Algorithm with variable coefficients for computing matrix inverses

TL;DR

This paper introduces a new efficient iterative method with variable coefficients for computing matrix inverses, which is proven to be optimal and numerically stable through theoretical analysis and numerical testing.

Contribution

It presents a novel generalized Schultz iterative method with dynamically chosen coefficients, improving efficiency and stability in matrix inversion.

Findings

Method is optimal in Frobenius norm

Numerical tests confirm theoretical predictions

Constructed method is numerically stable

Abstract

We present a general scheme for the construction of new eficient generalized Schultz iterative methods for computing the inverse matrix. These methods have the form $$ X_{k+1} = X_k(a_0^{(k)}I+a_1^{(k)}AX_k),\quad k\in\mathbb{N}, $$ where $A$ is square real matrix and $a_0^{(k)}$ and $a_0^{(k)}$ are dynamical coefficients. We are going to present basic case of the problem, while formulas are derived analogically in other cases but are more complicated. Constructed method is optimal, meaning that coefficients are chosen in optimal way in terms of Frobenius norm. We have done some numerical testing that confirm theoretical approach. Through construction and numerical testing of method we have considered numerical stability as well. In the end, constructed method in it's final form is numerically stable and optimal.