Iterative methods for the inclusion of the inverse matrix

TL;DR

This paper introduces a sixth-order iterative method combining point and interval iterations for efficiently computing the inverse or Moore-Penrose inverse of regular matrices, with proven convergence and high computational efficiency.

Contribution

The paper presents a novel sixth-order iterative method that unifies point and interval iterations for matrix inverse inclusion, applicable to full-rank and rectangular matrices.

Findings

Method achieves convergence with high efficiency.

Applicable to full-rank and rectangular matrices.

Computational efficiency surpasses existing hyper-power methods.

Abstract

In this paper we present an efficient iterative method of order six for the inclusion of the inverse of a given regular matrix. To provide the upper error bound of the outer matrix for the inverse matrix, we combine point and interval iterations. The new method is relied on a suitable matrix identity and a modification of a hyper-power method. This method is also feasible in the case of a full-rank $m\times n$ matrix, producing the interval sequence which converges to the Moore-Penrose inverse. It is shown that computational efficiency of the proposed method is equal or higher than the methods of hyper-power's type.