Implementation of a Superfast Algorithm for Symmetric Positive Deenite Linear Equations of Displacement Rank 2

A Superfast Algorithm for Toeplitz Systems of Linear Equations

In this paper we develop a new superfast solver for Toeplitz systems of linear equations. To solve Toeplitz systems many people use displacement equation methods. With displacement structures, Toeplitz matrices can be transformed into Cauchy-like matrices using the FFT or other trigonometric transformations. These Cauchy-like matrices have a special property, that is, their off-diagonal blocks ...

Superfast Solution of Linear Equations with Low Displacement Rank

Preconditioners for ill { conditioned Toeplitz matrices

This paper is concerned with the solution of systems of linear equations ANx = b, where fANg N2N denotes a sequence of positive deenite Hermitian ill{conditioned Toeplitz matrices arising from a (real{valued) nonnegative generating function f 2 C2 with zeros. We construct positive deenite Hermitian preconditioners MN such that the eigenvalues of M ?1 N AN are clustered at 1 and the correspondin...

cient Parallel Factorization and Solution ofStructured and Unstructured Linear

This paper gives improved parallel methods for several exact factoriza-tions of some classes of symmetric positive deenite (SPD) matrices. Our factorizations also provide us similarly eecient algorithms for exact computation of the solution of the corresponding linear systems (which need not be SPD), and for nding rank and determinant magnitude. We assume the input matrices have entries that ar...

Parallel Direct Methodsfor Sparse Linear

We present an overview of parallel direct methods for solving sparse systems of linear equations, focusing on symmetric positive deenite systems. We examine the performance implications of the important diierences between dense and sparse systems. Our main emphasis is on parallel implementation of the numerically intensive factorization process, but we also brieey consider the other major compo...