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ROOT 6.12/07 Reference Guide |
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ROOT 6.12/07 Reference Guide |
Public Member Functions | |
| virtual | ~RooMath () |
Static Public Member Functions | |
| static std::complex< double > | erf (const std::complex< double > z) |
| complex erf function More... | |
| static Double_t | erf (Double_t x) |
| static std::complex< double > | erf_fast (const std::complex< double > z) |
| complex erf function (fast version) More... | |
| static std::complex< double > | erfc (const std::complex< double > z) |
| complex erfc function More... | |
| static Double_t | erfc (Double_t x) |
| static std::complex< double > | erfc_fast (const std::complex< double > z) |
| complex erfc function (fast version) More... | |
| static std::complex< double > | faddeeva (std::complex< double > z) |
| evaluate Faddeeva function for complex argument More... | |
| static std::complex< double > | faddeeva_fast (std::complex< double > z) |
| evaluate Faddeeva function for complex argument (fast version) More... | |
| static Double_t | interpolate (Double_t yArr[], Int_t nOrder, Double_t x) |
| static Double_t | interpolate (Double_t xa[], Double_t ya[], Int_t n, Double_t x) |
#include <RooMath.h>
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complex erf function
Calculate erf(z) for complex z.
Definition at line 580 of file RooMath.cxx.
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complex erf function (fast version)
Calculate erf(z) for complex z. Use the code in faddeeva_fast to save some time.
Definition at line 592 of file RooMath.cxx.
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complex erfc function
Calculate erfc(z) for complex z.
Definition at line 556 of file RooMath.cxx.
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complex erfc function (fast version)
Calculate erfc(z) for complex z. Use the code in faddeeva_fast to save some time.
Definition at line 568 of file RooMath.cxx.
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evaluate Faddeeva function for complex argument
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 algotithmic 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 542 of file RooMath.cxx.
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evaluate Faddeeva function for complex argument (fast version)
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 549 of file RooMath.cxx.
Definition at line 605 of file RooMath.cxx.
Definition at line 641 of file RooMath.cxx.
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