A new stable and avoiding inversion iteration for computing matrix square root

TL;DR

This paper introduces a new stable iterative method that efficiently computes the principal matrix square root with sparse approximation, avoiding full matrix inversion and demonstrating high accuracy and computational efficiency.

Contribution

The paper presents a novel iterative scheme (SIAI) that avoids matrix inversion and combines it with filtering for improved efficiency in computing matrix square roots.

Findings

High computational efficiency demonstrated on various matrices

Accurate approximation of principal matrix square roots

Applicable to large sparse matrices

Abstract

The objective of this research was to compute the principal matrix square root with sparse approximation. A new stable iterative scheme avoiding fully matrix inversion (SIAI) is provided. The analysis on the sparsity and error of the matrices involved during the iterative process is given. Based on the bandwidth and error analysis, a more efficient algorithm combining the SIAI with the filtering technique is proposed. The high computational efficiency and accuracy of the proposed method are demonstrated by computing the principal square roots of different matrices to reveal its applicability over the existing methods.