16-feb-2005 NvE Support for user selectable split level and buffer size of the output...
[u/mrichter/AliRoot.git] / RALICE / AliMath.cxx
CommitLineData
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
40ClassImp(AliMath) // Class implementation to enable ROOT I/O
41
c72198f1 42AliMath::AliMath() : TObject()
d88f97cc 43{
44// Default constructor
45}
46///////////////////////////////////////////////////////////////////////////
47AliMath::~AliMath()
48{
49// Destructor
50}
51///////////////////////////////////////////////////////////////////////////
261c0caf 52AliMath::AliMath(const AliMath& m) : TObject(m)
c72198f1 53{
54// Copy constructor
55}
56///////////////////////////////////////////////////////////////////////////
261c0caf 57Double_t AliMath::Gamma(Double_t z) const
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///////////////////////////////////////////////////////////////////////////
261c0caf 78Double_t AliMath::Gamma(Double_t a,Double_t x) const
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///////////////////////////////////////////////////////////////////////////
261c0caf 109Double_t AliMath::LnGamma(Double_t z) const
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///////////////////////////////////////////////////////////////////////////
261c0caf 152Double_t AliMath::GamSer(Double_t a,Double_t x) const
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///////////////////////////////////////////////////////////////////////////
261c0caf 193Double_t AliMath::GamCf(Double_t a,Double_t x) const
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///////////////////////////////////////////////////////////////////////////
261c0caf 243Double_t AliMath::Erf(Double_t x) const
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///////////////////////////////////////////////////////////////////////////
261c0caf 252Double_t AliMath::Erfc(Double_t x) const
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///////////////////////////////////////////////////////////////////////////
261c0caf 286Double_t AliMath::Prob(Double_t chi2,Int_t ndf,Int_t mode) const
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///////////////////////////////////////////////////////////////////////////
261c0caf 356Double_t AliMath::BesselI0(Double_t x) const
29beb80d 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///////////////////////////////////////////////////////////////////////////
261c0caf 395Double_t AliMath::BesselK0(Double_t x) const
29beb80d 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,
7a086578 409 kp4= 3.488590e-2, kp5=2.62698e-3, kp6=1.0750e-4, kp7=7.4e-6;
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///////////////////////////////////////////////////////////////////////////
261c0caf 438Double_t AliMath::BesselI1(Double_t x) const
29beb80d 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///////////////////////////////////////////////////////////////////////////
261c0caf 478Double_t AliMath::BesselK1(Double_t x) const
29beb80d 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///////////////////////////////////////////////////////////////////////////
261c0caf 521Double_t AliMath::BesselK(Int_t n,Double_t x) const
29beb80d 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///////////////////////////////////////////////////////////////////////////
261c0caf 559Double_t AliMath::BesselI(Int_t n,Double_t x) const
29beb80d 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///////////////////////////////////////////////////////////////////////////