Accuracy and stability of inversion of power series

TL;DR

This paper analyzes the numerical stability of inverting power series and provides bounds on errors, highlighting issues in polynomial root deflation and connections to pseudozeros.

Contribution

It proves the stability of power series inversion, derives error bounds, and discusses implications for polynomial root finding and pseudozeros.

Findings

Inversion of power series is numerically stable with good error bounds.

Root deflation can cause instabilities in polynomial computations.

Accuracy relates to the pseudozeros of the polynomial.

Abstract

This article considers the numerical inversion of the power series $p(x)=1+b_{1}x+b_{2}x^{2}+\cdots$ 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). We prove the numerical stability of inversion of power series and obtain bounds on numerical error. A range of examples show these bounds to be quite good. 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 $p(x)$.