A General Algorithm to Calculate the Inverse Principal $p$-th Root of Symmetric Positive Definite Matrices
This work provides a more efficient algorithm for a fundamental matrix operation, benefiting numerical linear algebra and related fields.
The paper presents a new iterative method for computing the inverse p-th root of symmetric positive definite matrices, achieving at least quadratic convergence and outperforming previous schemes in efficiency across various matrix types.
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.