(Difference between revisions)
 Revision as of 04:38, 26 November 2012 (edit)Stevenj (Talk | contribs)← Previous diff Revision as of 16:49, 27 November 2012 (edit)Stevenj (Talk | contribs) Next diff → Line 6: Line 6: * '''erfcx''', the scaled complementary error function $\mathrm{erfcx}(z) = e^{z^2} \mathrm{erfc}(z) \!$ * '''erfcx''', the scaled complementary error function $\mathrm{erfcx}(z) = e^{z^2} \mathrm{erfc}(z) \!$ * '''erfi''', the imaginary error function $\mathrm{erfi}(z) = -i \mathrm{erf}(iz) \!$ * '''erfi''', the imaginary error function $\mathrm{erfi}(z) = -i \mathrm{erf}(iz) \!$ - * '''Dawson''', he [[w:Dawson function|Dawson function]] $\mathrm{Dawson}(z) = \frac{\sqrt{\pi}}{2} e^{-z^2} \mathrm{erfi}(z)$ + * '''Dawson''', the [[w:Dawson function|Dawson function]] $\mathrm{Dawson}(z) = \frac{\sqrt{\pi}}{2} e^{-z^2} \mathrm{erfi}(z)$ Given the Faddeeva function ''w''(''z''), one can also easily compute [[w:Voigt profile|Voigt functions]] and similar related functions as well. In benchmarks of our code, we find that it is comparable to or faster than most competing software for these functions in the complex plane (but we also have special-case optimizations for purely real or imaginary arguments), and we find that the accuracy is typically at at least 13 [[w:Significant figures|significant digits]] in both the real and imaginary parts. Given the Faddeeva function ''w''(''z''), one can also easily compute [[w:Voigt profile|Voigt functions]] and similar related functions as well. In benchmarks of our code, we find that it is comparable to or faster than most competing software for these functions in the complex plane (but we also have special-case optimizations for purely real or imaginary arguments), and we find that the accuracy is typically at at least 13 [[w:Significant figures|significant digits]] in both the real and imaginary parts.

## Revision as of 16:49, 27 November 2012

Steven G. Johnson has written free/open-source C++ code (with wrappers for Matlab, GNU Octave, and Python) to compute the various error functions of arbitrary complex arguments. In particular, we provide:

• w, the Faddeeva function $w(z) = e^{-z^2} \mathrm{erfc}(-iz) \!$, where erfc is the complementary error function.
• erf, the error function $\mathrm{erf}(z) \!$
• erfc, the complementary error function $\mathrm{erfc}(z) = 1 - \mathrm{erf}(z) \!$
• erfcx, the scaled complementary error function $\mathrm{erfcx}(z) = e^{z^2} \mathrm{erfc}(z) \!$
• erfi, the imaginary error function $\mathrm{erfi}(z) = -i \mathrm{erf}(iz) \!$
• Dawson, the Dawson function $\mathrm{Dawson}(z) = \frac{\sqrt{\pi}}{2} e^{-z^2} \mathrm{erfi}(z)$

Given the Faddeeva function w(z), one can also easily compute Voigt functions and similar related functions as well. In benchmarks of our code, we find that it is comparable to or faster than most competing software for these functions in the complex plane (but we also have special-case optimizations for purely real or imaginary arguments), and we find that the accuracy is typically at at least 13 significant digits in both the real and imaginary parts.

Because all of the algorithms are based on algorithms for the Faddeeva function, we call this the Faddeeva Package.

## Usage

To use the code, include the Faddeeva.hh header file:

#include "Faddeeva.hh"


and compile and link the Faddeeva.cc source code. You can then call various functions. For example:

extern std::complex<double> Faddeeva::w(std::complex<double> z, double relerr=0);


This function Faddeeva::w(z, relerr) computes w(z) to a desired relative error relerr.

Omitting the relerr argument, or passing relerr=0 (or any relerr less than machine precision ε≈10−16), corresponds to requesting machine precision, and in practice a relative error < 10−13 is usually achieved. Specifying a larger value of relerr may improve performance for some z (at the expense of accuracy).

