Fast algorithms for block Toeplitz matrices

Abstract

This paper presents the solution of linear systems with block Toeplitz matrices by constructing block versions of the Fade approximation. This approach yields a fast method of solving systems with low Toeplitz rank matrices. Also discussed are the results on the description of inverse matrices for some special classes of matrices and new possibilities of realizing the method of Toeplitz embedding'.

Structured preconditioners for operator equations

“…The use of circulants and block circulants as preconditioners for general (non-Toeplitz) matrices has been studied in [5,6]; application of circulants for some matrices of the boundary integral method is discussed in Reference [7]. Notice a detailed analysis of spectral equivalence and superlinearity in this case that can be found in Reference [8].…”

Transformation Du Problème De Résolution De Systèmes De Toeplitz Biniveaux À Un Problème Polynomial

“…A ce jour, il n'existe pas de résultats théoriques décrivant un algorithme rapide de résolution pour des systèmes de Toeplitz biniveaux généraux. Voir [11] pour plus d'informations sur les applications et les difficultés des matrices de Toeplitz biniveaux.…”

Constructive approach to develop vectorized and fast algorithms for special-type matrices

“…While constructing algorithms one does not have to seek the maximal but a 'reasonable' parallelism providing for certain structural properties. Therefore, a particular attention has been recently paid to the development of vectorized algorithms [1,6,[15][16][17]23] which sufficiently well meet the requirements of a wide class of computer systems such as vector pipelined computers, systolic architecture computers, etc.…”

“…In this paper, we consider the development of one of them suggested in [15,23] and actually dealing with a partially formalized transformation of a number of sequential algorithms into vectorized ones. Of particular interest is also the approach recently described in [11] where similar vectorized algorithms were constructed not by transforming known algorithms but in the direct manner using the special bordering of leading submatrices.…”