Wen-Wei Lin

Canonical Forms for Hamiltonian and Symplectic Matrices and Pencils

Abstract We study canonical forms for Hamiltonian and symplectic matrices or pencils under equivalence transformations which keep the class invariant. In contrast to other canonical forms our forms are as close as possible to a triangular structure in the same class. We give necessary and sufficient conditions for the existence of Hamiltonian and symplectic triangular Jordan, Kronecker and Schur forms. The presented results generalize results of Lin and Ho W.-W. Lin, T.-C. Ho, On Schur type decompositions for Hamiltonian and symplectic pencils, Technical report, Institute of Applied Mathematics, National Tsing Hua University, Taiwan, 1990 and simplify the proofs presented there.

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

In this paper, we introduce the doubling transformation, a structure-preserving transformation for symplectic pencils, and present its basic properties. Based on these properties, a unified convergence theory for the structure-preserving doubling algorithms for a class of Riccati-type matrix equations is established, using only elementary matrix theory.

A structure-preserving doubling algorithm for nonsymmetric algebraic Riccati equation

In this paper, we propose a structure-preserving doubling algorithm (SDA) for the computation of the minimal nonnegative solution to the nonsymmetric algebraic Riccati equation (NARE), based on the techniques developed for the symmetric cases. This method allows the simultaneous approximation to the minimal nonnegative solutions of the NARE and its dual equation, requiring only the solutions to two linear systems and several matrix multiplications per iteration. Similar to Newtons method and the fixed-point iteration methods for solving NAREs, we also establish global convergence for SDA under suitable conditions, using only elementary matrix theory. We show that sequences of matrices generated by SDA are monotonically increasing and quadratically convergent to the minimal nonnegative solutions of the NARE and its dual equation. Numerical experiments show that the SDA algorithm is feasible and effective, and outperforms Newtons iteration and the fixed-point iteration methods.

Nonsymmetric Algebraic Riccati Equations and Hamiltonian-like Matrices

We consider a nonsymmetric algebraic matrix Riccati equation arising from transport theory. The nonnegative solutions of the equation can be explicitly constructed via the inversion formula of a Cauchy matrix. An error analysis and numerical results are given. We also show a comparison theorem of the nonnegative solutions.

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

An inexact inverse iteration for large sparse eigenvalue problems

1/2

New Methods for Finite Element Model Updating Problems

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

Randomness Enhancement Using Digitalized Modified Logistic Map

In this paper, we propose an inverse inexact iteration method for the computation of the eigenvalue with the smallest modulus and its associated eigenvector for a large sparse matrix. The linear systems of the traditional inverse iteration are solved with accuracy that depends on the eigenvalue with the second smallest modulus and iteration numbers. We prove that this approach preserves the linear convergence of inverse iteration. We also propose two practical formulas for the accuracy bound which are used in actual implementation.

Normwise Scaling of Second Order Polynomial Matrices

We consider two finite element model updating problems, which incorporate the measured modal data into the analytical finite element model, producing an adjusted model on the (mass) damping and stiffness, that closely matches the experimental modal data. We develop two efficient numerical algorithms for solving these problems. The new algorithms are direct methods that require O(nk2) and O(nk2 + k6) flops, respectively, and employ sparse matrix techniques when the analytic model is sparse. Here n is the dimension of the coefficient matrices defining the analytical model, and k is the number of measured eigenpairs.

Solutions of the Partially Described Inverse Quadratic Eigenvalue Problem

In this brief, a nonlinear digitalized modified logistic map-based pseudorandom number generator (DMLM-PRNG) is proposed for randomness enhancement. Two techniques, i.e., constant parameter selection and output sequence scrambling, are employed to reduce the computation cost without sacrificing the complexity of the output sequence. Statistical test results show that with only one multiplication, DMLM-PRNG passes all cases in SP800-22. Moreover, it passes most of the cases in Crush, one of the test suites of TesuU01. When compared with solutions based on digitized pseudochaotic maps previously proposed in the literature, in terms of randomness quality, our system is as good as a Rényi-map-based PRNG and better than a logistic-map-based PRNG. Moreover, compared with solutions based on a Rényi-map-based PRNG, DMLM-PRNG is better scalable to high digital resolutions with reasonable area overhead.