$AB$-algorithm and its application for solving matrix square roots

TL;DR

This paper introduces an iterative method based on matrix pencils to compute stable subspaces and matrix square roots, offering a convergent approach with adjustable order for both nonsingular and singular matrices.

Contribution

It proposes a novel iterative algorithm for stable subspace computation and matrix square root calculation, with proven convergence properties and the ability to achieve any desired order of convergence.

Findings

The method effectively computes matrix square roots for nonsingular and singular matrices.

It preserves a discrete flow depending only on the initial matrix pencil.

The approach can be accelerated to converge with any specified order.

Abstract

This work is to propose an iterative method of choice to compute a stable subspace of a regular matrix pencil. This approach is to define a sequence of matrix pencils via particular left null spaces. We show that this iteration preserves a discrete-type flow depending only on the initial matrix pencil. Via this recursion relationship, we propose an accelerated iterative method to compute the stable subspace and use it to provide a theoretical result to solve the principal square root of a given matrix, both nonsingular and singular. We show that this method can not only find out the matrix square root, but also construct an iterative approach which converges to the square root with any desired order.