In this paper we propose a method for computing the Faddeeva function $w(z) := e^{-z^2}\mathrm{erfc}(-i z)$ via truncated modified trapezoidal rule approximations to integrals on the real line. Our starting point is the method due to Matta and Reichel (Math. Comp. 25 (1971), pp. 339-344) and Hunter and Regan (Math. Comp. 26 (1972), pp. 339-541). Addressing shortcomings flagged by Weideman (SIAM. J. Numer. Anal. 31 (1994), pp. 1497-1518), we construct approximations which we prove are exponentially convergent as a function of $N+1$, the number of quadrature points, obtaining error bounds which show that accuracies of $2\times 10^{-15}$ in the computation of $w(z)$ throughout the complex plane are achieved with $N = 11$, this confirmed by computations. These approximations, moreover, provably achieve small relative errors throughout the upper complex half-plane where $w(z)$ is non-zero. Numerical tests suggest that this new method is competitive, in accuracy and computation times, with existing methods for computing $w(z)$ for complex $z$.
翻译:在本文中,我们提出了一个计算Faddeeva函数$w(z)的方法 : = e ⁇ - z ⁇ 2 ⁇ mathrm{erfc}(- iz)$ : = e ⁇ - z ⁇ 2 ⁇ mathrem{erfc}(- iz)$, (i z) 。 我们的起点是Matta 和 Reichel (Math. comp.25(1971),pp.339-344) 和Hunter and Regan (Math. comp.26(1972),pp.339-541) 的计算方法。 解决Weideman(SIAM. J. Numer. 31(1994), Anal. (1994) (1994) pp.1497-1518) 所标出的缺点, 我们所建近似的近似近似值以指数指数趋近值为$+1美元, 四重点数, 获得误差界限显示在计算整个复杂平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面, 。