ROOT   Reference Guide
RooMath Class Reference

Definition at line 25 of file RooMath.h.

## Public Member Functions

virtual ~RooMath ()

## Static Public Member Functions

static std::complex< doubleerf (const std::complex< double > z)
complex erf function More...

static double erf (double x)

static std::complex< doubleerf_fast (const std::complex< double > z)
complex erf function (fast version) More...

static std::complex< doubleerfc (const std::complex< double > z)
complex erfc function More...

static double erfc (double x)

static std::complex< doubleerfc_fast (const std::complex< double > z)
complex erfc function (fast version) More...

static std::complex< doublefaddeeva (std::complex< double > z)
evaluate Faddeeva function for complex argument More...

static std::complex< doublefaddeeva_fast (std::complex< double > z)
evaluate Faddeeva function for complex argument (fast version) More...

static double interpolate (double xa[], double ya[], Int_t n, double x)

static double interpolate (double yArr[], Int_t nOrder, double x)

#include <RooMath.h>

## ◆ ~RooMath()

 virtual RooMath::~RooMath ( )
inlinevirtual

Definition at line 27 of file RooMath.h.

## ◆ erf() [1/2]

 std::complex< double > RooMath::erf ( const std::complex< double > z )
static

complex erf function

Date
2013-02-21

Calculate erf(z) for complex z.

Definition at line 60 of file RooMath.cxx.

## ◆ erf() [2/2]

 static double RooMath::erf ( double x )
inlinestatic

Definition at line 172 of file RooMath.h.

## ◆ erf_fast()

 std::complex< double > RooMath::erf_fast ( const std::complex< double > z )
static

complex erf function (fast version)

Date
2013-02-21

Calculate erf(z) for complex z. Use the code in faddeeva_fast to save some time.

Definition at line 69 of file RooMath.cxx.

## ◆ erfc() [1/2]

 std::complex< double > RooMath::erfc ( const std::complex< double > z )
static

complex erfc function

Date
2013-02-21

Calculate erfc(z) for complex z.

Definition at line 41 of file RooMath.cxx.

## ◆ erfc() [2/2]

 static double RooMath::erfc ( double x )
inlinestatic

Definition at line 174 of file RooMath.h.

## ◆ erfc_fast()

 std::complex< double > RooMath::erfc_fast ( const std::complex< double > z )
static

complex erfc function (fast version)

Date
2013-02-21

Calculate erfc(z) for complex z. Use the code in faddeeva_fast to save some time.

Definition at line 50 of file RooMath.cxx.

 std::complex< double > RooMath::faddeeva ( std::complex< double > z )
static

evaluate Faddeeva function for complex argument

Date
2013-02-21

Calculate the value of the Faddeeva function $$w(z) = \exp(-z^2) \mathrm{erfc}(-i z)$$.

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:

$w(z) \approx \frac{i}{2\sqrt{\pi}}\left( \sum^N_{n=0} a_n \tau_m\left( \frac{1-e^{i(n\pi+\tau_m z)}}{n\pi + \tau_m z} - \frac{1-e^{i(-n\pi+\tau_m z)}}{n\pi - \tau_m z} \right) - a_0 \frac{1-e^{i \tau_m z}}{z} \right)$

The coefficients $$a_b$$ are given by:

$a_n=\frac{2\sqrt{\pi}}{\tau_m} \exp\left(-\frac{n^2\pi^2}{\tau_m^2}\right)$

To achieve machine accuracy in double precision floating point arithmetic for most of the upper half of the complex plane, chose $$t_m=12$$ and $$N=23$$ 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 $$z = n * pi / 12$$ (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 $$|z|$$, the familiar continued fraction approximation

$w(z)=\frac{-iz/\sqrt{\pi}}{-z^2+\frac{1/2}{1+\frac{2/2}{-z^2 + \frac{3/2}{1+\frac{4/2}{-z^2+\frac{5/2}{1+\frac{6/2}{-z^2+\frac{7/2 }{1+\frac{8/2}{-z^2+\frac{9/2}{1+\ldots}}}}}}}}}}$

is used, truncated at the ellipsis ("...") in the formula; for $$|z| > 12$$, $$Im(z)>0$$ it will give full double precision at a smaller computational cost than the method described above. (For $$|z|>12$$, $$Im(z)<0$$, the symmetry property $$w(x-iy)=2e^{-(x+iy)^2-w(x+iy)}$$ is used.

Definition at line 31 of file RooMath.cxx.

 std::complex< double > RooMath::faddeeva_fast ( std::complex< double > z )
static

evaluate Faddeeva function for complex argument (fast version)

Date
2013-02-21

Calculate the value of the Faddeeva function $$w(z) = \exp(-z^2) \mathrm{erfc}(-i z)$$.

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 $$t_m=8$$ and $$N=11$$ 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 $$|z|$$, the familiar continued fraction approximation

$w(z)=\frac{-iz/\sqrt{\pi}}{-z^2+\frac{1/2}{1+\frac{2/2}{-z^2 + \frac{3/2}{1+\ldots}}}}$

is used, truncated at the ellipsis ("...") in the formula; for $$|z| > 8$$, $$Im(z)>0$$ it will give full float precision at a smaller computational cost than the method described above. (For $$|z|>8$$, $$Im(z)<0$$, the symmetry property $$w(x-iy)=2e^{-(x+iy)^2-w(x+iy)}$$ is used.

Definition at line 36 of file RooMath.cxx.

## ◆ interpolate() [1/2]

 double RooMath::interpolate ( double xa[], double ya[], Int_t n, double x )
static

Definition at line 113 of file RooMath.cxx.

## ◆ interpolate() [2/2]

 double RooMath::interpolate ( double yArr[], Int_t nOrder, double x )
static

Definition at line 79 of file RooMath.cxx.

The documentation for this class was generated from the following files: