Algorithm with variable coefficients for computing matrix inverses

TL;DR

Through construction and numerical testing of method, constructed method in it's final form is numerically stable and optimal, meaning that coefficients are chosen in optimal way in terms of Frobenius norm.

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.