Convergence Analysis of Structure-Preserving Doubling Algorithms for Riccati-Type Matrix Equations

Abstract

In this paper, a structure-preserving transformation of a symplectic pencil is introduced, referred to as the doubling transformation, and its some basic properties are presented. Based on the nice properties of this kind of transformations, a unified convergence theory for the structure-preserving doubling algorithms for solving a class of Riccati-type matrix equations is established by using only the knowledge from elementary matrix theory.

A numerical method for quadratic eigenvalue problems of gyroscopic systems

“…Several numerical methods have been proposed, including the cyclic reduction method [8] and the structure-preserving doubling algorithm [9]. These algorithms are similar, but the convergence analysis of the structure-preserving doubling algorithm is simpler.…”

“…The method is quadratically convergent if ψ(λ) ≡ λ 2 A + λQ + A has no unimodular eigenvalues, and is linearly convergent if all unimodular eigenvalues of Z −1 + A are semisimple. A more efficient method for solving the NME based on the SDA algorithm [9] is shown to be linearly convergent if the unimodular eigenvalues of Z −1 + A have half of the partial multiplicity of the associated unimodular eigenvalue of ψ(λ).…”

“…In Section 2, we review some results in [1] on how to solve the QEP by the solvent approach, and the SDA algorithm and the related convergence analysis in [10,9]. The main technique of shifting the purely imaginary eigenvalues of the QEP is developed in Section 3.…”

“…The main purpose of this paper is to provide a new technique to shift the purely imaginary eigenvalues, while keeping the other eigenpairs unchanged. We then apply the structure-preserving methods in [1] or [9] to find the maximal solution of the resulted NME with quadratic convergence.…”

On the numerical solution of large-scale sparse discrete-time Riccati equations

“…The usual solution methods for DAREs such as the Schur vector method [46], symplectic SR methods [12,23], the matrix sign function [7,15,26,51], the matrix disk function [7,15,39,55] or the doubling method [52,42] do not make (full) use of the sparse structure of A, E and require in general O(n 3 ) flops and workspace of size O(n 2 ) even for sparse problems, and are therefore not suitable here.…”