Zolotarev Iterations for the Matrix Square Root

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.