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We develop a novel algorithm for the parallel factorization of an unsymmetric sparse matrix using a Markowitz threshold algorithm. Our algorithm uses a significant extension of a maximum independent set algorithm of Luby to select a block of independent pivots in parallel. We then use this block to update the matrix again in parallel. This algorithm does not require any symmetry in the matrix and is particularly efficacious when the matrix is structurally very unsymmetric using metrics defined in the paper. We implement this algorithm using OpenMP and show its performance against other state-of-the-art codes on a standard set of sparse matrix test problems. [ABSTRACT FROM AUTHOR]

Abstract: We consider the factorization of sparse skew symmetric matrices. We see that the pivoting strategies are similar, but simpler, to those used in the factorization of sparse symmetric indefinite matrices, and we briefly describe the algorithms used in a forthcoming direct code based on multifrontal techniques for the factorization of real skew symmetric matrices. We show how this factorization can be very efficient for preconditioning matrices that have a large skew component. [Copyright &y& Elsevier]

Abstract: We design, analyse and test a class of incomplete orthogonal factorization preconditioners constructed from Givens rotations, incorporating some dropping strategies and updating tricks, for the solution of large sparse systems of linear equations. Comprehensive accounts about how the preconditioners are coded, what storage is required and how the computation is executed for a given accuracy are presented. A number of numerical experiments show that these preconditioners are competitive with standard incomplete triangular factorization preconditioners when they are applied to accelerate Krylov subspace iteration methods such as GMRES and BiCGSTAB. [Copyright &y& Elsevier]

We consider the direct solution of general sparse linear systems baseds on a multifrontal method. The approach combines partial static scheduling of the task dependency graph during the symbolic factorization and distributed dynamic scheduling during the numerical factorization to balance the work among the processes of a distributed memory computer. We show that to address clusters of Symmetric Multi-Processor (SMP) architectures, and more generally non-uniform memory access multiprocessors, our algorithms for both the static and the dynamic scheduling need to be revisited to take account of the non-uniform cost of communication. The performance analysis on an IBM SP3 with 16 processors per SMP node and up to 128 processors shows that we can significantly reduce both the amount of inter-node communication and the solution time. [Copyright &y& Elsevier]

We present an incremental approach to 2-norm estimation for triangular matrices. Our investigation covers both dense and sparse matrices which can arise for example from a QR, a Cholesky or a LU factorization. If the explicit inverse of a triangular factor is available, as in the case of an implicit version of the LU factorization, we can relate our results to incremental condition estimation (ICE). Incremental norm estimation (INE) extends directly from the dense to the sparse case without needing the modifications that are necessary for the sparse version of ICE. INE can be applied to complement ICE, since the product of the two estimates gives an estimate for the matrix condition number. Furthermore, when applied to matrix inverses, INE can be used as the basis of a rank-revealing factorization. [ABSTRACT FROM AUTHOR]

We present a class of incomplete orthogonal factorization methods based on Givens rotations for large sparse unsymmetric matrices. These methods include: Incomplete Givens Orthogonalization (IGO-method) and its generalisation (GIGO-method), which drop entries from the incomplete orthogonal and upper triangular factors by position; Threshold Incomplete Givens Orthogonalization (TIGO(τ)-method), which drops entries dynamically by their magnitudes; and its generalisation (GTIGO(τ,p)-method), which drops entries dynamically by both their magnitudes and positions. Theoretical analyses show that these methods can produce a nonsingular sparse incomplete upper triangular factor and either a complete orthogonal factor or a sparse nonsingular incomplete orthogonal factor for a general nonsingular matrix. Therefore, these methods can potentially generate effficient preconditioners for Krylov subspace methods for solving large sparse systems of linear equations. Moreover, the upper triangular factor is an incomplete Cholesky factorization preconditioner for the normal equations matrix from least-squares problems. [ABSTRACT FROM AUTHOR]

Sparse matrix factorization algorithms for general problems are typically characterized by irregular memory access patterns that limit their performance on parallel-vector supercomputers. For symmetric problems, methods such as the multifrontal method avoid indirect addressing in the innermost loops by using dense matrix kernels. However, no efficient LU factorization algorithm based primarily on dense matrix kernels exists for matrices whose pattern is very unsymmetric. We address this deficiency and present a new unsymmetric-pattern multifrontal method based on dense matrix kernels. As in the classical multifrontal method, advantage is taken of repetitive structure in the matrix by factorizing more than one pivot in each frontal matrix, thus enabling the use of Level 2 and Level 3 BLAS. The performance is compared with the classical multifrontal method and other unsymmetric solvers on a CRAY C-98. [ABSTRACT FROM AUTHOR]