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Reference Guide
ProbFuncMathCore.cxx
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1// @(#)root/mathcore:$Id$
2// Authors: L. Moneta, A. Zsenei 06/2005
3
4
5
6#include "Math/Math.h"
7#include "Math/Error.h"
10#include <stdio.h>
11#include <limits>
12using namespace std;
13namespace ROOT {
14namespace Math {
15
16
17
18 static const double kSqrt2 = 1.41421356237309515; // sqrt(2.)
19
20 double beta_cdf_c(double x, double a, double b)
21 {
22 // use the fact that I(x,a,b) = 1. - I(1-x,b,a)
23 return ROOT::Math::inc_beta(1-x, b, a);
24 }
25
26
27 double beta_cdf(double x, double a, double b )
28 {
29 return ROOT::Math::inc_beta(x, a, b);
30 }
31
32
33 double breitwigner_cdf_c(double x, double gamma, double x0)
34 {
35 return 0.5 - std::atan(2.0 * (x-x0) / gamma) / M_PI;
36 }
37
38
39 double breitwigner_cdf(double x, double gamma, double x0)
40 {
41 return 0.5 + std::atan(2.0 * (x-x0) / gamma) / M_PI;
42 }
43
44
45 double cauchy_cdf_c(double x, double b, double x0)
46 {
47 return 0.5 - std::atan( (x-x0) / b) / M_PI;
48 }
49
50
51 double cauchy_cdf(double x, double b, double x0)
52 {
53 return 0.5 + std::atan( (x-x0) / b) / M_PI;
54 }
55
56
57 double chisquared_cdf_c(double x, double r, double x0)
58 {
59 return ROOT::Math::inc_gamma_c ( 0.5 * r , 0.5* (x-x0) );
60 }
61
62
63 double chisquared_cdf(double x, double r, double x0)
64 {
65 return ROOT::Math::inc_gamma ( 0.5 * r , 0.5* (x-x0) );
66 }
67
68
69 double crystalball_cdf(double x, double alpha, double n, double sigma, double mean )
70 {
71 if (n <= 1.) {
72 MATH_ERROR_MSG("crystalball_cdf","CrystalBall cdf not defined for n <=1");
73 return std::numeric_limits<double>::quiet_NaN();
74 }
75
76 double abs_alpha = std::abs(alpha);
77 double C = n/abs_alpha * 1./(n-1.) * std::exp(-alpha*alpha/2.);
78 double D = std::sqrt(M_PI/2.)*(1.+ROOT::Math::erf(abs_alpha/std::sqrt(2.)));
79 double totIntegral = sigma*(C+D);
80
81 double integral = crystalball_integral(x,alpha,n,sigma,mean);
82 return (alpha > 0) ? 1. - integral/totIntegral : integral/totIntegral;
83 }
84 double crystalball_cdf_c(double x, double alpha, double n, double sigma, double mean )
85 {
86 if (n <= 1.) {
87 MATH_ERROR_MSG("crystalball_cdf_c","CrystalBall cdf not defined for n <=1");
88 return std::numeric_limits<double>::quiet_NaN();
89 }
90 double abs_alpha = std::abs(alpha);
91 double C = n/abs_alpha * 1./(n-1.) * std::exp(-alpha*alpha/2.);
92 double D = std::sqrt(M_PI/2.)*(1.+ROOT::Math::erf(abs_alpha/std::sqrt(2.)));
93 double totIntegral = sigma*(C+D);
94
95 double integral = crystalball_integral(x,alpha,n,sigma,mean);
96 return (alpha > 0) ? integral/totIntegral : 1. - (integral/totIntegral);
97 }
98 double crystalball_integral(double x, double alpha, double n, double sigma, double mean)
99 {
100 // compute the integral of the crystal ball function (ROOT::Math::crystalball_function)
101 // If alpha > 0 the integral is the right tail integral.
102 // If alpha < 0 is the left tail integrals which are always finite for finite x.
