Accuracy and stability of inversion of power series

作者：Navarrete Raymundo; Viswanath Divakar^*

来源：IMA Journal of Numerical Analysis, 2016, 36(1): 421-436.

DOI：10.1093/imanum/drv005

• power series
• root deflation
• rounding errors
• pseudozeros

This article considers the numerical inversion of the power series p(x) = 1 + b(1)x + b(2)x(2) + center dot center dot center dot to compute the inverse series q(x) satisfying p(x)q(x) = 1. Numerical inversion is a special case of triangular back-substitution, which has been known for its beguiling numerical stability since the classic work of Wilkinson (1961, Error analysis of direct methods of matrix inversion. J. Assoc. Comput. Mach., 8, 281-330). We prove the numerical stability of inversion of power series and obtain bounds on numerical error. A range of examples show that these bounds overestimate the error by only a few digits. When p(x) is a polynomial and x = a is a root with p(a) = 0, we show that root deflation via the simple division p(x)/(x - a) can trigger instabilities relevant to polynomial root finding and computation of finite-difference weights. When p(x) is a polynomial, the accuracy of the computed inverse q(x) is connected to the pseudozeros of p(x).

• 出版日期2016-1