Hung Yuan Fan

Normwise Scaling of Second Order Polynomial Matrices

We propose a minimax scaling procedure for second order polynomial matrices that aims to minimize the backward errors incurred in solving a particular linearized generalized eigenvalue problem. We give numerical examples to illustrate that it can significantly improve the backward errors of the computed eigenvalue-eigenvector pairs.

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

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.

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

We consider the numerical solution of the generalized Lyapunov and Stein equations in ℝn

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

\mathbb {R}^{n}

Perturbation analysis of the stochastic algebraic Riccati equation

, arising respectively from stochastic optimal control in continuous- and discrete-time. Generalizing the Smith method, our algorithms converge quadratically and have an O(n3) computational complexity per iteration and an O(n2) memory requirement. For large-scale problems, when the relevant matrix operators are “sparse”, our algorithm for generalized Stein (or Lyapunov) equations may achieve the complexity and memory requirement of O(n) (or similar to that of the solution of the linear systems associated with the sparse matrix operators). These efficient algorithms can be applied to Newton’s method for the solution of the rational Riccati equations. This contrasts favourably with the naive Newton algorithms of O(n6) complexity or the slower modified Newton’s methods of O(n3) complexity. The convergence and error analysis will be considered and numerical examples provided.

Numerical solution to a linear equation with tensor product structure

Abstract We consider the solution of the large-scale nonsymmetric algebraic Riccati equation XCX - XD - AX + B = 0 from transport theory (Juang 1995), with M ≡ [ D , - C ; - B , A ] ∈ R 2 n × 2 n being a nonsingular M-matrix. In addition, A , D are rank-1 updates of diagonal matrices, with the products A - 1 u , A - ⊤ u , D - 1 v and D - ⊤ v computable in O ( n ) complexity, for some vectors u and v, and B, C are rank 1. The structure-preserving doubling algorithm by Guo et al. (2006) is adapted, with the appropriate applications of the Sherman–Morrison–Woodbury formula and the sparse-plus-low-rank representations of various iterates. The resulting large-scale doubling algorithm has an O ( n ) computational complexity and memory requirement per iteration and converges essentially quadratically, as illustrated by the numerical examples.

Projected nonsymmetric algebraic Riccati equations and refining estimates of invariant and deflating subspaces

In this paper we study a general class of stochastic algebraic Riccati equations (SARE) arising from the indefinite linear quadratic control and stochastic H∞ problems. Using the Brouwer fixed point theorem, we provide sufficient conditions for the existence of a stabilizing solution of the perturbed SARE. We obtain a theoretical perturbation bound for measuring accurately the relative error in the exact solution of the SARE. Moreover, we slightly modify the condition theory developed by Rice and provide explicit expressions of the condition number with respect to the stabilizing solution of the SARE. A numerical example is applied to illustrate the sharpness of the perturbation bound and its correspondence with the condition number.MSC: Primary 15A24; 65F35; secondary 47H10, 47H14.

Homotopy for rational riccati equations arising in stochastic optimal control

Summary We consider the numerical solution of a c-stable linear equation in the tensor product space Rn1×⋯×nd, arising from a discretized elliptic partial differential equation in Rd. Utilizing the stability, we produce an equivalent d-stable generalized Stein-like equation, which can be solved iteratively. For large-scale problems defined by sparse and structured matrices, the methods can be modified for further efficiency, producing algorithms of O(∑ini)+O(ns) computational complexity, under appropriate assumptions (with ns being the flop count for solving a linear system associated with Ai−γIni). Illustrative numerical examples will be presented.

The Rayleigh-Ritz method, refinement and Arnoldi process for periodic matrix pairs

We consider the numerical solution of the projected nonsymmetric algebraic Riccati equations or their associated Sylvester equations via Newtons method, arising in the refinement of estimates of invariant (or deflating subspaces) for a large and sparse real matrix A (or pencil A - λ B ). The engine of the method is the inversion of the matrix P 2 P 2 ź A - γ I n or P l 2 P l 2 ź ( A - γ B ) , for some orthonormal P 2 or P l 2 from R n × ( n - m ) , making use of the structures in A or A - λ B and the Sherman-Morrison-Woodbury formula. Our algorithms are efficient, under appropriate assumptions, as shown in our error analysis and illustrated by numerical examples.

A structure-preserving doubling algorithm for continuous-time algebraic Riccati equations

We consider the numerical solution of the rational algebraic Riccati equations in