Highly Efficient Computation of Generalized Inverse of a Matrix

TL;DR

This paper introduces a highly efficient hyperpower iteration method for computing the outer generalized inverse of a matrix, achieving the highest known convergence order with minimal matrix multiplications, suitable for various applications.

Contribution

The paper presents a novel 18th-order convergence hyperpower iteration for the outer generalized inverse, surpassing existing methods in efficiency and stability, with practical numerical validation.

Findings

Achieves 18th order convergence with only seven matrix multiplications per iteration

Stabilized the algorithm with an additional matrix multiplication, maintaining high efficiency

Numerical tests confirm effectiveness across different matrix cases

Abstract

We propose a hyperpower iteration for numerical computation of the outer generalized inverse of a matrix which achieves the 18th order of convergence by using only seven matrix multiplication per iteration loop. This is the record high efficiency for that computational task. The algorithm has a relatively mild numerical instability, and we stabilize it at the price of adding one extra matrix multiplication per iteration loop. This imlplies an efficiency index that significantly exceeds the known record for numerically stable iterations for this task. Our numerical tests cover a variety of examples such as Drazin case, rectangular case, and preconditioning of linear systems. The test results are in good accordance with our formal study and indicate that our algorithms can be of interest for the user.