103 // parameters:
104 // alpha : is non equal to zero, define the # of sigma from which it becomes a power-law function (from mean-alpha*sigma)
105 // n > 1 : is integrer, is the power of the low tail
106 // add a value xmin for cases when n <=1 the integral diverges
107 if (sigma == 0) return 0;
108 if (alpha==0)
109 {
110 MATH_ERROR_MSG("crystalball_integral","CrystalBall function not defined at alpha=0");
111 return 0.;
112 }
113 bool useLog = (n == 1.0);
114 if (n<=0) MATH_WARN_MSG("crystalball_integral","No physical meaning when n<=0");
115
116 double z = (x-mean)/sigma;
117 if (alpha < 0 ) z = -z;
118
119 double abs_alpha = std::abs(alpha);
120
121 //double D = *(1.+ROOT::Math::erf(abs_alpha/std::sqrt(2.)));
122 //double N = 1./(sigma*(C+D));
123 double intgaus = 0.;
124 double intpow = 0.;
125
126 const double sqrtpiover2 = std::sqrt(M_PI/2.);
127 const double sqrt2pi = std::sqrt( 2.*M_PI);
128 const double oneoversqrt2 = 1./sqrt(2.);
129 if (z <= -abs_alpha)
130 {
131 double A = std::pow(n/abs_alpha,n) * std::exp(-0.5 * alpha*alpha);
132 double B = n/abs_alpha - abs_alpha;
133
134 if (!useLog) {
135 double C = (n/abs_alpha) * (1./(n-1)) * std::exp(-alpha*alpha/2.);
136 intpow = C - A /(n-1.) * std::pow(B-z,-n+1) ;
137 }
138 else {
139 // for n=1 the primitive of 1/x is log(x)
140 intpow = -A * std::log( n / abs_alpha ) + A * std::log( B -z );
141 }
142 intgaus = sqrtpiover2*(1.+ROOT::Math::erf(abs_alpha*oneoversqrt2));
143 }
144 else
145 {
146 intgaus = ROOT::Math::gaussian_cdf_c(z, 1);
147 intgaus *= sqrt2pi;
148 intpow = 0;
149 }
150 return sigma * (intgaus + intpow);
151 }
152
153
154 double exponential_cdf_c(double x, double lambda, double x0)
155 {
156 if ((x-x0) < 0) return 1.0;
157 else return std::exp(- lambda * (x-x0));
158 }
159
160
161 double exponential_cdf(double x, double lambda, double x0)
162 {
163 if ((x-x0) < 0) return 0.0;
164 else // use expm1 function to avoid errors at small x
165 return - ROOT::Math::expm1( - lambda * (x-x0) ) ;
166 }
167
168
169 double fdistribution_cdf_c(double x, double n, double m, double x0)
170 {
171 // f distribution is defined only for both n and m > 0
172 if (n < 0 || m < 0) return std::numeric_limits<double>::quiet_NaN();
173
174 double z = m/(m + n*(x-x0));
175 // fox z->1 and large a and b IB looses precision use complement function
176 if (z > 0.9 && n > 1 && m > 1) return 1.- fdistribution_cdf(x,n,m,x0);
177
178 // for the complement use the fact that IB(x,a,b) = 1. - IB(1-x,b,a)
179 return ROOT::Math::inc_beta(m/(m + n*(x-x0)), .5*m, .5*n);
180 }
181
182
183 double fdistribution_cdf(double x, double n, double m, double x0)
184 {
185 // f distribution is defined only for both n and m > 0
186 if (n < 0 || m < 0)
187 return std::numeric_limits<double>::quiet_NaN();
188
189 double z = n*(x-x0)/(m + n*(x-x0));
190 // fox z->1 and large a and b IB looses precision use complement function
191 if (z > 0.9 && n > 1 && m > 1)
