Efficient multiple-precision computation of the scaled complementary error function and the Dawson integral
Mofreh R. Zaghloul
Abstract
We present algorithms to approximate the scaled complementary error function, $$\mathit{exp}\left({x}^{2}\right)erfc(x)$$ , and the Dawson integral, $${e}^{{-x}^{2}}\underset{0}{\overset{x}{\int }}{e}^{{t}^{2}}dt$$ , to the best accuracy in the standard single, double, and quadruple precision arithmetic. The algorithms are based on expansion in Chebyshev subinterval polynomial approximations together with expansion in terms of Taylor series and/or Laplace continued fraction. The present algorithms, implemented as Fortran elemental modules, have been benchmarked versus competitive algorithms available in the literature and versus functions built-in in modern Fortran compilers, in addition to comprehensive tables generated with variable precision computations using the Matlab™ symbolic toolbox. The present algorithm for calculating the scaled complementary error function showed an overall significant efficiency improvement (factors between 1.3 and 20 depending on the compiler and tested dataset) compared to the built-in function “Erfc_Scaled” in modern Fortran compilers, whereas the algorithm for calculating the Dawson integral is exceptional in calculating the function to 32 significant digits (compared to 19 significant digits reported in the literature) while being more efficient than competitive algorithms as well.