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

Abstract

In [E. S. Gawlik, Zolotarev iterations for the matrix square root, arXiv preprint 1804.11000, (2018)], a family of iterations for computing the matrix square root was constructed by exploiting a recursion obeyed by Zolotarev's rational minimax approximants of the function z 1/2 . The present paper generalizes this construction by deriving rational minimax iterations for the matrix p th root, where p ≥ 2 is an integer. The analysis of these iterations is considerably different from the case p = 2, owing to the fact that when p > 2, rational minimax approximants of the function z 1/p do not obey a recursion. Nevertheless, we show that several of the salient features of the Zolotarev iterations for the matrix square root, including equioscillatory error, order of convergence, and stability, carry over to case p > 2. A key role in the analysis is played by the asymptotic behavior of rational minimax approximants on short intervals. Numerical examples are presented to illustrate the predictions of the theory.

Approximating the pth Root by Composite Rational Functions

“…This observation generalizes earlier work on rational approximation of the square root with optimally scaled Newton iterations [2,13,15,18]. Moreover, an extension was derived in [5], which shows that the pth root can be approximated efficiently on intervals [δ, 1] ⊂ (0, 1], although not with minimax quality.…”

“…Our analysis will focus on the lowest-order version of the iteration (9-10), obtained by choosing (m, ) = (1, 0). It is shown in [5,Proposition 5] (and elsewhere [9,11]) that for this choice of m and , r1,0 (x, α,…”

“…To approximate x 1/p on an interval [α p , 1] ⊂ (0, 1], Gawlik [5] considers the recursively defined rational function…”

“…It turns out that the rational function ( 9) does a good job approximating on [0, 1], when α is chosen carefully: when it is too small, the error is large on [α p , 1] (in fact it is maximal at x = 1 [5]). Conversely if α is too large, the error is large on [0, α p ] (in fact it is O(α) at x = 0, as we show below).…”

“…This paper is organized as follows. In Section 2, we review some theory from [5] concerning composite rational approximants of the pth root on positive real intervals. In Section 3, we study the behavior of these approximants near the origin.…”