |
479 | { | |
480 | // Computation of the modified Bessel function K_1(x) for positive real x. | |
481 | // | |
482 | // The algorithm is based on the article by Abramowitz and Stegun [1] | |
483 | // as denoted in Numerical Recipes 2nd ed. on p. 230 (W.H.Press et al.). | |
484 | // | |
485 | // [1] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions, | |
486 | // Applied Mathematics Series vol. 55 (1964), Washington. | |
487 | // | |
488 | //--- NvE 12-mar-2000 UU-SAP Utrecht | |
489 | ||
490 | // Parameters of the polynomial approximation | |
387a745b | 491 | const Double_t kp1= 1., kp2= 0.15443144, kp3=-0.67278579, |
492 | kp4=-0.18156897, kp5=-1.919402e-2, kp6=-1.10404e-3, kp7=-4.686e-5; | |
29beb80d | 493 | |
387a745b | 494 | const Double_t kq1= 1.25331414, kq2= 0.23498619, kq3=-3.655620e-2, |
495 | kq4= 1.504268e-2, kq5=-7.80353e-3, kq6= 3.25614e-3, kq7=-6.8245e-4; | |
29beb80d | 496 | |
497 | if (x <= 0) | |
498 | { | |
499 | cout << " *BesselK1* Invalid argument x = " << x << endl; | |
500 | return 0; | |
501 | } | |
502 | ||
503 | Double_t y=0,result=0; | |
504 | ||
505 | if (x <= 2) | |
506 | { | |
507 | y=x*x/4.; | |
387a745b | 508 | result=(log(x/2.)*BesselI1(x)) |
509 | +(1./x)*(kp1+y*(kp2+y*(kp3+y*(kp4+y*(kp5+y*(kp6+y*kp7)))))); | |
29beb80d | 510 | } |
511 | else | |
512 | { | |
513 | y=2./x; | |
387a745b | 514 | result=(exp(-x)/sqrt(x)) |
515 | *(kq1+y*(kq2+y*(kq3+y*(kq4+y*(kq5+y*(kq6+y*kq7)))))); | |
29beb80d | 516 | } |
517 | ||
518 | return result; | |
519 | } | |
520 | /////////////////////////////////////////////////////////////////////////// | |
521 | Double_t AliMath::BesselK(Int_t n,Double_t x) | |
522 | { | |
523 | // Computation of the Integer Order Modified Bessel function K_n(x) | |
524 | // for n=0,1,2,... and positive real x. | |
525 | // | |
526 | // The algorithm uses the recurrence relation | |
527 | // | |
528 | // K_n+1(x) = (2n/x)*K_n(x) + K_n-1(x) | |
529 | // | |
530 | // as denoted in Numerical Recipes 2nd ed. on p. 232 (W.H.Press et al.). | |
531 | // | |
532 | //--- NvE 12-mar-2000 UU-SAP Utrecht | |
533 | ||
534 | if (x <= 0 || n < 0) | |
535 | { | |
536 | cout << " *BesselK* Invalid argument(s) (n,x) = (" << n << " , " << x << ")" << endl; | |
537 | return 0; | |
538 | } | |
539 | ||
540 | if (n==0) return BesselK0(x); | |
541 | ||
542 | if (n==1) return BesselK1(x); | |
543 | ||
544 | // Perform upward recurrence for all x | |
545 | Double_t tox=2./x; | |
546 | Double_t bkm=BesselK0(x); | |
547 | Double_t bk=BesselK1(x); | |
548 | Double_t bkp=0; | |
549 | for (Int_t j=1; j<n; j++) | |
550 | { | |
551 | bkp=bkm+double(j)*tox*bk; | |
552 | bkm=bk; | |
553 | bk=bkp; | |
554 | } | |
555 | ||
556 | return bk; | |
557 | } | |
558 | /////////////////////////////////////////////////////////////////////////// | |
559 | Double_t AliMath::BesselI(Int_t n,Double_t x) | |
560 | { | |
561 | // Computation of the Integer Order Modified Bessel function I_n(x) | |
562 | // for n=0,1,2,... and any real x. | |
563 | // | |
564 | // The algorithm uses the recurrence relation | |
565 | // | |
566 | // I_n+1(x) = (-2n/x)*I_n(x) + I_n-1(x) | |
567 | // | |
568 | // as denoted in Numerical Recipes 2nd ed. on p. 232 (W.H.Press et al.). | |
569 | // | |
570 | //--- NvE 12-mar-2000 UU-SAP Utrecht | |
571 | ||
572 | Int_t iacc=40; // Increase to enhance accuracy | |
573 | Double_t bigno=1.e10, bigni=1.e-10; | |
574 | ||
575 | if (n < 0) | |
576 | { | |
577 | cout << " *BesselI* Invalid argument (n,x) = (" << n << " , " << x << ")" << endl; | |
578 | return 0; | |
579 | } | |
580 | ||
581 | if (n==0) return BesselI0(x); | |
582 | ||
583 | if (n==1) return BesselI1(x); | |
584 | ||
585 | if (fabs(x) < 1.e-10) return 0; | |
586 | ||
587 | Double_t tox=2./fabs(x); | |
588 | Double_t bip=0,bim=0; | |
589 | Double_t bi=1; | |
590 | Double_t result=0; | |
591 | Int_t m=2*((n+int(sqrt(float(iacc*n))))); // Downward recurrence from even m | |
592 | for (Int_t j=m; j<=1; j--) | |
593 | { | |
594 | bim=bip+double(j)*tox*bi; | |
595 | bip=bi; | |
596 | bi=bim; | |
597 | if (fabs(bi) > bigno) // Renormalise to prevent overflows | |
598 | { | |
599 | result*=bigni; | |
600 | bi*=bigni; | |
601 | bip*=bigni; | |
602 | } | |
603 | if (j==n) result=bip; | |
604 | } | |
605 | ||
606 | result*=BesselI0(x)/bi; // Normalise with I0(x) | |
607 | if ((x < 0) && (n%2 == 1)) result=-result; | |
608 | ||
609 | return result; | |
610 | } | |
611 | /////////////////////////////////////////////////////////////////////////// |