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

TL;DR

This paper introduces a new iterative algorithm for efficiently computing the inverse p-th root of symmetric positive definite matrices, with quadratic or better convergence, outperforming previous methods.

Contribution

A novel iterative method for calculating inverse p-th roots of matrices with adaptive parameter adjustment and proven quadratic convergence.

Findings

Quadratic or faster convergence achieved.

Method outperforms existing schemes in efficiency.

Effective for matrices with various properties.

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 adaptively adjusting a parameter $q$ always 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.