论文标题
使用修改的梯形规则计算复杂错误函数
Computation of the Complex Error Function using Modified Trapezoidal Rules
论文作者
论文摘要
在本文中，我们提出了一种用于计算Faddeeva函数$ W（z）的方法：= E^{ - z^2} \ MathRM {erfc}（ - I z）$通过截断的修改的梯形规则近似值近似于真实线上的积分。我们的起点是由于Matta和Reichel引起的方法（Math。Comp。25（1971），第339-344页）和Hunter and Regan（Math。Comp。26（1972），第339-541页）。解决Weideman（Siam。J。Numer。Anal。31（1994），第1497-1518页）标记的缺点，我们构建了近似值，我们证明这些近似值是$ n+1 $的函数，该函数是$ n+1 $的函数，该函数的数量是正交点的数量，并显示了整个$ 2 \ times $ 2 \ times times $ 10^$ 15^$ 15^$ 15}的准确误差界限使用$ n = 11 $，这是通过计算确认的。此外，这些近似值在整个上层复杂的半平面中实现了小的相对错误，其中$ w（z）$是非零的。数值测试表明，这种新方法在准确性和计算时间上具有竞争力，并具有用于计算复杂$ z $的$ W（z）$的现有方法。
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 $z$.
