A study of Schr\"oder's method for the matrix $p$th root using power series expansions

TL;DR

This paper analyzes Schr"oder's method for computing the matrix pth root, providing new error estimates, convergence properties, and structure preservation results using power series expansions.

Contribution

It introduces novel error bounds, monotonic convergence results, and structure preservation insights for Schr"oder's method applied to matrix pth roots.

Findings

New error estimate for the matrix sequence

Monotonic convergence for nonsingular M-matrices

Structure preservation for M-matrices and H-matrices

Abstract

When $A$ is a matrix with all eigenvalues in the disk $|z-1|<1$, the principal $p$th root of $A$ can be computed by Schr\"oder's method, among many other methods. In this paper we present a further study of Schr\"oder's method for the matrix $p$th root, through an examination of power series expansions of some sequences of scalar functions. Specifically, we obtain a new and informative error estimate for the matrix sequence generated by the Schr\"oder's method, a monotonic convergence result when $A$ is a nonsingular $M$-matrix, and a structure preserving result when $A$ is a nonsingular $M$-matrix or a real nonsingular $H$-matrix with positive diagonal entries.