RooMath.h

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/*****************************************************************************
 * Project: RooFit                                                           *
 * Package: RooFitCore                                                       *
 *    File: $Id: RooMath.h,v 1.16 2007/05/11 09:11:30 verkerke Exp $
 * Authors:                                                                  *
 *   WV, Wouter Verkerke, UC Santa Barbara, verkerke@slac.stanford.edu       *
 *   DK, David Kirkby,    UC Irvine,         dkirkby@uci.edu                 *
 *                                                                           *
 * Copyright (c) 2000-2005, Regents of the University of California          *
 *                          and Stanford University. All rights reserved.    *
 *                                                                           *
 * Redistribution and use in source and binary forms,                        *
 * with or without modification, are permitted according to the terms        *
 * listed in LICENSE (http://roofit.sourceforge.net/license.txt)             *
 *****************************************************************************/
#ifndef ROO_MATH
#define ROO_MATH
 
#include <TMath.h>
 
#include <complex>
 
typedef double *pdouble;
 
class RooMath {
public:
   virtual ~RooMath(){};
 
   /** @brief evaluate Faddeeva function for complex argument
    *
    * @author Manuel Schiller <manuel.schiller@nikhef.nl>
    * @date 2013-02-21
    *
    * Calculate the value of the Faddeeva function @f$w(z) = \exp(-z^2)
    * \mathrm{erfc}(-i z)@f$.
    *
    * The method described in
    *
    * S.M. Abrarov, B.M. Quine: "Efficient algorithmic implementation of
    * Voigt/complex error function based on exponential series approximation"
    * published in Applied Mathematics and Computation 218 (2011) 1894-1902
    * doi:10.1016/j.amc.2011.06.072
    *
    * is used. At the heart of the method (equation (14) of the paper) is the
    * following Fourier series based approximation:
    *
    * @f[ w(z) \approx \frac{i}{2\sqrt{\pi}}\left(
    * \sum^N_{n=nullptr} a_n \tau_m\left(
    * \frac{1-e^{i(n\pi+\tau_m z)}}{n\pi + \tau_m z} -
    * \frac{1-e^{i(-n\pi+\tau_m z)}}{n\pi - \tau_m z}
    * \right) - a_0 \frac{1-e^{i \tau_m z}}{z}
    * \right) @f]
    *
    * The coefficients @f$a_b@f$ are given by:
    *
    * @f[ a_n=\frac{2\sqrt{\pi}}{\tau_m}
    * \exp\left(-\frac{n^2\pi^2}{\tau_m^2}\right) @f]
    *
    * To achieve machine accuracy in double precision floating point arithmetic
    * for most of the upper half of the complex plane, chose @f$t_m=12@f$ and
    * @f$N=23@f$ as is done in the paper.
    *
    * There are two complications: For Im(z) negative, the exponent in the
    * equation above becomes so large that the roundoff in the rest of the
    * calculation is amplified enough that the result cannot be trusted.
    * Therefore, for Im(z) < 0, the symmetry of the erfc function under the
    * transformation z --> -z is used to avoid accuracy issues for Im(z) < 0 by
    * formulating the problem such that the calculation can be done for Im(z) > 0
    * where the accuracy of the method is fine, and some postprocessing then
    * yields the desired final result.
    *
    * Second, the denominators in the equation above become singular at
    * @f$z = n * pi / 12@f$ (for 0 <= n < 24). In a tiny disc around these
    * points, Taylor expansions are used to overcome that difficulty.
    *
    * This routine precomputes everything it can, and tries to write out complex
    * operations to minimise subroutine calls, e.g. for the multiplication of
    * complex numbers.
    *
    * In the square -8 <= Re(z) <= 8, -8 <= Im(z) <= 8, the routine is accurate
    * to better than 4e-13 relative, the average relative error is better than
    * 7e-16. On a modern x86_64 machine, the routine is roughly three times as
    * fast than the old CERNLIB implementation and offers better accuracy.
    *
    * For large @f$|z|@f$, the familiar continued fraction approximation
    *
    * @f[ w(z)=\frac{-iz/\sqrt{\pi}}{-z^2+\frac{1/2}{1+\frac{2/2}{-z^2 +
    * \frac{3/2}{1+\frac{4/2}{-z^2+\frac{5/2}{1+\frac{6/2}{-z^2+\frac{7/2
    * }{1+\frac{8/2}{-z^2+\frac{9/2}{1+\ldots}}}}}}}}}} @f]
    *
    * is used, truncated at the ellipsis ("...") in the formula; for @f$|z| >
    * 12@f$, @f$Im(z)>0@f$ it will give full double precision at a smaller
    * computational cost than the method described above. (For @f$|z|>12@f$,
    * @f$Im(z)<0@f$, the symmetry property @f$w(x-iy)=2e^{-(x+iy)^2-w(x+iy)}@f$
    * is used.
    */
   static std::complex<double> faddeeva(std::complex<double> z);
   /** @brief evaluate Faddeeva function for complex argument (fast version)
    *
    * @author Manuel Schiller <manuel.schiller@nikhef.nl>
    * @date 2013-02-21
    *
    * Calculate the value of the Faddeeva function @f$w(z) = \exp(-z^2)
    * \mathrm{erfc}(-i z)@f$.
    *
    * This is the "fast" version of the faddeeva routine above. Fast means that
    * is takes roughly half the amount of CPU of the slow version of the
    * routine, but is a little less accurate.
    *
    * To be fast, chose @f$t_m=8@f$ and @f$N=11@f$ which should give accuracies
    * around 1e-7.
    *
    * In the square -8 <= Re(z) <= 8, -8 <= Im(z) <= 8, the routine is accurate
    * to better than 4e-7 relative, the average relative error is better than
    * 5e-9. On a modern x86_64 machine, the routine is roughly five times as
    * fast than the old CERNLIB implementation, or about 30% faster than the
    * interpolation/lookup table based fast method used previously in RooFit,
    * and offers better accuracy than the latter (the relative error is roughly
    * a factor 280 smaller than the old interpolation/table lookup routine).
    *
    * For large @f$|z|@f$, the familiar continued fraction approximation
    *
    * @f[ w(z)=\frac{-iz/\sqrt{\pi}}{-z^2+\frac{1/2}{1+\frac{2/2}{-z^2 +
    * \frac{3/2}{1+\ldots}}}} @f]
    *
    * is used, truncated at the ellipsis ("...") in the formula; for @f$|z| >
    * 8@f$, @f$Im(z)>0@f$ it will give full float precision at a smaller
    * computational cost than the method described above. (For @f$|z|>8@f$,
    * @f$Im(z)<0@f$, the symmetry property @f$w(x-iy)=2e^{-(x+iy)^2-w(x+iy)}@f$
    * is used.
    */
   static std::complex<double> faddeeva_fast(std::complex<double> z);
 
