Error Functions – Approximations and Implementations with modern FORTRAN

The Error Function is used in many areas of mathematics, statistics, science and scientific applications such as error probability in signal detection, option pricing, diffusion, heat equation, modeling of magnetization, transitions between two planes, nonlinearities in the amplifier, rubbery materials and soft tissue. Error Functions, Volume I: erf, erfc and erfcx is the first comprehensive collection of multi-precision implementations in modern module-oriented FORTRAN for GFortran and Silverfrost FTN 95. The accuracy of many approximate values is up to 32 digits. New approximations, especially for the scaled complementary error function (erfcx), show better performance than some standard intrinsic functions. The implementation methods are mainly rational functions, Chebyshev series and rational Chebyshev functions, series expansions and continued fractions. A menu-driven test program guides through the various error functions. The complete source code of all functions and the test program is available at https://cuvillier.de/de/error-functions