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RooMath.h
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1/*****************************************************************************
2 * Project: RooFit *
3 * Package: RooFitCore *
4 * File: $Id: RooMath.h,v 1.16 2007/05/11 09:11:30 verkerke Exp$
5 * Authors: *
6 * WV, Wouter Verkerke, UC Santa Barbara, verkerke@slac.stanford.edu *
7 * DK, David Kirkby, UC Irvine, dkirkby@uci.edu *
8 * *
9 * Copyright (c) 2000-2005, Regents of the University of California *
10 * and Stanford University. All rights reserved. *
11 * *
12 * Redistribution and use in source and binary forms, *
13 * with or without modification, are permitted according to the terms *
14 * listed in LICENSE (http://roofit.sourceforge.net/license.txt) *
15 *****************************************************************************/
16#ifndef ROO_MATH
17#define ROO_MATH
18
19#include <cmath>
20#include <complex>
21
22#include "Rtypes.h"
23#include "TMath.h"
24
26
27class RooMath {
28public:
29
30 virtual ~RooMath() {};
31
32 /** @brief evaluate Faddeeva function for complex argument
33 *
34 * @author Manuel Schiller <manuel.schiller@nikhef.nl>
35 * @date 2013-02-21
36 *
37 * Calculate the value of the Faddeeva function @f$w(z) = \exp(-z^2) 38 * \mathrm{erfc}(-i z)@f$.
39 *
40 * The method described in
41 *
42 * S.M. Abrarov, B.M. Quine: "Efficient algotithmic implementation of
43 * Voigt/complex error function based on exponential series approximation"
44 * published in Applied Mathematics and Computation 218 (2011) 1894-1902
45 * doi:10.1016/j.amc.2011.06.072
46 *
47 * is used. At the heart of the method (equation (14) of the paper) is the
48 * following Fourier series based approximation:
49 *
50 * @f[ w(z) \approx \frac{i}{2\sqrt{\pi}}\left(
51 * \sum^N_{n=0} a_n \tau_m\left(
52 * \frac{1-e^{i(n\pi+\tau_m z)}}{n\pi + \tau_m z} -
53 * \frac{1-e^{i(-n\pi+\tau_m z)}}{n\pi - \tau_m z}
54 * \right) - a_0 \frac{1-e^{i \tau_m z}}{z}
55 * \right) @f]
56 *
57 * The coefficients @f$a_b@f$ are given by:
58 *
59 * @f[ a_n=\frac{2\sqrt{\pi}}{\tau_m}
60 * \exp\left(-\frac{n^2\pi^2}{\tau_m^2}\right) @f]
61 *
62 * To achieve machine accuracy in double precision floating point arithmetic
63 * for most of the upper half of the complex plane, chose @f$t_m=12@f$ and
64 * @f$N=23@f$ as is done in the paper.
65 *
66 * There are two complications: For Im(z) negative, the exponent in the
67 * equation above becomes so large that the roundoff in the rest of the
68 * calculation is amplified enough that the result cannot be trusted.
69 * Therefore, for Im(z) < 0, the symmetry of the erfc function under the
70 * transformation z --> -z is used to avoid accuracy issues for Im(z) < 0 by
71 * formulating the problem such that the calculation can be done for Im(z) > 0
72 * where the accuracy of the method is fine, and some postprocessing then
73 * yields the desired final result.
74 *
75 * Second, the denominators in the equation above become singular at
76 * @f$z = n * pi / 12@f$ (for 0 <= n < 24). In a tiny disc around these
77 * points, Taylor expansions are used to overcome that difficulty.
78 *
79 * This routine precomputes everything it can, and tries to write out complex
80 * operations to minimise subroutine calls, e.g. for the multiplication of
81 * complex numbers.
82 *
83 * In the square -8 <= Re(z) <= 8, -8 <= Im(z) <= 8, the routine is accurate
84 * to better than 4e-13 relative, the average relative error is better than
85 * 7e-16. On a modern x86_64 machine, the routine is roughly three times as
86 * fast than the old CERNLIB implementation and offers better accuracy.
87 *
88 * For large @f$|z|@f$, the familiar continued fraction approximation
89 *
90 * @f[ w(z)=\frac{-iz/\sqrt{\pi}}{-z^2+\frac{1/2}{1+\frac{2/2}{-z^2 +
91 * \frac{3/2}{1+\frac{4/2}{-z^2+\frac{5/2}{1+\frac{6/2}{-z^2+\frac{7/2
92 * }{1+\frac{8/2}{-z^2+\frac{9/2}{1+\ldots}}}}}}}}}} @f]
93 *
94 * is used, truncated at the ellipsis ("...") in the formula; for @f$|z| > 95 * 12@f$, @f$Im(z)>0@f$ it will give full double precision at a smaller
96 * computational cost than the method described above. (For @f$|z|>12@f$,
97 * @f$Im(z)<0@f$, the symmetry property @f$w(x-iy)=2e^{-(x+iy)^2-w(x+iy)}@f$
98 * is used.
99 */
100 static std::complex<double> faddeeva(std::complex<double> z);
101 /** @brief evaluate Faddeeva function for complex argument (fast version)
102 *
103 * @author Manuel Schiller <manuel.schiller@nikhef.nl>
104 * @date 2013-02-21
105 *
106 * Calculate the value of the Faddeeva function @f$w(z) = \exp(-z^2) 107 * \mathrm{erfc}(-i z)@f$.
