Symmetry of cyclic weighted shift matrices with pivot-reversible weights

We prove that every cyclic weighted shift matrix with pivot-reversible weights is unitarily similar to a complex symmetric matrix.

Parametric randomization, complex symplectic factorizations, and quadratic-exponential functionals for Gaussian quantum states

This paper combines probabilistic and algebraic techniques for computing quantum expectations of operator exponentials (and their products) of quadratic forms of quantum variables in Gaussian states. Such quadratic-exponential functionals (QEFs) resemble quantum statistical mechanical partition functions with quadratic Hamiltonians and are also used as performance criteria in quantum risk-sensitive filtering and control problems for linear quantum stochastic systems. We employ a Lie-algebraic correspondence between complex symplectic matrices and quadratic-exponential functions of system variables of a quantum harmonic oscillator. The complex symplectic factorizations are used together with a parametric randomization of the quasi-characteristic or moment-generating functions according to an auxiliary classical Gaussian distribution. This reduces the QEF to an exponential moment of a quadratic form of classical Gaussian random variables with a complex symmetric matrix and is applicable to recursive computation of such moments.

A note on multilevel Toeplitz matrices

Chien, Liu, Nakazato and Tam proved that all n × n classical Toeplitz matrices (one-level Toeplitz matrices) are unitarily similar to complex symmetric matrices via two types of unitary matrices and the type of the unitary matrices only depends on the parity of n. In this paper we extend their result to multilevel Toeplitz matrices that any multilevel Toeplitz matrix is unitarily similar to a complex symmetric matrix. We provide a method to construct the unitary matrices that uniformly turn any multilevel Toeplitz matrix to a complex symmetric matrix by taking tensor products of these two types of unitary matrices for one-level Toeplitz matrices according to the parity of each level of the multilevel Toeplitz matrices. In addition, we introduce a class of complex symmetric matrices that are unitarily similar to some p-level Toeplitz matrices.

Frequency response computation of structures including non-proportional damping in a shared memory environment

With the increased size of the finite element model for improved accuracy, the modal frequency response analysis has been one of the common practices of evaluating the performance of vehicle dynamics. However, there is difficulty in predicting the vehicle dynamics response with non-proportional damping regarding performance. The fast frequency response analysis algorithm (FFRA) has been proved to be very effective for partially damped structural system in the modal frequency response analysis. In the fast frequency response analysis algorithm, performance depends mainly on the complex symmetric matrix eigenvalue problem. Therefore, an efficient complex symmetric matrix eigenvalue problem solver is developed in this article. This approach also uses parallel processing in a shared memory machine for more efficient analysis. Numerical examples show that the new complex symmetric matrix eigensolver provides good accuracy and high performance. Then, the fast frequency response analysis algorithm is applied to a full scale vehicle system that includes only a few viscous damping finite elements. The fast frequency response analysis algorithm significantly improves the performance of the modal frequency response analysis compared to conventional method. In addition, parallel processing improves the efficiency of the overall simulation.