A General Algorithm to Calculate the Inverse Principal p-th Root of Symmetric Positive Definite Matrices (Communications in Computational Physics, 2018)
We address the general mathematical problem of computing the inversep-th root of a given matrix in an efficient way. A new method to construct iterationfunctions that allow calculating arbitrary p-th roots and their inverses of symmetricpositive definite matrices is presented. We show that the order of convergence is atleast 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 isachieved. The efficiency of the iterative functions is demonstrated for various matriceswith 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.