Component-wise accurate computation of the square root of an M-matrix

TL;DR

This paper introduces component-wise accurate algorithms for computing the principal square root of M-matrices using triplet representations, ensuring numerical stability regardless of matrix singularity or condition number.

Contribution

It develops new triplet-based Cyclic Reduction and Incremental Newton algorithms for stable, component-wise computation of M-matrix square roots, extending existing methods.

Findings

Algorithms are component-wise numerically stable regardless of matrix singularity.

Numerical experiments confirm the stability of the proposed algorithms.

Principal square root of an M-matrix exists and can be represented by a triplet.

Abstract

Component-wise accurate algorithms for computing the principal square root of an M-matrix are designed in terms of triplet representations. A triplet representation of an M-matrix $A$ is the triple $(P, {\bf u},{\bf v})$, where the matrix $P$ is such that $p_{ij}=-a_{ij}$ for $i\ne j$, $p_{ii}=0$, and ${\bf u}>0$, ${\bf v}\ge 0$ are two vectors such that $A{\bf u}={\bf v}$. It is shown that if $A$ is an M-matrix representable by a triplet, then its principal square root exists and is an M-matrix represented by a triplet as well. New versions of the Cyclic Reduction and the Incremental Newton iterations are provided in terms of triplets, to compute the principal matrix square root of $A$. It is shown that these algorithms are component-wise numerically stable independently of the singularity of $A$ and of its condition number. Numerical experiments are shown to confirm the component-wise stability.