Rational Minimax Iterations for Computing the Matrix pth Root

2019/03/14 by Gawlik, Evan S.
#FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1903.06268

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.

Related

• Zolotarev Iterations for the Matrix Square Root
• Approximating the pth Root by Composite Rational Functions
• Rational Minimax Approximations for Matrix-Valued Functions: Existence, Optimality and Algorithms
• Rational minimax approximation of matrix-valued functions
• A convergence analysis of Lawson's iteration for computing polynomial and rational minimax approximations