Similarly, the erf, erfc, erfcx, erfi, and Dawson functions are computed by calling:

extern std::complex<double> Faddeeva::erf(std::complex<double> z, double relerr=0);
extern std::complex<double> Faddeeva::erfc(std::complex<double> z, double relerr=0);
extern std::complex<double> Faddeeva::erfcx(std::complex<double> z, double relerr=0);
extern std::complex<double> Faddeeva::erfi(std::complex<double> z, double relerr=0);
extern std::complex<double> Faddeeva::Dawson(std::complex<double> z, double relerr=0);


Since these functions are purely real for real arguments z=x, we provide the following specialized interfaces for convenience (and a slight performance gain, although the complex functions above automatically execute specialized code for purely real arguments):

extern double Faddeeva::erf(double x);


(These functions always compute to maximum accuracy, usually near machine precision.)

It is also sometimes useful to compute Im[w(x)] for real x, since $\mathrm{Im}[w(x)] = e^{-x^2} \mathrm{erfi}(x)$ in that case (like the Dawson function but without the √π/2 factor). [Note that Re[w(x)] is simply exp(−x2) for real x.] Im[w(x)] can be computed efficiently to nearly machine precision by calling:

extern double Faddeeva::w_im(double x);


## Wrappers: Matlab, GNU Octave, and Python

Wrappers are available for this function in other languages.

• Matlab (also available here): We provide source code for compiled Matlab plugins (MEX files) to interface all of the error functions above from Matlab.
• The provided functions are called Faddeeva_w, Faddeeva_erf, Faddeeva_erfc, Faddeeva_erfi, Faddeeva_erfcx, and Faddeeva_Dawson, equivalent to the C++ functions above. All have usage of the form w = Faddeeva_w(z) [or w = Faddeeva_w(z, relerr) to pass the optional relative error], to compute the function value from an array or matrix z of complex (or real) inputs.
• For convenience, a script to compile all of the plugins using the mex command in Matlab is included. Assuming you have a C++ compiler installed (and have run mex -setup to tell Matlab to use it), you can simply run the Faddeeva_build.m script in Matlab to compile all of the Faddeeva functions.
• Install the resulting *.mex* files, along with the *.m help files, into your Matlab path
• GNU Octave: Similar to Matlab, above, we provide source code for compiled GNU Octave plugins (.oct files) for all of the error functions above.
• The provided functions are called Faddeeva_w, Faddeeva_erf, Faddeeva_erfc, Faddeeva_erfi, Faddeeva_erfcx, and Faddeeva_Dawson, with usage identical to the Matlab plugins above.
• A Makefile is included. Assuming you have a C++ compiler and the mkoctfile command installed (mkoctfile comes with Octave, possibly in an octave-devel or similarly named package in GNU/Linux distributions), you can simply run make to compile the plugins, and sudo make install to install them system-wide (assuming you have system administrator privileges); otherwise put the compiled .oct files somewhere in your octave path.
• Python: Our code is used to provide scipy.special.erf, scipy.special.wofz, and the other error functions in SciPy starting in version 0.12.0 (see here).

## Algorithms

Our implementation uses a combination of different algorithms, mostly centering around computing the Faddeeva function w(z).

To compute the Faddeeva function for sufficiently large |z|, we use a continued-fraction expansion for w(z) similar to those described in

Unlike those papers, however, we switch to a completely different algorithm for smaller |z| or for z close to the real axis:

(I initially used this algorithm for all z, but the continued-fraction expansion turned out to be faster for larger |z|. On the other hand, Algorithm 916 is competitive or faster for smaller |z|, and appears to be significantly more accurate than the Poppe & Wijers code in some regions, e.g. in the vicinity of |z|=1 [although comparison with other compilers suggests that this may be a problem specific to gfortran]. Algorithm 916 also has better relative accuracy in Re[z] for some regions near the real-z axis. You can switch back to using Algorithm 916 for all z by changing USE_CONTINUED_FRACTION to 0 in the code.)

