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We propose a scaling scheme for Newton's iteration for calculating the polar decomposition. The scaling factors are generated by a simple scalar iteration in which the initial value depends only on estimates of the extreme singular values of the original matrix, which can for example be the Frobenius norms of the matrix and its inverse. In exact arithmetic,… (More)

We discuss the numerical solution of structured generalized eigenvalue problems that arise from linear-quadratic optimal control problems, H∞ optimization, multibody systems, and many other areas of applied mathematics, physics, and chemistry. The classical approach for these problems requires computing invariant and deflating subspaces of matrices and… (More)

We derive formulas for the minimal positive solution of a particular non-symmetric Riccati equation arising in transport theory. The formulas are based on the eigenvalues of an associated matrix. We use the formulas to explore some new properties of the minimal positive solution and to derive fast and highly accurate numerical methods. Some numerical tests… (More)

Bounds are developed for the condition number of the linear system resulting from the finite element discretization of an anisotropic diffusion problem with arbitrary meshes. These bounds are shown to depend on three major factors: a factor representing the base order corresponding to the condition number for a uniform mesh, a factor representing the… (More)

- Weishi Liu, Hongguo Xu
- 2014

In this work, we examine the stationary one-dimensional classical Poisson-Nernst-Planck (cPNP) model for ionic flow – a singularly perturbed boundary value problem (BVP). For the case of zero permanent charge, we provide a complete answer concerning the existence and uniqueness of the BVP. The analysis relies on a number of ingredients: a geometric singular… (More)

The existence, uniqueness, and parametrization of Lagrangian invariant subspaces for Hamiltonian matrices is studied. Necessary and sufficient conditions and a complete parametrization are given. Some necessary and sufficient conditions for the existence of Hermitian solutions of algebraic Riccati equations follow as simple corollaries.

We present formulas for the construction of optimal H∞ controllers that can be implemented in a numerically robust way. We base the formulas on the γ-iteration developed in [6]. The controller formulas proposed here avoid the solution of algebraic Riccati equations with their problematic matrix inverses and matrix products. They are also applicable in the… (More)

We present a numerical method to compute the SVD-like decomposition B = QDS −1 , where Q is orthogonal, S is symplectic and D is a permuted diagonal matrix. The method can be applied directly to compute the canonical form of the Hamiltonian matrices of the form JB T B, where J = 0 −I I 0. It can also be applied to solve the related application problems such… (More)

The classical singular value decomposition for a matrix A ∈ C m×n is a canonical form for A that also displays the eigenvalues of the Hermitian matrices AA * and A * A. In this paper, we develop a corresponding decomposition for A that provides the Jordan canonical forms for the complex symmetric matrices AA T and A T A. More generally, we consider the… (More)

In this work, we are interested in effects of a simple profile of permanent charges on ionic flows. We determine when a permanent charge produces current reversal. We adopt the classical Poisson-Nernst-Planck models of ionic flows for this study. The starting point of our analysis is the recently developed geometric singular perturbation approach for… (More)