192 return 1. - fdistribution_cdf_c(x,n,m,x0);
193
194 return ROOT::Math::inc_beta(z, .5*n, .5*m);
195 }
196
197
198 double gamma_cdf_c(double x, double alpha, double theta, double x0)
199 {
200 return ROOT::Math::inc_gamma_c(alpha, (x-x0)/theta);
201 }
202
203
204 double gamma_cdf(double x, double alpha, double theta, double x0)
205 {
206 return ROOT::Math::inc_gamma(alpha, (x-x0)/theta);
207 }
208
209
210 double lognormal_cdf_c(double x, double m, double s, double x0)
211 {
212 double z = (std::log((x-x0))-m)/(s*kSqrt2);
213 if (z > 1.) return 0.5*ROOT::Math::erfc(z);
214 else return 0.5*(1.0 - ROOT::Math::erf(z));
215 }
216
217
218 double lognormal_cdf(double x, double m, double s, double x0)
219 {
220 double z = (std::log((x-x0))-m)/(s*kSqrt2);
221 if (z < -1.) return 0.5*ROOT::Math::erfc(-z);
222 else return 0.5*(1.0 + ROOT::Math::erf(z));
223 }
224
225
226 double normal_cdf_c(double x, double sigma, double x0)
227 {
228 double z = (x-x0)/(sigma*kSqrt2);
229 if (z > 1.) return 0.5*ROOT::Math::erfc(z);
230 else return 0.5*(1.-ROOT::Math::erf(z));
231 }
232
233
234 double normal_cdf(double x, double sigma, double x0)
235 {
236 double z = (x-x0)/(sigma*kSqrt2);
237 if (z < -1.) return 0.5*ROOT::Math::erfc(-z);
238 else return 0.5*(1.0 + ROOT::Math::erf(z));
239 }
240
241
242 double tdistribution_cdf_c(double x, double r, double x0)
243 {
244 double p = x-x0;
245 double sign = (p>0) ? 1. : -1;
246 return .5 - .5*ROOT::Math::inc_beta(p*p/(r + p*p), .5, .5*r)*sign;
247 }
248
249
250 double tdistribution_cdf(double x, double r, double x0)
251 {
252 double p = x-x0;
253 double sign = (p>0) ? 1. : -1;
254 return .5 + .5*ROOT::Math::inc_beta(p*p/(r + p*p), .5, .5*r)*sign;
255 }
256
257
258 double uniform_cdf_c(double x, double a, double b, double x0)
259 {
260 if ((x-x0) < a) return 1.0;
261 else if ((x-x0) >= b) return 0.0;
262 else return (b-(x-x0))/(b-a);
263 }
264
265
266 double uniform_cdf(double x, double a, double b, double x0)
267 {
268 if ((x-x0) < a) return 0.0;
269 else if ((x-x0) >= b) return 1.0;
270 else return ((x-x0)-a)/(b-a);
271 }
272
273 /// discrete distributions
274
275 double poisson_cdf_c(unsigned int n, double mu)
276 {
277 // mu must be >= 0 . Use poisson - gamma relation
278 // Pr ( x <= n) = Pr( y >= a) where x is poisson and y is gamma distributed ( a = n+1)
279 double a = (double) n + 1.0;
280 return ROOT::Math::gamma_cdf(mu, a, 1.0);
281 }
282
283
284 double poisson_cdf(unsigned int n, double mu)
285 {
286 // mu must be >= 0 . Use poisson - gamma relation
287 // Pr ( x <= n) = Pr( y >= a) where x is poisson and y is gamma distributed ( a = n+1)
288 double a = (double) n + 1.0;
289 return ROOT::Math::gamma_cdf_c(mu, a, 1.0);
290 }
291
292
293 double binomial_cdf_c(unsigned int k, double p, unsigned int n)
294 {
295 // use relation with in beta distribution
296 // p must be 0 <=p <= 1
297 if ( k >= n) return 0;
298 double a = (double) k + 1.0;
299 double b = (double) n - k;
