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

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.

References in corpus (4)

• An Efficient and Accurate Car-Parrinello-like Approach to Born-Oppenheimer Molecular Dynamics
• Accurate and efficient linear scaling DFT calculations with universal applicability
• Challenges in Large Scale Quantum Mechanical Calculations
• An efficient and accurate decomposition of the Fermi operator

Cited by in corpus (4)

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