Faddeeva Package

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Faddeeva / complex error function

Steven G. Johnson has written free/open-source C++ code to compute the scaled complex error function w(z) = ez2erfc(−iz), also called the Faddeeva function (and also the plasma dispersion function), for arbitrary complex arguments z to a given accuracy. Download the source code from:

Usage

To use the code, add the following declaration to your C++ source (or header file):

#include <complex>
extern std::complex<double> Faddeeva_w(std::complex<double> z, double relerr=0);

The 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 generally improves performance (at the expense of accuracy).

You should also compile Faddeeva_w.cc and link it with your program, of course.

Algorithm

This implementation uses a combination of different algorithms. 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|:

(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, it is competitive for smaller |z|, and is significantly more accurate than the Poppe & Wijers code in some regions, e.g. in the vicinity of |z|=1.)

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 include an erfcx function derived from the DERFC routine in SLATEC (modified by SGJ to compute erfcx instead of erfc), originally written by W. Fullerton at Los Alamos National Laboratory.

Test program

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

License

The software is distributed under the "MIT License", a simple permissive free/open-source license:

Copyright © 2012 Massachusetts Institute of Technology
[Also included are functions derived from derfc in SLATEC (netlib.org/slatec), which "is in the public domain" and hence may be redistributed under these or any terms.]
Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
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