Many of the currently popular \block algorithms" are scalar algorithms in which the operations have been grouped and reordered into matrix operations. One genuine block algorithm in practical use is block LU factorization, and this has recently been shown by Demmel and Higham to be unstable in general. It is shown here that block LU factorization is stable if A is block diagonally dominant by c...

Given a square matrix and single right and left starting vectors, the classical nonsymmetric Lanczos process generates two sequences of biorthogonal basis vectors for the right and left Krylov subspaces induced by the given matrix and vectors. In this paper, we propose a Lanczos-type algorithm that extends the classical Lanczos process for single starting vectors to multiple starting vectors. G...

We consider the SIMPLE preconditioning for block two-by-two generalized saddle point problems; this is the general nonsymmetric, nonsingular case where the 1,2 block needs not to equal the transposed 2,1 block, and the 2,2 block may not be zero. The eigenvalue analysis of the SIMPLE preconditioned matrix is presented. The relationship between the two different formulations spectrum of the SIMPL...

The row-by-row frontal method may be used to solve general large sparse linear systems of equations. By partitioning the matrix into (nearly) independent blocks and applying the frontal method to each block, a coarse-grained parallel frontal algorithm is obtained. The success of this approach depends on preordering the matrix. This can be done in two stages, (1) order the matrix to bordered blo...

In graph theory a partition of the vertex set of a graph is called equitable if for all pairs of cells all vertices in one cell have an equal number of neighbours in the other cell. Considering the implications for the adjacency matrix one may generalize that concept as a block partition of a complex square matrix s.t. each block has constant row sum. It is well known that replacing each block ...

In many applications, e.g. digital signal processing, discrete inverse scattering, linear prediction etc., Toeplitz-plus-Hankel (T + H) matrices need to be inverted. (For further applications see [1] and references therein). Firstly the T +H matrix inversion problem has been solved in [2] where it was reduced to the inversion problem of the block Toeplitz matrix (the so-called mosaic matrix). T...

We discuss formal orthogonal polynomials with respect to a moment matrix that has no structure whatsoever. In the classical case the moment matrix is often a Hankel or a Toeplitz matrix. We link this to block factorization of the moment matrix and its inverse, the block Hessenberg matrix of the recurrence relation, the computation of successive Schur complements and general subspace iterative m...

A block Toeplitz algorithm is proposed to perform the J-spectral factorization of a para-Hermitian polynomial matrix. The input matrix can be singular or indefinite, and it can have zeros along the imaginary axis. The key assumption is that the finite zeros of the input polynomial matrix are given as input data. The algorithm is based on numerically reliable operations only, namely computation ...

Reduction of a matrix to triangular form plays a crucial role in the solution of linear equations. In this chapter, I analyze a pipeline algorithm for Householder reduction (Brinch Hansen 1990). The pipeline is folded several times across an array of processors to achieve approximate load balancing. The pipeline inputs, transforms, and outputs a matrix, column by column. During the computation,...

In block-partitioned parallel matrix factorization algorithms, where the matrix is distributed over a logical torus processor grid with an rs block-cyclic matrix distribution, the greatest scope for optimization exists in the formation of (block) panels. Let ! be the panel width, with ! m being an optimal value based on the characteristics a single processor's memory hierarchy. To date, two wel...