We describe the main features and discuss the tuning of algorithms for the direct solution of sparse linear systems on distributed memory computers developed in the context of PARASOL ESPRIT IV LTR Project No 20160. The algorithms use a multifrontal approach and are especially designed to cover a large class of problems. The problems can be symmetric positive de nite, general symmetric, or unsymmetric matrices, all possibly rank de cient, and they can be provided by the user in several formats. The algorithms achieve high performance by exploiting parallelism coming from the sparsity in the problem and that available for dense matrices. The algorithms use a dynamic distributed task scheduling technique to accommodate numerical pivoting and to allow the migration of computational tasks to lightly loaded processors. Large computational tasks are divided into subtasks to enhance parallelism. Asynchronous communication is used throughout the solution process for the e cient o verlap of communication and computation. We illustrate our design choices by experimental results obtained on a Cray SGI Origin 2000 and an IBM SP2 for test matrices provided by industrial partners in the PARASOL project.

This paper describes an extension to the set of Basic Linear Algebra Subprograms. The extensions are targeted at matrix-vector operations that should provide for efficient and portable implementations of algorithms for high-performance computers

An Approximate Minimum Degree ordering algorithm (AMD) for preordering a symmetric sparse matrix prior to numerical factorization is presented. We use techniques based on the quotient graph for matrix factorization that allow us to obtain computationally cheap bounds for the minimum degree. We s h o w that these bounds are often equal to the actual degree. The resulting algorithm is typically much faster than previous minimum degree ordering algorithms, and produces results that are comparable in quality with the best orderings from other minimum degree algorithms.

We consider the solution of both symmetric and unsymmetric systems of sparse linear equations. A new parallel distributed memory multifrontal approach is described. To handle numerical pivoting e ciently, a parallel asynchronous algorithm with dynamic scheduling of the computing tasks has been developed. We discuss some of the main algorithmic choices and compare both implementation issues and the performance of the LDL T and LU factorizations. Performance analysis on an IBM SP2 shows the e ciency and the potential of the method. The test problems used are from the Rutherford-Boeing collection and from the PARASOL end users.

We extend the frontal method for solving linear systems of equations by permitting more than one front to occur at the same time. This enables us to develop code for general symmetric systems. We discuss the orgamzation and implementatmn of a multifrontal code which uses the minimum-degree ordering and indicate how we can solve indefinite systems in a stable manner We illustrate the performance of our code both on the IBM 3033 and on the CRAY-1.