A quartically fast iteration solver with convergence analysis for numerically determining the sign of a matrix

The calculation of the matrix sign function (MSF) is pivotal in numerous mathematical contexts, offering a matrix-based transformation that identifies the sign for every eigenvalue within an invertible matrix. This paper introduces a new iteration procedure tailored to effectively compute the MSF, with a particular focus on expanding the order of convergence. Our proposed solver achieves fourth-order convergence, rendering it effective for a broad spectrum of matrices. Numerical experiments for both real and complex matrices are included to support the derivations.