We i n v estigate several ways to improve the performance of sparse LU factorization with partial pivoting, as used to solve unsymmetric linear systems. To perform most of the numerical computation in dense matrix kernels, we i n troduce the notion of unsymmetric supernodes. To better exploit the memory hierarchy, w e i n troduce unsymmetric supernode-panel updates and two-dimensional data partitioning. To speed up symbolic factorization, we use Gilbert and Peierls's depth-rst search with Eisenstat and Liu's symmetric structural reductions. We h a v e implemented a sparse LU code using all these ideas. We present experiments demonstrating that it is signi cantly faster than earlier partial pivoting codes. We also compare performance with Umfpack, which uses a multifrontal approach; our code is usually faster.Keywords: sparse matrix algorithms; unsymmetric linear systems; supernodes; column elimination tree; partial pivoting. AMSMOS subject classi cations: 65F05, 65F50. Computing Reviews descriptors: G.1.3 Numerical Analysis : Numerical Linear Algebra | Linear systems direct and iterative methods, Sparse and very large systems.

Abstract. In this paper we generalize the hierarchically semiseparable (HSS) representations and propose some fast algorithms for HSS matrices. We provide a new linear complexity U LV T factorization algorithm for symmetric positive definite HSS matrices with small off-diagonal ranks. The corresponding factors can be used to solve compact HSS systems also in linear complexity. Numerical examples demonstrate the efficiency of the solver. We also present fast algorithms including new HSS structure generation, HSS form Cholesky factorization, and model compression. These algorithms are useful for problems where off-diagonal blocks have small numerical ranks.

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This paper describes a new software package for performing arithmetic with an arbitrarily high level of numeric precision. It is based on the earlier MPFUN package [2], enhanced with special IEEE floating-point numerical techniques and several new functions. This package is written in C++ code for high performance and broad portability and includes both C++ and Fortran-90 translation modules, so that conventional C++ and Fortran-90 programs can utilize the package with only very minor changes. This paper includes a survey of some of the interesting applications of this package and its predecessors.

Although Gaussian elimination with partial pivoting is a robust algorithm to solve unsymmetric sparse linear systems of equations, it is di cult to implement e ciently on parallel machines, because of its dynamic and somewhat unpredictable way of generating work and intermediate results at run time. In this paper, we present an e cient parallel algorithm that overcomes this di culty. The high performance of our algorithm is achieved through 1 using a graph reduction technique and a supernode-panel computational kernel for high single processor utilization, and 2 scheduling two t ypes of parallel tasks for a high level of concurrency. One such task is factoring the independent panels on the disjoint subtrees in the column elimination tree of A. Another task is updating a panel by previously computed supernodes. A scheduler assigns tasks to free processors dynamically and facilitates the smooth transition between the two t ypes of parallel tasks. No global synchronization is used in the algorithm. The algorithm is well suited for shared memory machines SMP with a modest number of processors. We demonstrate 4 7 fold speedups on a range of 8 processor SMPs, and more on larger SMPs. One realistic problem arising from a 3-D ow calculation achieves factorization rates of 1.0, 2.5, 0.8 and 0.8 Giga ops, on the 12 processor Power Challenge, 8 processor Cray C90, 16 processor Cray J90, and 8 processor AlphaServer 8400, respectively.