A Unifying Framework for Doubling Algorithms

TL;DR

This work addresses robustly solving nonlinear matrix equations by exploiting eigenspaces of regular matrix pencils $\mathscr{A}-\lambda\mathscr{B}$ via structure-preserving doubling. It introduces the Q-doubling algorithm, a generalization that subsumes existing doubling methods and removes the requirement that a basis matrix must take the form $Z=IX$, thereby improving numerical stability when eigenbasis magnitudes are large. The paper also analyzes the eigenvalue structure, showing connections between the original pencil and derived matrices, and demonstrates through numerical experiments that Q-doubling offers superior robustness. The contribution provides a more versatile and stable tool for eigenvalue problems and related control and system-theory applications where nonlinear matrix equations arise.

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.