Convergence analysis of a Pad\'{e} family of iterations for the matrix sector function

TL;DR

This paper proves a conjecture about the convergence of a Padé family of iterative methods for the matrix sector function, using a refined Schwarz's lemma to strengthen the original conjecture.

Contribution

It provides a proof of a conjecture on the convergence of Padé-based iterations for the matrix sector function and enhances the conjecture with a sharpened Schwarz's lemma.

Findings

Confirmed convergence of the Padé family of iterations

Strengthened the original conjecture with a new lemma

Provided theoretical foundation for matrix sector function computations

Abstract

The main purpose of this paper is to give a solution to a conjecture concerning a Pad\'{e} family of iterations for the matrix sector function that was recently raised by B. Laszkiewicz et al in [A Pad\'{e} family of iterations for the matrix sector function and the matrix $p$th root, Numer. Linear Algebra Appl. 2009; 16:951-970]. Using a sharpened version Schwarz's lemma, we also demonstrate a strengthening of the conjecture.