300 return ROOT::Math::beta_cdf(p, a, b);
301 }
302
303
304 double binomial_cdf(unsigned int k, double p, unsigned int n)
305 {
306 // use relation with in beta distribution
307 // p must be 0 <=p <= 1
308 if ( k >= n) return 1.0;
309
310 double a = (double) k + 1.0;
311 double b = (double) n - k;
312 return ROOT::Math::beta_cdf_c(p, a, b);
313 }
314
315
316 double negative_binomial_cdf(unsigned int k, double p, double n)
317 {
318 // use relation with in beta distribution
319 // p must be 0 <=p <= 1
320 if (n < 0) return 0;
321 if (p < 0 || p > 1) return 0;
322 return ROOT::Math::beta_cdf(p, n, k+1.0);
323 }
324
325
326 double negative_binomial_cdf_c(unsigned int k, double p, double n)
327 {
328 // use relation with in beta distribution
329 // p must be 0 <=p <= 1
330 if ( n < 0) return 0;
331 if ( p < 0 || p > 1) return 0;
332 return ROOT::Math::beta_cdf_c(p, n, k+1.0);
333 }
334
335
336 double landau_cdf(double x, double xi, double x0)
337 {
338 // implementation of landau distribution (from DISLAN)
339 //The algorithm was taken from the Cernlib function dislan(G110)
340 //Reference: K.S.Kolbig and B.Schorr, "A program package for the Landau
341 //distribution", Computer Phys.Comm., 31(1984), 97-111
342
343 static double p1[5] = {0.2514091491e+0,-0.6250580444e-1, 0.1458381230e-1,-0.2108817737e-2, 0.7411247290e-3};
344 static double q1[5] = {1.0 ,-0.5571175625e-2, 0.6225310236e-1,-0.3137378427e-2, 0.1931496439e-2};
345
346 static double p2[4] = {0.2868328584e+0, 0.3564363231e+0, 0.1523518695e+0, 0.2251304883e-1};
347 static double q2[4] = {1.0 , 0.6191136137e+0, 0.1720721448e+0, 0.2278594771e-1};
348
349 static double p3[4] = {0.2868329066e+0, 0.3003828436e+0, 0.9950951941e-1, 0.8733827185e-2};
350 static double q3[4] = {1.0 , 0.4237190502e+0, 0.1095631512e+0, 0.8693851567e-2};
351
352 static double p4[4] = {0.1000351630e+1, 0.4503592498e+1, 0.1085883880e+2, 0.7536052269e+1};
353 static double q4[4] = {1.0 , 0.5539969678e+1, 0.1933581111e+2, 0.2721321508e+2};
354
355 static double p5[4] = {0.1000006517e+1, 0.4909414111e+2, 0.8505544753e+2, 0.1532153455e+3};
356 static double q5[4] = {1.0 , 0.5009928881e+2, 0.1399819104e+3, 0.4200002909e+3};
357
358 static double p6[4] = {0.1000000983e+1, 0.1329868456e+3, 0.9162149244e+3,-0.9605054274e+3};
359 static double q6[4] = {1.0 , 0.1339887843e+3, 0.1055990413e+4, 0.5532224619e+3};
360
361 static double a1[4] = {0 ,-0.4583333333e+0, 0.6675347222e+0,-0.1641741416e+1};
362 static double a2[4] = {0 , 1.0 ,-0.4227843351e+0,-0.2043403138e+1};
363
364 double v = (x - x0)/xi;
365 double u;
366 double lan;
367
368 if (v < -5.5)
369 {
370 u = std::exp(v+1);
371 lan = 0.3989422803*std::exp(-1./u)*std::sqrt(u)*(1+(a1[1]+(a1[2]+a1[3]*u)*u)*u);
372 }
373 else if (v < -1 )
374 {
375 u = std::exp(-v-1);
376 lan = (std::exp(-u)/std::sqrt(u))*(p1[0]+(p1[1]+(p1[2]+(p1[3]+p1[4]*v)*v)*v)*v)/
377 (q1[0]+(q1[1]+(q1[2]+(q1[3]+q1[4]*v)*v)*v)*v);
378 }
379 else if (v < 1)