   /** @brief complex erf function
    *
    * @author Manuel Schiller <manuel.schiller@nikhef.nl>
    * @date 2013-02-21
    *
    * Calculate erf(z) for complex z.
    */
   static std::complex<double> erf(const std::complex<double> z);
 
   /** @brief complex erf function (fast version)
    *
    * @author Manuel Schiller <manuel.schiller@nikhef.nl>
    * @date 2013-02-21
    *
    * Calculate erf(z) for complex z. Use the code in faddeeva_fast to save some time.
    */
   static std::complex<double> erf_fast(const std::complex<double> z);
   /** @brief complex erfc function
    *
    * @author Manuel Schiller <manuel.schiller@nikhef.nl>
    * @date 2013-02-21
    *
    * Calculate erfc(z) for complex z.
    */
   static std::complex<double> erfc(const std::complex<double> z);
   /** @brief complex erfc function (fast version)
    *
    * @author Manuel Schiller <manuel.schiller@nikhef.nl>
    * @date 2013-02-21
    *
    * Calculate erfc(z) for complex z. Use the code in faddeeva_fast to save some time.
    */
   static std::complex<double> erfc_fast(const std::complex<double> z);
 
   // 1-D nth order polynomial interpolation routines
   static double interpolate(double yArr[], Int_t nOrder, double x);
   static double interpolate(double xa[], double ya[], Int_t n, double x);
 
   static inline double erf(double x) { return TMath::Erf(x); }
 
   static inline double erfc(double x) { return TMath::Erfc(x); }
};
 
#endif