Thomas Huckle

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The computation of selected eigenvalues and eigenvectors of a symmetric (Hermitian) matrix is an important subtask in many contexts, for example in electronic structure calculations. If a significant portion of the eigensystem is required then typically direct eigensolvers are used. The central three steps are: reduce the matrix to tridiagonal form, compute(More)
We extend the theory of Multigrid methods developed for PDE, Toeplitz and related matrices to the Block Toeplitz case. Prolongations and restrictions are defined depending on the zeroes of the generating function of the Block Toeplitz matrix. On numerical examples we compare different choices for prolongations and restrictions.
A parallel preconditioner is presented for the solution of general sparse linear systems of equations. A sparse approximate inverse is computed explicitly and then applied as a preconditioner to an iterative method. The computation of the preconditioner is inherently parallel, and its application only requires a matrix-vector product. The sparsity pattern(More)
In this paper we discuss Multigrid methods for Toeplitz matrices. Then the restriction and prolongation operator can be seen as projected Toeplitz matrices. Because of the intimate connection between such matrices and trigonometric series we can express the Multigrid algorithm in terms of the underlying functions with special zeroes. This shows how to(More)
In recent papers the use of sparse approximate inverses for the preconditioning of linear equations Ax=b is examined. The minimization of ||AM−I|| in the Frobenius norm generates good preconditioners without any a priori knowledge on the pattern of M. For symmetric positive definite A and a given a priori pattern there exist methods for computing factorized(More)
In a recent paper Chan and Chan study the use of circulant preconditioners for the solution of elliptic problems. They prove that circulant preconditioners can be chosen so that the condition number of the preconditioned system can be reduced fromO(n 2 ) toO(n). In addition, using the Fast Fourier Transform, the computation of the preconditioner is highly(More)
In this paper we will give a survey on iterative methods for solving linear equations with Toeplitz matrices. We introduce a new class of Toeplitz matrices for which clustering of eigenvalues and singular values can be proved. We consider optimal (ω)circulant preconditioners as a generalization of the circulant preconditioner. For positive definite Toeplitz(More)
We present an efficient implementation of the Modified SParse Approximate Inverse (MSPAI) preconditioner. MSPAI generalizes the class of preconditioners based on Frobenius norm minimizations, the class of modified preconditioners such as MILU, as well as interface probing techniques in domain decomposition: it adds probing constraints to the basic SPAI(More)