380 lan = (p2[0]+(p2[1]+(p2[2]+p2[3]*v)*v)*v)/(q2[0]+(q2[1]+(q2[2]+q2[3]*v)*v)*v);
381
382 else if (v < 4)
383 lan = (p3[0]+(p3[1]+(p3[2]+p3[3]*v)*v)*v)/(q3[0]+(q3[1]+(q3[2]+q3[3]*v)*v)*v);
384
385 else if (v < 12)
386 {
387 u = 1./v;
388 lan = (p4[0]+(p4[1]+(p4[2]+p4[3]*u)*u)*u)/(q4[0]+(q4[1]+(q4[2]+q4[3]*u)*u)*u);
389 }
390 else if (v < 50)
391 {
392 u = 1./v;
393 lan = (p5[0]+(p5[1]+(p5[2]+p5[3]*u)*u)*u)/(q5[0]+(q5[1]+(q5[2]+q5[3]*u)*u)*u);
394 }
395 else if (v < 300)
396 {
397 u = 1./v;
398 lan = (p6[0]+(p6[1]+(p6[2]+p6[3]*u)*u)*u)/(q6[0]+(q6[1]+(q6[2]+q6[3]*u)*u)*u);
399 }
400 else
401 {
402 u = 1./(v-v*std::log(v)/(v+1));
403 lan = 1-(a2[1]+(a2[2]+a2[3]*u)*u)*u;
404 }
405 return lan;
406 }
407
408
409 double landau_xm1(double x, double xi, double x0)
410 {
411 // implementation of first momentum of Landau distribution
412 // translated from Cernlib (XM1LAN function) by Benno List
413
414 static double p1[5] = {-0.8949374280E+0, 0.4631783434E+0,-0.4053332915E-1,
415 0.1580075560E-1,-0.3423874194E-2};
416 static double q1[5] = { 1.0 , 0.1002930749E+0, 0.3575271633E-1,
417 -0.1915882099E-2, 0.4811072364E-4};
418 static double p2[5] = {-0.8933384046E+0, 0.1161296496E+0, 0.1200082940E+0,
419 0.2185699725E-1, 0.2128892058E-2};
420 static double q2[5] = { 1.0 , 0.4935531886E+0, 0.1066347067E+0,
421 0.1250161833E-1, 0.5494243254E-3};
422 static double p3[5] = {-0.8933322067E+0, 0.2339544896E+0, 0.8257653222E-1,
423 0.1411226998E-1, 0.2892240953E-3};
424 static double q3[5] = { 1.0 , 0.3616538408E+0, 0.6628026743E-1,
425 0.4839298984E-2, 0.5248310361E-4};
426 static double p4[4] = { 0.9358419425E+0, 0.6716831438E+2,-0.6765069077E+3,
427 0.9026661865E+3};
428 static double q4[4] = { 1.0 , 0.7752562854E+2,-0.5637811998E+3,
429 -0.5513156752E+3};
430 static double p5[4] = { 0.9489335583E+0, 0.5561246706E+3, 0.3208274617E+5,
431 -0.4889926524E+5};
432 static double q5[4] = { 1.0 , 0.6028275940E+3, 0.3716962017E+5,
433 0.3686272898E+5};
434 static double a0[6] = {-0.4227843351E+0,-0.1544313298E+0, 0.4227843351E+0,
435 0.3276496874E+1, 0.2043403138E+1,-0.8681296500E+1};
436 static double a1[4] = { 0, -0.4583333333E+0, 0.6675347222E+0,
437 -0.1641741416E+1};
438 static double a2[5] = { 0, -0.1958333333E+1, 0.5563368056E+1,
439 -0.2111352961E+2, 0.1006946266E+3};
440
441 double v = (x-x0)/xi;
442 double xm1lan;
443 if (v < -4.5)
444 {
445 double u = std::exp(v+1);
446 xm1lan = v-u*(1+(a2[1]+(a2[2]+(a2[3]+a2[4]*u)*u)*u)*u)/
447 (1+(a1[1]+(a1[2]+a1[3]*u)*u)*u);
448 }
449 else if (v < -2)
450 {
451 xm1lan = (p1[0]+(p1[1]+(p1[2]+(p1[3]+p1[4]*v)*v)*v)*v)/
452 (q1[0]+(q1[1]+(q1[2]+(q1[3]+q1[4]*v)*v)*v)*v);
453 }
454 else if (v < 2)
455 {
456 xm1lan = (p2[0]+(p2[1]+(p2[2]+(p2[3]+p2[4]*v)*v)*v)*v)/
457 (q2[0]+(q2[1]+(q2[2]+(q2[3]+q2[4]*v)*v)*v)*v);
458 }
459 else if (v < 10)
460 {
461 xm1lan = (p3[0]+(p3[1]+(p3[2]+(p3[3]+p3[4]*v)*v)*v)*v)/
462 (q3[0]+(q3[1]+(q3[2]+(q3[3]+q3[4]*v)*v)*v)*v);
463 }