108 *
109 * This is the "fast" version of the faddeeva routine above. Fast means that
110 * is takes roughly half the amount of CPU of the slow version of the
111 * routine, but is a little less accurate.
112 *
113 * To be fast, chose @f$t_m=8@f$ and @f$N=11@f$ which should give accuracies
114 * around 1e-7.
115 *
116 * In the square -8 <= Re(z) <= 8, -8 <= Im(z) <= 8, the routine is accurate
117 * to better than 4e-7 relative, the average relative error is better than
118 * 5e-9. On a modern x86_64 machine, the routine is roughly five times as
119 * fast than the old CERNLIB implementation, or about 30% faster than the
120 * interpolation/lookup table based fast method used previously in RooFit,
121 * and offers better accuracy than the latter (the relative error is roughly
122 * a factor 280 smaller than the old interpolation/table lookup routine).
123 *
124 * For large @f$|z|@f$, the familiar continued fraction approximation
125 *
126 * @f[ w(z)=\frac{-iz/\sqrt{\pi}}{-z^2+\frac{1/2}{1+\frac{2/2}{-z^2 +
127 * \frac{3/2}{1+\ldots}}}} @f]
128 *
129 * is used, truncated at the ellipsis ("...") in the formula; for @f$|z| > 130 * 8@f$, @f$Im(z)>0@f$ it will give full float precision at a smaller
131 * computational cost than the method described above. (For @f$|z|>8@f$,
132 * @f$Im(z)<0@f$, the symmetry property @f$w(x-iy)=2e^{-(x+iy)^2-w(x+iy)}@f$
133 * is used.
134 */
135 static std::complex<double> faddeeva_fast(std::complex<double> z);
136
137 /** @brief complex erf function
138 *
139 * @author Manuel Schiller <manuel.schiller@nikhef.nl>
140 * @date 2013-02-21
141 *
142 * Calculate erf(z) for complex z.
143 */
144 static std::complex<double> erf(const std::complex<double> z);
145
146 /** @brief complex erf function (fast version)
147 *
148 * @author Manuel Schiller <manuel.schiller@nikhef.nl>
149 * @date 2013-02-21
150 *
151 * Calculate erf(z) for complex z. Use the code in faddeeva_fast to save some time.
152 */
153 static std::complex<double> erf_fast(const std::complex<double> z);
154 /** @brief complex erfc function
155 *
156 * @author Manuel Schiller <manuel.schiller@nikhef.nl>
157 * @date 2013-02-21
158 *
159 * Calculate erfc(z) for complex z.
160 */
161 static std::complex<double> erfc(const std::complex<double> z);
162 /** @brief complex erfc function (fast version)
163 *
164 * @author Manuel Schiller <manuel.schiller@nikhef.nl>
165 * @date 2013-02-21
166 *
167 * Calculate erfc(z) for complex z. Use the code in faddeeva_fast to save some time.
168 */
169 static std::complex<double> erfc_fast(const std::complex<double> z);
170
171 // 1-D nth order polynomial interpolation routines
172 static Double_t interpolate(Double_t yArr[],Int_t nOrder, Double_t x) ;
173 static Double_t interpolate(Double_t xa[], Double_t ya[], Int_t n, Double_t x) ;
174
175 static inline Double_t erf(Double_t x)
176 { return TMath::Erf(x); }
177
178 static inline Double_t erfc(Double_t x)
179 { return TMath::Erfc(x); }
180
181};
182
183#endif
Double_t * pDouble_t
Definition: RooMath.h:25
int Int_t
Definition: RtypesCore.h:43
double Double_t
Definition: RtypesCore.h:57
static Double_t erfc(Double_t x)
Definition: RooMath.h:178
static std::complex< double > erfc(const std::complex< double > z)
complex erfc function
Definition: RooMath.cxx:556
static std::complex< double > erf(const std::complex< double > z)
complex erf function
Definition: RooMath.cxx:580
static std::complex< double > faddeeva(std::complex< double > z)
evaluate Faddeeva function for complex argument
Definition: RooMath.cxx:542
static Double_t erf(Double_t x)
Definition: RooMath.h:175
virtual ~RooMath()
Definition: RooMath.h:30
static std::complex< double > faddeeva_fast(std::complex< double > z)
evaluate Faddeeva function for complex argument (fast version)
Definition: RooMath.cxx:549
static std::complex< double > erfc_fast(const std::complex< double > z)
complex erfc function (fast version)
Definition: RooMath.cxx:568
static std::complex< double > erf_fast(const std::complex< double > z)
complex erf function (fast version)
Definition: RooMath.cxx:592
static Double_t interpolate(Double_t yArr[], Int_t nOrder, Double_t x)
Definition: RooMath.cxx:605
Double_t x[n]
Definition: legend1.C:17
const Int_t n
Definition: legend1.C:16
Double_t Erf(Double_t x)
Computation of the error function erf(x).
Definition: TMath.cxx:184
Double_t Erfc(Double_t x)
Compute the complementary error function erfc(x).
Definition: TMath.cxx:194