A Unifying Framework for Doubling Algorithms

TL;DR

This paper introduces a new unifying doubling algorithm, called the Q-doubling algorithm, which generalizes existing methods and improves robustness in solving eigenvalue problems without requiring specific basis matrix forms.

Contribution

The paper proposes the Q-doubling algorithm that encompasses existing doubling algorithms as special cases and removes the need for specific basis matrix forms, enhancing robustness.

Findings

Q-doubling algorithm includes existing algorithms as special cases

Demonstrates superior robustness in numerical experiments

Applicable to eigenvalue problems in engineering

Abstract

The existing doubling algorithms have been proven efficient for several important nonlinear matrix equations arising from real-world engineering applications. In a nutshell, the algorithms iteratively compute a basis matrix, in one of the two particular forms, for the eigenspace of some matrix pencil associated with its eigenvalues in certain complex region such as the left-half plane or the open unit disk, and their success critically depends on that the interested eigenspace do have a basis matrix taking one of the two particular forms. However, that requirement in general cannot be guaranteed. In this paper, a new doubling algorithm, called the $Q$-doubling algorithm, is proposed. It includes the existing doubling algorithms as special cases and does not require that the basis matrix takes one of the particular forms. An application of the $Q$-doubling algorithm to solve eigenvalue problems is investigated with numerical experiments that demonstrate its superior robustness to the existing doubling algorithms.