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 √ 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 Padé iterations is recovered. Theoretical results and numerical experiments indicate that the iterations perform especially well on matrices having eigenvalues with widely varying magnitudes.

Rational Minimax Iterations for Computing the Matrix $p$th Root

“…Introduction. In recent years, a growing body of literature has highlighted the usefulness of rational minimax iterations for computing functions of matrices [25,26,7,8,4]. In these studies, f (A) is approximated by a rational function r of A possessing two properties: r closely (and often optimally) approximates f in the uniform norm over a subset of the real line, and r can be generated from a recursion.…”

Approximating the pth Root by Composite Rational Functions

“…That is, as in the square root approximation, the minimax rational approximant is contained in the class of (here purely) composite rational functions. Below we derive estimates for the maximum weighted error |z(g k (z) − sect p (z))| on the sets S p , S p,α ⊂ C defined in ( 5) and (6). As before, it will be convenient to work not with g k (z) but with the rescaled function…”

“…Functions related to the sign function, such as |x| (via |x| = x/sign(x)) and √ x (via |x| ≈ p(x 2 )/q(x 2 ) then √ x ≈ p(x)/q(x)) can similarly be approximated by composite rational functions. Gawlik [6] does this for the square root and shows that a composite rational function yields the minimax rational approximant (in the relative sense) on intervals [δ, 1] ⊂ (0, 1], and that the approximation extends far into the complex plane. This observation generalizes earlier work on rational approximation of the square root with optimally scaled Newton iterations [2,13,15,18].…”

Dressing for generalised linear Hamiltonian systems depending rationally on the spectral parameter and some applications

“…Among important works on this topic one could mention [4,10,12,41]. See also interesting recent works [2,13,29] and references therein.…”