Accuracy and stability of inversion of power series.

This article considers the numerical inversion of the power series p(x)=1 + b1x + b²x² +· · · 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). [ABSTRACT FROM AUTHOR]