Chern Shuh Wang

The shift-inverted J-Lanczos algorithm for the numerical solutions of large sparse algebraic Riccati equations

Abstract The goal of solving an algebraic Riccati equation is to find the stable invariant subspace corresponding to all the eigenvalues lying in the open left-half plane. The purpose of this paper is to propose a structure-preserving Lanczos-type algorithm incorporated with shift and invert techniques, named shift-inverted J -Lanczos algorithm, for computing the stable invariant subspace for large sparse Hamiltonian matrices. The algorithm is based on the J -tridiagonalization procedure of a Hamiltonian matrix using symplectic similarity transformations. We give a detailed analysis on the convergence behavior of the J -Lanczos algorithm and present error bound analysis and Paige-type theorem. Numerical results for the proposed algorithm applied to a practical example arising from the position and velocity control for a string of high-speed vehicles are reported.

Nonequivalence transformation of lambda-matrix eigenproblems and model embedding approach to model tuning

In this paper we present some theory for a non-equivalence transformation of matrix eigenvalues for -matrix polynomials. Application of this transformation to eigenvalue embedding for model tuning on a realistic industry problem is illustrated. The new approach has several advantages such as exibility, e ciency, and structure-preservation. A numerical experiment on a pseudosimulation data set from The Boeing Company is reported. Copyright ? 2001 John Wiley & Sons, Ltd.

SYNCHRONIZATION IN A LATTICE OF COUPLED VAN DER POL SYSTEMS

We consider a lattice of coupled van der Pol systems with external periodic forces and Dirichlet boundary conditions. Under the assumption of bounded dissipativeness, we prove that the asymptotic synchronization occurs provided that the coupling coefficients are sufficiently large. Furthermore, we demonstrate from numerical simulations that asymptotic synchronization is affected by the coupling coefficients and the size of lattice.

SYNCHRONIZATION IN COUPLED MAP LATTICES WITH PERIODIC BOUNDARY CONDITION

We consider a lattice of coupled logistic maps with periodic boundary condition. We prove that synchronization and almost synchronization occur for the case of 1D lattice with lattice size n=2, 3, ...

Synchronization in lattices of coupled oscillators with various boundary conditions

where index i is the ordered pair given by i= (i1; i2), for 1≤ i1; i2 ≤ n; i; i are constants with i i ≤ 0; x; y are two vectors in Rn2 with components xi and yi, respectively, fi is a C2 function with fi(0; 0) = 0; gi(t) is a periodic function of t; (c1; c2) denote the coupling coe7cients. L is a diagonalizable operator on Rn2 with max{ i}= 0 ≤ 0, for all i ∈ (L) ≡ the spectrum of L; (Lx)i and (Ly)i denote the ith components of Lx and Ly, respectively. If c1 =c2 =0 then system (1.1) represents the motion of the ith oscillator in an n×n uncoupled squared lattice. The individual dynamics for the ith node in the lattice is

Numerical algorithms for undamped gyroscopic systems

Abstract The solutions of a gyroscopic vibrating system oscillating about an equilibrium position, with no external applied forces and no damping forces, are completely determined by the quadratic eigenvalue problem (− λ 2 i M + λ i G + K ) x i = 0, for i = 1, …, 2 n , where M , G , and K are real n × n matrices, and M is symmetric positive definite (denoted by M > 0), G is skew symmetric, and either K > 0 or − K > 0. Gyroscopic system in motion about a stable equilibrium position (with − K > 0) are well understood. Two Lanczos-type algorithms, the pseudo skew symmetric Lanczos algorithm and the J -Lanczos algorithm, are studied for computing some extreme eigenpairs for solving gyroscopic systems in motion about an unstable equilibrium position (with K > 0). Shift and invert strategies, error bounds, implementation issues, and numerical results for both algorithms are presented in details.

On Computing Stable Lagrangian Subspaces of Hamiltonian Matrices and Symplectic Pencils

This paper presents algorithms for computing stable Lagrangian invariant subspaces of a Hamiltonian matrix and a symplectic pencil, respectively, having purely imaginary and unimodular eigenvalues. The problems often arise in solving continuous- or discrete-time

Perturbation results related to palindromic eigenvalue problems

H^{\infty}

A structure-preserving Lanczos-type algorithm with application to control problems

-optimal control, linear-quadratic control and filtering theory, etc. The main approach of our algorithms is to determine an isotropic Jordan subbasis corresponding to purely imaginary (unimodular) eigenvalues by using the associated Jordan basis of the square of the Hamiltonian matrix (the

Symmetry preserving eigenvalue embedding in finite-element model updating of vibrating structures

S+S^{-1}