SIAM Journal on Numerical Analysis, Volume 59, Issue 5, Page 2346-2367, January 2021.
In this paper we propose a method for computing the Faddeeva function $w(z) := {e}^{-z^2}{erfc}(-{i}\,z)$ 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 is confirmed by computations. These approximations, moreover, provably achieve small relative errors throughout the upper complex half-plane where $w(z)$ is nonzero. Numerical tests suggest that this new method is competitive, in accuracy and computation times, with existing methods for computing $w(z)$ for complex $z$.

中文翻译：

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使用修正梯形规则计算复杂误差函数

SIAM 数值分析杂志，第 59 卷，第 5 期，第 2346-2367 页，2021 年 1 月。
在本文中，我们提出了一种计算 Faddeeva 函数 $w(z) := {e}^{-z^2}{erfc}(-{i}\,z)$ 的方法，通过截断修正梯形规则逼近积分在真正的线上。我们的出发点是 Matta 和 Reichel [Math. Comp., 25 (1971), pp. 339--344] 和 Hunter 和 Regan [Math. Comp., 26 (1972), pp. 339--541]。解决 Weideman [SIAM. J. 数字。Anal., 31 (1994), pp. 1497--1518]，我们构造了我们证明作为 $N+1$ 的函数呈指数收敛的近似值，正交点的数量，获得误差界限，表明 $2 的精度\times 10^{-15}$ 在整个复平面的 $w(z)$ 计算中是通过 $N = 11$ 实现的；计算证实了这一点。此外，这些近似值，可证明在 $w(z)$ 非零的整个上复半平面中实现较小的相对误差。数值测试表明，这种新方法在准确性和计算时间方面与现有的计算复杂 $z$ 的 $w(z)$ 方法相比具有竞争力。