Many algebraic preconditioners rely on incomplete factorization, where nonzero entries are dropped, based on some rule, during the factorization process. In a series of papers by Bollhöfer and Saad [2, 3, 4] it was shown that robust rules for both dropping and pivoting can be obtained from information about the inverses of the submatrices that are consecutively constructed. In particular, monit...

We consider the multi-objective optimization problem of choosing bottom left block-entry a block lower triangular matrix to minimize ranks all sub-matrices. provide proof that there exists simultaneous rank-minimizer by constructing complete set minimizers.

Sparse linear systems Kx = b are considered where K is a specially structured symmetric indeenite matrix. These systems arise frequently, e.g., from mixed nite element discretiza-tions of PDE problems. The LDL T factorization of K with diagonal D and unit lower triangular L is known to exist for natural ordering of K but the resulting triangular factors can be rather dense. On the other hand, f...

A tridiagonal matrix with entries given by square matrices is a block tridiagonal matrix; the matrix is banded if off-diagonal blocks are upper or lower triangular. Such matrices are of great importance in numerical analysis and physics, and to obtain general properties is of great utility. The blocks of the inverse matrix of a block tridiagonal matrix can be factored in terms of two sets of ma...

We describe a new extension to ScaLAPACK [2] for computing with symmetric (Hermi-tian) matrices stored in a packed form. The new code is built upon the ScaLAPACK routines for full dense storage for a high degree of software reuse. The original ScaLAPACK stores a symmetric matrix as a full matrix but accesses only the lower or upper triangular part. The new code enables more efficient use of mem...

We describe a new extension to ScaLAPACK [2] for computing with symmetric (Hermi-tian) matrices stored in a packed form. The new code is built upon the ScaLAPACK routines for full dense storage for a high degree of software reuse. The original ScaLAPACK stores a symmetric matrix as a full matrix but accesses only the lower or upper triangular part. The new code enables more efficient use of mem...

This paper explores the problem of solving triangular linear systems on parallel distributed-memory machines. Working within the LogP model, tight asymptotic bounds for solving these systems using forward/backward substitution are presented. Specifically, lower bounds on execution time independent of the data layout, lower bounds for data layouts in which the number of data items per processor ...

Associating to each pre-order on the indices 1; :::;n the corresponding structural matrix ring, or incidence algebra, embeds the lattice of n-element pre-orders into the lattice of n n matrix rings. Rings within the order-convex hull of the embedding, i.e. matrix rings that contain the ring of diagonal matrices, can be viewed as incidence algebras of ideal-valued, generalized pre-order relation...

We present our work on the sparse Cholesky factorization using a hypermatrix data structure. First, we provide some background on the sparse Cholesky factorization and explain the hypermatrix data structure. Next, we present the matrix test suite used. Afterwards, we present the techniques we have developed in pursuit of performance improvements for the sparse hypermatrix Cholesky factorization...