Note that this is SGJ's independent re-implementation of these algorithms, based on the descriptions in the papers only. In particular, we did not refer to the authors' Fortran or Matlab implementations (respectively), which are under restrictive "semifree" ACM copyright terms and are therefore unusable in free/open-source software.

Algorithm 916 requires an external complementary error function erfc(x) function for real arguments x to be supplied as a subroutine. More precisely, it requires the scaled function erfcx(x) = ex2erfc(x). Here, we use an erfcx routine written by SGJ that uses a combination of two algorithms: a continued-fraction expansion for large x and a lookup table of Chebyshev polynomials for small x. (I initially used an erfcx function derived from the DERFC routine in SLATEC, modified by SGJ to compute erfcx instead of erfc, but the new erfcx routine is much faster, and also seems to be faster than the calerf rational-Chebyshev code by W. J. Cody.)

Similarly, we also implement special-case code for real z, where the imaginary part of w is Dawson's integral. Similar to erfcx, this is also computed by a continued-fraction expansion for large |x|, a lookup table of Chebyshev polynomials for smaller |x|, and finally a Taylor expansion for very small |x|. (This seems to be faster than the dawsn function in the Cephes library, and is substantially faster than the gsl_sf_dawson function in the GNU Scientific Library.)

The other error functions can be computed in terms of w(z). The basic equations are:

$\mathrm{erfcx}(z) = e^{z^2} \mathrm{erfc}(z) = w(iz)$ (scaled complementary error function)
$\mathrm{erfc}(z) = e^{-z^2} w(iz) = \begin{cases} e^{-z^2} w(iz) & \mathrm{Re}\,z \geq 0 \\ 2 - e^{-z^2} w(-iz)) & \mathrm{Re}\,z < 0 \end{cases}$ (complementary error function)
$\mathrm{erf}(z) = 1 - \mathrm{erfc}(z) = \begin{cases} 1 - e^{-z^2} w(iz) & \mathrm{Re}\,z \geq 0 \\ e^{-z^2} w(-iz) - 1 & \mathrm{Re}\,z < 0 \end{cases}$ (error function)
$\mathrm{erfi}(z) = -i\mathrm{erf}(iz) \!$; for real x, $\mathrm{erfi}(x) = e^{x^2} \mathrm{Im}[w(x)] = \frac{\mathrm{Im}[w(x)]}{\mathrm{Re}[w(x)]}$ (imaginary error function)
$\mathrm{Dawson}(z) = \frac{\sqrt{\pi}}{2} e^{-z^2} \mathrm{erfi}(z) = \frac{i\sqrt{\pi}}{2} \begin{cases} e^{-z^2} - w(z) & \mathrm{Re}\,z \geq 0 \\ w(-z) - e^{-z^2} & \mathrm{Re}\,z < 0 \end{cases}$; for real x, $F(x) = \frac{\sqrt{\pi}}{2}\mathrm{Im}[w(x)]$ (Dawson function)

Note that we sometimes employ different equations for positive and negative Re(z) in order to avoid numerical problems arising from multiplying exponentially large and small quantities. For erfi and the Dawson function, there are simplifications that occur for real x as noted. In some cases, however, there are additional complications that require our implementation to go beyond these simple formulas. For erf, large cancellation errors occur in these formulas near |z|=0 where w(z) is nearly 1, as well as near the imaginary axis for Re[erf], and in these regimes we switch to a Taylor expansion. Similarly, for the Dawson function we switch to a Taylor expansion near the origin or near the real axis. (Similar problems occur for erfi, but our erfi implementation simply calls our erf code.)

## Test program

To test the code, a small test program is included at the end of Faddeeva.cc which tests w(z) against several known results (from Wolfram Alpha) and prints the relative errors obtained. To compile the test program, #define TEST_FADDEEVA in the file (or compile with -DTEST_FADDEEVA on Unix) and compile Faddeeva.cc. The resulting program prints SUCCESS at the end of its output if the errors were acceptable.

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