464 else if (v < 40)
465 {
466 double u = 1/v;
467 xm1lan = std::log(v)*(p4[0]+(p4[1]+(p4[2]+p4[3]*u)*u)*u)/
468 (q4[0]+(q4[1]+(q4[2]+q4[3]*u)*u)*u);
469 }
470 else if (v < 200)
471 {
472 double u = 1/v;
473 xm1lan = std::log(v)*(p5[0]+(p5[1]+(p5[2]+p5[3]*u)*u)*u)/
474 (q5[0]+(q5[1]+(q5[2]+q5[3]*u)*u)*u);
475 }
476 else
477 {
478 double u = v-v*std::log(v)/(v+1);
479 v = 1/(u-u*(u+ std::log(u)-v)/(u+1));
480 u = -std::log(v);
481 xm1lan = (u+a0[0]+(-u+a0[1]+(a0[2]*u+a0[3]+(a0[4]*u+a0[5])*v)*v)*v)/
482 (1-(1-(a0[2]+a0[4]*v)*v)*v);
483 }
484 return xm1lan*xi + x0;
485 }
486
487
488
489 double landau_xm2(double x, double xi, double x0)
490 {
491 // implementation of second momentum of Landau distribution
492 // translated from Cernlib (XM2LAN function) by Benno List
493
494 static double p1[5] = { 0.1169837582E+1,-0.4834874539E+0, 0.4383774644E+0,
495 0.3287175228E-2, 0.1879129206E-1};
496 static double q1[5] = { 1.0 , 0.1795154326E+0, 0.4612795899E-1,
497 0.2183459337E-2, 0.7226623623E-4};
498 static double p2[5] = { 0.1157939823E+1,-0.3842809495E+0, 0.3317532899E+0,
499 0.3547606781E-1, 0.6725645279E-2};
500 static double q2[5] = { 1.0 , 0.2916824021E+0, 0.5259853480E-1,
501 0.3840011061E-2, 0.9950324173E-4};
502 static double p3[4] = { 0.1178191282E+1, 0.1011623342E+2,-0.1285585291E+2,
503 0.3641361437E+2};
504 static double q3[4] = { 1.0 , 0.8614160194E+1, 0.3118929630E+2,
505 0.1514351300E+0};
506 static double p4[5] = { 0.1030763698E+1, 0.1216758660E+3, 0.1637431386E+4,
507 -0.2171466507E+4, 0.7010168358E+4};
508 static double q4[5] = { 1.0 , 0.1022487911E+3, 0.1377646350E+4,
509 0.3699184961E+4, 0.4251315610E+4};
510 static double p5[4] = { 0.1010084827E+1, 0.3944224824E+3, 0.1773025353E+5,
511 -0.7075963938E+5};
512 static double q5[4] = { 1.0 , 0.3605950254E+3, 0.1392784158E+5,
513 -0.1881680027E+5};
514 static double a0[7] = {-0.2043403138E+1,-0.8455686702E+0,-0.3088626596E+0,
515 0.5821346754E+1, 0.4227843351E+0, 0.6552993748E+1,
516 -0.1076714945E+2};
517 static double a1[4] = { 0. ,-0.4583333333E+0, 0.6675347222E+0,
518 -0.1641741416E+1};
519 static double a2[4] = {-0.1958333333E+1, 0.5563368056E+1,-0.2111352961E+2,
520 0.1006946266E+3};
521 static double a3[4] = {-1.0 , 0.4458333333E+1,-0.2116753472E+2,
522 0.1163674359E+3};
523
524 double v = (x-x0)/xi;
525 double xm2lan;
526 if (v < -4.5)
527 {
528 double u = std::exp(v+1);
529 xm2lan = v*v-2*u*u*
530 (v/u+a2[0]*v+a3[0]+(a2[1]*v+a3[1]+(a2[2]*v+a3[2]+
531 (a2[3]*v+a3[3])*u)*u)*u)/
532 (1+(a1[1]+(a1[2]+a1[3]*u)*u)*u);
533 }
534 else if (v < -2)
535 {
536 xm2lan = (p1[0]+(p1[1]+(p1[2]+(p1[3]+p1[4]*v)*v)*v)*v)/
537 (q1[0]+(q1[1]+(q1[2]+(q1[3]+q1[4]*v)*v)*v)*v);
538 }
539 else if (v < 2)
540 {
541 xm2lan = (p2[0]+(p2[1]+(p2[2]+(p2[3]+p2[4]*v)*v)*v)*v)/
542 (q2[0]+(q2[1]+(q2[2]+(q2[3]+q2[4]*v)*v)*v)*v);
543 }
544 else if (v < 5)
545 {
546 double u = 1/v;
547 xm2lan = v*(p3[0]+(p3[1]+(p3[2]+p3[3]*u)*u)*u)/
548 (q3[0]+(q3[1]+(q3[2]+q3[3]*u)*u)*u);
