Zolotarev Iterations for the Matrix Square Root

TL;DR

This paper introduces a new family of rational iterations based on Zolotarev's minimax approximants for efficiently computing the principal square root of matrices, with proven convergence and superior performance on matrices with diverse eigenvalues.

Contribution

It develops a novel iterative method for matrix square roots using Zolotarev rational functions, unifying and extending known methods like Newton and Padé iterations.

Findings

Converges for matrices with no nonpositive real eigenvalues.

Performs well on matrices with widely varying eigenvalues.

Includes special cases reducing to Newton and Padé iterations.

Abstract

We construct a family of iterations for computing the principal square root of a square matrix $A$ using Zolotarev's rational minimax approximants of the square root function. We show that these rational functions obey a recursion, allowing one to iteratively generate optimal rational approximants of $\sqrt{z}$ of high degree using compositions and products of low-degree rational functions. The corresponding iterations for the matrix square root converge to $A^{1/2}$ for any input matrix $A$ having no nonpositive real eigenvalues. In special limiting cases, these iterations reduce to known iterations for the matrix square root: the lowest-order version is an optimally scaled Newton iteration, and for certain parameter choices, the principal family of Pad\'e iterations is recovered. Theoretical results and numerical experiments indicate that the iterations perform especially well on matrices having eigenvalues with widely varying magnitudes.