A General Algorithm to Calculate the Inverse Principal p-th Root of Symmetric Positive Definite Matrices

A General Algorithm to Calculate the Inverse Principal p-th Root of Symmetric Positive Definite Matrices

Year:    2019

Communications in Computational Physics, Vol. 25 (2019), Iss. 2 : pp. 564–585

Abstract

We address the general mathematical problem of computing the inverse p-th root of a given matrix in an efficient way. A new method to construct iteration functions that allow calculating arbitrary p-th roots and their inverses of symmetric positive definite matrices is presented. We show that the order of convergence is at least quadratic and that adjusting a parameter q leads to an even faster convergence. In this way, a better performance than with previously known iteration schemes is achieved. The efficiency of the iterative functions is demonstrated for various matrices with different densities, condition numbers and spectral radii.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.OA-2018-0053

Communications in Computational Physics, Vol. 25 (2019), Iss. 2 : pp. 564–585

Published online:    2019-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    22

Keywords:    Matrix $p$-th root iteration function order of convergence symmetric positive definite matrices Newton-Schulz Altman hyperpower method.