549 }
550 else if (v < 50)
551 {
552 double u = 1/v;
553 xm2lan = v*(p4[0]+(p4[1]+(p4[2]+(p4[3]+p4[4]*u)*u)*u)*u)/
554 (q4[0]+(q4[1]+(q4[2]+(q4[3]+q4[4]*u)*u)*u)*u);
555 }
556 else if (v < 200)
557 {
558 double u = 1/v;
559 xm2lan = v*(p5[0]+(p5[1]+(p5[2]+p5[3]*u)*u)*u)/
560 (q5[0]+(q5[1]+(q5[2]+q5[3]*u)*u)*u);
561 }
562 else
563 {
564 double u = v-v*std::log(v)/(v+1);
565 v = 1/(u-u*(u+log(u)-v)/(u+1));
566 u = -std::log(v);
567 xm2lan = (1/v+u*u+a0[0]+a0[1]*u+(-u*u+a0[2]*u+a0[3]+
568 (a0[4]*u*u+a0[5]*u+a0[6])*v)*v)/(1-(1-a0[4]*v)*v);
569 }
570 if (x0 == 0) return xm2lan*xi*xi;
571 double xm1lan = ROOT::Math::landau_xm1(x, xi, x0);
572 return xm2lan*xi*xi + (2*xm1lan-x0)*x0;
573 }
574
575} // namespace Math
576} // namespace ROOT
577
578
579
double
Definition: Converters.cxx:939
#define MATH_ERROR_MSG(loc, str)
Definition: Error.h:83
#define MATH_WARN_MSG(loc, str)
Definition: Error.h:80
ROOT::R::TRInterface & r
Definition: Object.C:4
#define b(i)
Definition: RSha256.hxx:100
#define M_PI
Definition: Rotated.cxx:105
double pow(double, double)
double atan(double)
double sqrt(double)
double exp(double)
double log(double)
double poisson_cdf(unsigned int n, double mu)
Cumulative distribution function of the Poisson distribution Lower tail of the integral of the poisso...
double crystalball_integral(double x, double alpha, double n, double sigma, double x0=0)
Integral of the not-normalized Crystal Ball function.
double uniform_cdf(double x, double a, double b, double x0=0)
Cumulative distribution function of the uniform (flat) distribution (lower tail).
double binomial_cdf_c(unsigned int k, double p, unsigned int n)
Complement of the cumulative distribution function of the Binomial distribution.
double lognormal_cdf(double x, double m, double s, double x0=0)
Cumulative distribution function of the lognormal distribution (lower tail).
double cauchy_cdf_c(double x, double b, double x0=0)
Complement of the cumulative distribution function (upper tail) of the Cauchy distribution which is a...
double fdistribution_cdf(double x, double n, double m, double x0=0)
Cumulative distribution function of the F-distribution (lower tail).
double binomial_cdf(unsigned int k, double p, unsigned int n)
Cumulative distribution function of the Binomial distribution Lower tail of the integral of the binom...
double landau_cdf(double x, double xi=1, double x0=0)
Cumulative distribution function of the Landau distribution (lower tail).
double lognormal_cdf_c(double x, double m, double s, double x0=0)
Complement of the cumulative distribution function of the lognormal distribution (upper tail).
double uniform_cdf_c(double x, double a, double b, double x0=0)
Complement of the cumulative distribution function of the uniform (flat) distribution (upper tail).
double fdistribution_cdf_c(double x, double n, double m, double x0=0)
Complement of the cumulative distribution function of the F-distribution (upper tail).
