Carla Ferreira

University of Minho · Centro de Matemática

We consider the Alternating Direction Implicit (ADI) method to compute the numerical solution of a continuous Sylvester equation \(AX+XB=C\), based on the recently developed inexact ADI iteration, and we propose classical acceleration techniques to enhance its convergence rate. An extrapolated variant (EADI) and a block successive overrelaxation va...

We present a circulant and skew-circulant splitting (CSCS) iterative method for solving large sparse continuous Sylvester equations AX+XB=C, where the coefficient matrices A and B are Toeplitz matrices. A theoretical study shows that if the circulant and skew-circulant splitting factors of A and B are positive semi-definite (not necessarily Hermiti...

The paper discusses the following topics: attractions of the real tridiagonal case, relative eigenvalue condition number for matrices in factored form, dqds, triple dqds, error analysis, new criteria for splitting and deflation, eigenvectors of the balanced form, twisted factorizations and generalized Rayleigh quotient iteration. We present our fas...

We present a circulant and skew-circulant splitting (CSCS) iterative method for solving large sparse continuous Sylvester equations $AX + XB = C$, where the coefficient matrices $A$ and $B$ are Toeplitz matrices. A theoretical study shows that if the circulant and skew-circulant splitting factors of $A$ and $B$ are positive semi-definite and at lea...

Stationary splitting iterative methods for solving AXB=C are considered in this paper. The main tool to derive our new method is the induced splitting of a given nonsingular matrix A=M−N by a matrix H such that (I−H)−1 exists. Convergence properties of the proposed method are discussed and numerical experiments are presented to illustrate its compu...

A little known property of a pair of eigenvectors (column and row) of a real tridiagonal matrix is presented. With its help we can define necessary and sufficient conditions for the unique real tridiagonal matrix for which an approximate pair of complex eigenvectors is exact. Similarly, we can designate the unique real tridiagonal matrix for which...

For given Z,B ∈ ℂn×k, the problem of finding A ∈ ℂn×n, in some prescribed class W, that minimizes ǁAZ − Bǁ (Frobenius norm) has been considered by different authors for distinct classes W. Here, this minimization problem is studied for two other classes, which include the symmetric Hamiltonian, symmetric skew-Hamiltonian, real orthogonal symplectic...

Several relative eigenvalue condition numbers that exploit tridiagonal form are derived. Some of them use triangular factorizations instead of the matrix entries and so they shed light on when eigenvalues are less sensitive to perturbations of factored forms than to perturbations of the matrix entries. A novel empirical condition number is used to...

The contribution in this paper is two-folded. First, a complete characterization is given of the square roots of a real nonsingular skew-Hamiltonian matrix W. Using the known fact that every real skew-Hamiltonian matrix has infinitely many real Hamiltonian square roots, such square roots are described. Second, a structure-exploiting method is propo...

We present a new transform, triple dqds, to help to compute the eigenvalues of a real tridiagonal matrix C using real arithmetic. The algorithm uses the real dqds transform to shift by a real number and triple dqds to shift by a complex conjugate pair. We present what seems to be a new criteria for splitting the current pair L,U. The algorithm reje...

We propose an algorithm for solving the inverse eigenvalue problem for real symmetric block Toeplitz matrices with symmetric Toeplitz blocks. It is based upon an algorithm which has been used before by others to solve the inverse eigenvalue problem for general real symmetric matrices and also for Toeplitz matrices. First we expose the structure of...

We prove convergence for the basic LR algorithm on a real unreduced tridiagonal matrix with a one-point spectrum—the Jordan form is one big Jordan block. First we develop properties of eigenvector matrices. We also show how to deal with the singular case.