Eric King-wah Chu

Convergence Analysis of the Doubling Algorithm for Several Nonlinear Matrix Equations in the Critical Case

In this paper, we review two types of doubling algorithm and some techniques for analyzing them. We then use the techniques to study the doubling algorithm for three different nonlinear matrix equations in the critical case. We show that the convergence of the doubling algorithm is at least linear with rate

Pole assignment via the schur form

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Projected Generalized Discrete-Time Periodic Lyapunov Equations and Balanced Realization of Periodic Descriptor Systems

. As compared to earlier work on this topic, the results we present here are more general, and the analysis here is much simpler.

Perturbation of Eigenvalues for Matrix Polynomials via The Bauer--Fike Theorems

Abstract We propose an algorithm for the state-feedback pole assignment problem. The algorithm is the first of its kind, making direct use of the Schur form, and minimizing the departure from normality of the closed-loop poles for a given first Schur vector x 1 . The robust pole assignment problem can then be solved via choosing x 1 optimally. Several numerical examples were presented to illustrate the feasibility of the algorithm.

Large-scale Stein and Lyapunov equations, Smith method, and applications

From the necessary and sufficient conditions for complete reachability and observability of periodic descriptor systems with time-varying dimensions, the symmetric positive semidefinite reachability/observability Gramians are defined. These Gramians can be shown to satisfy some projected generalized discrete-time periodic Lyapunov equations. We propose a numerical method for solving these projected Lyapunov equations, and give an illustrative numerical example. As an application of our results, the balanced realization of periodic descriptor systems is discussed.

A generalized structure-preserving doubling algorithm for generalized discrete-time algebraic Riccati equations

In earlier papers, the Bauer--Fike technique [F. L. Bauer and C. T. Fike, Numer. Math., 2 (1960), pp. 137--144] was applied to the eigenvalue problem

Numerical solution to generalized Lyapunov/Stein and rational Riccati equations in stochastic control

A{\bf x} = \lambda {\bf x}

SOLVING LARGE-SCALE NONSYMMETRIC ALGEBRAIC RICCATI EQUATIONS BY DOUBLING ∗

[E. K.-W. Chu, Numer. Math., 49 (1986), pp. 685--691] and the generalized eigenvalue problem

Low-rank approximation to the solution of a nonsymmetric algebraic Riccati equation from transport theory

A{\bf x} = \lambda B{\bf x}

A nonmonotone inexact Newton algorithm for nonlinear systems of equations

[E. K.-W. Chu, SIAM J. Numer. Anal., 24 (1987), pp. 1114--1125]. General multiple eigenvalues were dealt with and perturbation results were obtained for individual as well as clusters of eigenvalues. In this paper, we shall generalize the technique to the eigenvalue problem for matrix polynomials. Multiple eigenvalues for monic as well as regular matrix polynomials will be considered.