double crystalball_cdf_c(double x, double alpha, double n, double sigma, double x0=0)
Complement of the Cumulative distribution for the Crystal Ball distribution.
double normal_cdf_c(double x, double sigma=1, double x0=0)
Complement of the cumulative distribution function of the normal (Gaussian) distribution (upper tail)...
double chisquared_cdf(double x, double r, double x0=0)
Cumulative distribution function of the distribution with degrees of freedom (lower tail).
double negative_binomial_cdf_c(unsigned int k, double p, double n)
Complement of the cumulative distribution function of the Negative Binomial distribution.
double gamma_cdf_c(double x, double alpha, double theta, double x0=0)
Complement of the cumulative distribution function of the gamma distribution (upper tail).
double beta_cdf(double x, double a, double b)
Cumulative distribution function of the beta distribution Upper tail of the integral of the beta_pdf.
double crystalball_cdf(double x, double alpha, double n, double sigma, double x0=0)
Cumulative distribution for the Crystal Ball distribution function.
double negative_binomial_cdf(unsigned int k, double p, double n)
Cumulative distribution function of the Negative Binomial distribution Lower tail of the integral of ...
double breitwigner_cdf_c(double x, double gamma, double x0=0)
Complement of the cumulative distribution function (upper tail) of the Breit_Wigner distribution and ...
double normal_cdf(double x, double sigma=1, double x0=0)
Cumulative distribution function of the normal (Gaussian) distribution (lower tail).
double breitwigner_cdf(double x, double gamma, double x0=0)
Cumulative distribution function (lower tail) of the Breit_Wigner distribution and it is similar (jus...
double exponential_cdf_c(double x, double lambda, double x0=0)
Complement of the cumulative distribution function of the exponential distribution (upper tail).
double tdistribution_cdf(double x, double r, double x0=0)
Cumulative distribution function of Student's t-distribution (lower tail).
double beta_cdf_c(double x, double a, double b)
Complement of the cumulative distribution function of the beta distribution.
double poisson_cdf_c(unsigned int n, double mu)
Complement of the cumulative distribution function of the Poisson distribution.
double chisquared_cdf_c(double x, double r, double x0=0)
Complement of the cumulative distribution function of the distribution with degrees of freedom (upp...
double cauchy_cdf(double x, double b, double x0=0)
Cumulative distribution function (lower tail) of the Cauchy distribution which is also Lorentzian dis...
double tdistribution_cdf_c(double x, double r, double x0=0)
Complement of the cumulative distribution function of Student's t-distribution (upper tail).
double gamma_cdf(double x, double alpha, double theta, double x0=0)
Cumulative distribution function of the gamma distribution (lower tail).
double exponential_cdf(double x, double lambda, double x0=0)
Cumulative distribution function of the exponential distribution (lower tail).
double inc_gamma_c(double a, double x)
Calculates the normalized (regularized) upper incomplete gamma function (upper integral)
double inc_beta(double x, double a, double b)
Calculates the normalized (regularized) incomplete beta function.
double erfc(double x)
Complementary error function.
double inc_gamma(double a, double x)
Calculates the normalized (regularized) lower incomplete gamma function (lower integral)
double erf(double x)
Error function encountered in integrating the normal distribution.
double landau_xm1(double x, double xi=1, double x0=0)
First moment (mean) of the truncated Landau distribution.
double landau_xm2(double x, double xi=1, double x0=0)
Second moment of the truncated Landau distribution.
const Double_t sigma
Double_t x[n]
Definition: legend1.C:17
const Int_t n
Definition: legend1.C:16
Namespace for new Math classes and functions.
static double B[]
static double A[]
static double C[]
double gamma(double x)
double expm1(double x)
exp(x) -1 with error cancellation when x is small
Definition: Math.h:110
VecExpr< UnaryOp< Sqrt< T >, VecExpr< A, T, D >, T >, T, D > sqrt(const VecExpr< A, T, D > &rhs)
double gaussian_cdf_c(double x, double sigma=1, double x0=0)
Alternative name for same function.
static const double kSqrt2
tbb::task_arena is an alias of tbb::interface7::task_arena, which doesn't allow to forward declare tb...
static constexpr double s
auto * m
Definition: textangle.C:8
auto * a
Definition: textangle.C:12