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Component-wise accurate fixed point iterations for computing the square root of a singular M-matrix
Abstract
We analyze two fixed-point iterations for computing the principal square root of an M-matrix A. Although these iterations, with customary initialization, converge sublinearly when A is a singular M-matrix, we show that, under suitable mild conditions on the initial approximation, the convergence is linear. Moreover, we provide component-wise accurate versions of these iterations, which allow us to approximate the principal square root with a component-wise relative error uniformly bounded by a small multiple of the machine precision. Numerical experiments demonstrating the effectiveness of the proposed algorithms for certain classes of problems are presented.
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BibTeX
@misc{bini2026componentwise,
title = {{Component-wise accurate fixed point iterations for computing the square root of a singular M-matrix}},
author = {Dario Andrea Bini and Bruno Iannazzo and Beatrice Meini and Jie Meng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27520},
eprint = {2609.27520},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27520},
}
RIS
TY - UNPB
TI - Component-wise accurate fixed point iterations for computing the square root of a singular M-matrix
AU - Dario Andrea Bini
AU - Bruno Iannazzo
AU - Beatrice Meini
AU - Jie Meng
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27520
UR - https://arxiv.org/abs/2609.27520
AB - We analyze two fixed-point iterations for computing the principal square root of an M-matrix A. Although these iterations, with customary initialization, converge sublinearly when A is a singular M-matrix, we show that, under suitable mild conditions on the initial approximation, the convergence is linear. Moreover, we provide component-wise accurate versions of these iterations, which allow us to approximate the principal square root with a component-wise relative error uniformly bounded by a small multiple of the machine precision. Numerical experiments demonstrating the effectiveness of the proposed algorithms for certain classes of problems are presented.
ER -