Computation of the Complex Error Function

1994/10/01 by J. A. C. Weideman · 3 citations
Computer Science · Mathematics · #Digital Filter Design and Implementation #Numerical Methods and Algorithms #Numerical methods for differential equations

paper · doi:10.1137/0731077

openalex publication_date 1994/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Rational expansions for computing the complex error function w(z) = e ^{- z2} erfc( - iz) are presented. These expansions have the following attractive properties: (1) they can be evaluated using a polynomial evaluation routine such as Homer’s method, (2) the polynomial coefficients can be computed once and for all by a single Fast Fourier Transform (FFT), and (3) high accuracy is achieved uniformly in the complex plane with only a small number of terms. Comparisons reveal that in some parts of the complex plane certain competitors may be more efficient. However, the difference in efficiency is never great, and the new algorithms are simpler than existing ones: a complete program takes eight lines of Matlab code.

Citations

• Efficient Computation of the Complex Error Function
• Rational Chebyshev approximations for the error function

Cited by

• A rational approximation for the Dawson's integral of real argument
• Quasi-Monte Carlo Feature Maps for Shift-Invariant Kernels
• Representation of the Fourier transform as a weighted sum of the complex error functions

Related

• Efficient Computation of the Complex Error Function
• Computation of the Complex Error Function using Modified Trapezoidal Rules
• Efficient application of the Chiarella and Reichel series approximation of the complex error function
• A rapid and highly accurate approximation for the error function of complex argument
• Evaluation of complex-valued error-like functions by the exponentially-convergent trapezoidal rule