This paper considers key ideas in the design of out-of-core dense LU factorization routines. A left-looking variant of the LU factorization algorithm is shown to require less I/O to disk than the rightlooking variant, and is used to develop a parallel, out-of-core implementation. This implementation makes use of a small library of parallel I/O routines, together with ScaLAPACK and PBLAS routine...

In this paper we consider a direct method to solve a sparse unsymmetric system of linear equations Ax = b, which is the Gaussian elimination. This elimination consists in explicitly factoring the matrix A into the product of L and U , where L is a unit lower triangular matrix, and U is an upper triangular matrix, followed by solving LUx = b one factor at a time. One of the main characteristics ...

Complexity/Cost We stated last time that for a system of n equations in n unknowns (represented by an N × N matrix A), both LU and QR-factorization take time proportional to the cube of n (O(n3)). LU-factorization takes less time by a constant factor of around 2 as n gets large, this constant factor is dwarfed by the third power of the matrix dimension, however. Note that the theoretical “best-...

Iterative methods are often suitable for solving least-squares problems minkAx bk2 , where A 2 R n is large and sparse. The use of the conjugate gradient method with a nonsingular square submatrix A1 2 R n of A as preconditioner was first suggested by Läuchli in 1961. This conjugate gradient method has recently been extended by Yuan to generalized least-squares problems. In this paper we consid...

In this paper we present a new incomplete LU decomposition which is based on an existing sparse direct solver. In contrast to many incomplete LU decompositions this ILU incorporates information about the inverse factors L−1 and U−1 which have direct influence on the dropping strategy. We demonstrate in several large scale examples that this implementation constructs a robust preconditioner.

In this paper, we make eecient use of pipelining on LU decomposition with pivoting and a column-scattered data decomposition to derive precise variations of the computational complexities. We then compare these results with experiments on the Intel iPSC/860 and Paragon machines.

The paper describes a parallel algorithm for the LU fac-torization of sparse matrices on distributed memory machines by using SPMD as programming model and PVM as message passing interface. We address all the diiculties arising in sparse codes, as the ll-in or the dynamic movement of data inside the matrix. The cyclic distribution has been used to evenly distribute the elements onto a mesh of p...

This paper presents a new fine-grained parallel algorithm for computing an incomplete LU factorization. All nonzeros in the incomplete factors can be computed in parallel and asynchronously, using one or more sweeps that iteratively improve the accuracy of the factorization. Unlike existing parallel algorithms, the new algorithm does not depend on reordering the matrix. Numerical tests show tha...

Since the cost of communication (moving data) greatly exceeds the cost of doing arithmetic on current and future computing platforms, we are motivated to devise algorithms that communicate as little as possible, even if they do slightly more arithmetic, and as long as they still get the right answer. This paper is about getting the right answer for such an algorithm. It discusses CALU, a commun...

Sparse LU factorization with partial pivoting is important for many scienti c applications and delivering high perfor mance for this problem is di cult on distributed memory machines Our previous work has developed an approach called S that incorporates static symbolic factorization supernode partitioning and graph scheduling This paper studies the properties of elimination forests and uses the...