Volume 25, Issue 2
A General Algorithm to Calculate the Inverse Principal p-th Root of Symmetric Positive Definite Matrices

Commun. Comput. Phys., 25 (2019), pp. 564-585.

Published online: 2018-10

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• 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.

• Keywords

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

15A09, 15A16, 65F25, 65F50, 65F60, 65N25

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@Article{CiCP-25-564, author = {}, title = {A General Algorithm to Calculate the Inverse Principal p-th Root of Symmetric Positive Definite Matrices}, journal = {Communications in Computational Physics}, year = {2018}, volume = {25}, number = {2}, pages = {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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2018-0053}, url = {http://global-sci.org/intro/article_detail/cicp/12763.html} }
TY - JOUR T1 - A General Algorithm to Calculate the Inverse Principal p-th Root of Symmetric Positive Definite Matrices JO - Communications in Computational Physics VL - 2 SP - 564 EP - 585 PY - 2018 DA - 2018/10 SN - 25 DO - http://doi.org/10.4208/cicp.OA-2018-0053 UR - https://global-sci.org/intro/article_detail/cicp/12763.html KW - Matrix $p$-th root, iteration function, order of convergence, symmetric positive definite matrices, Newton-Schulz, Altman hyperpower method. AB -

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.

Dorothee Richters, Michael Lass, Andrea Walther, Christian Plessl & Thomas D. Kühne. (2020). A General Algorithm to Calculate the Inverse Principal p-th Root of Symmetric Positive Definite Matrices. Communications in Computational Physics. 25 (2). 564-585. doi:10.4208/cicp.OA-2018-0053
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