TY - JOUR
T1 - A General Algorithm to Calculate the Inverse Principal p-th Root of Symmetric Positive Definite Matrices
AU - Dorothee Richters, Michael Lass, Andrea Walther, Christian Plessl & Thomas D. Kühne
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 - <p style="text-align: justify;">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.</p>