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

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A set of level 3 basic linear algebra subprograms

SUMMARY This paper describes the LINPACK Benchmark and some of its variations commonly used to assess the performance of computer systems. Aside from the LINPACK Benchmark suite, the TOP500 and the HPL codes are presented. The latter is frequently used to obtained results for TOP500 submissions. Information is also given on how to interpret the results of the benchmark and how the results fit into the performance evaluation process. Copyright c � 2003 John Wiley & Sons, Ltd.

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The LINPACK Benchmark: past, present and future

This paper presents an overview of the multifrontal method for the solution of large sparse symmetric positive definite linear systems. The method is formulated in terms of frontal matrices, update matrices, and an assembly tree. Formal definitions of these notions are given based on the sparse matrix structure. Various advances to the basic method are surveyed. They include the role of matrix reorderings, the use of supernodes, and other implementatjon techniques. The use of the method in different computational environments is also described.

The multifrontal method for sparse matrix solution: theory and practice

This paper surveys recent progress in the development of parallel algorithms for solving sparse linear systems on computer architectures having multiple processors. Attention is focused on direct methods for solving sparse symmetric positive definite systems, specifically by Cholesky factorization. Recent progress on parallel algorithms is surveyed for all phases of the solution process, including ordering, symbolic factorization, numeric factorization, and triangular solution.

Parallel algorithms for sparse linear systems

Wilkinson defined a sparse matrix as one with enough zeros that it pays to take advantage of them.1 This informal yet practical definition captures the essence of the goal of direct methods for solving sparse matrix problems. They exploit the sparsity of a matrix to solve problems economically: much faster and using far less memory than if all the entries of a matrix were stored and took part in explicit computations. These methods form the backbone of a wide range of problems in computational science. A glimpse of the breadth of applications relying on sparse solvers can be seen in the origins of matrices in published matrix benchmark collections (Duff and Reid 1979a, Duff, Grimes and Lewis 1989a, Davis and Hu 2011). The goal of this survey article is to impart a working knowledge of the underlying theory and practice of sparse direct methods for solving linear systems and least-squares problems, and to provide an overview of the algorithms, data structures, and software available to solve these problems, so that the reader can both understand the methods and know how best to use them.

A survey of direct methods for sparse linear systems

The CRAY-2 is sometimes viewed as a CRAY-1 with a faster cycle time. We show that this viewpoint is not completely appropriate by describing some of the important architectural differences between the machines and indicating how they can be used to facilitate numerical calculations. We have been testing kernels and codes on a CRAY-2 prior to the delivery of a machine to Harwell in 1987 and report some results on the solution of sparse equations which indicate that high efficiency can be obtained. We give details of one of the techniques for attaining this performance. We also comment on the use of parallelism in the solution of sparse linear equations on the CRAY-2.

Ameet K Dave

Papers

Sparse matrix calculations on the CRAY-2