Progress in Sparse Matrix Methods for Large Linear Systems On Vector Supercomputers

TL;DR: Progress in the use of direct methods for solving very large sparse symmetric positive definite systems of linear equations on vector supercomputers is summarized.

Abstract: This paper summarizes progress in the use of direct methods for solving very large sparse symmetric positive definite systems of linear equations on vector supercomputers. Sparse di rect solvers based on the multifrontal method or the general sparse method now outperform band or envelope solvers on vector supercomputers such as the CRAY X-MP. This departure from conventional wisdom is due to several advances. The hardware gather/scatter feature or indirect address feature of some recent vector super computers permits vectorization of the general sparse factorization. Other advances are algo rithmic. The new multiple minimum degree algo rithm calculates a powerful ordering much faster than its predecessors. Exploiting the supernode structure of the factored matrix provides vectori zation over nested loops, giving greater speed in the factorization module for the multifrontal and general sparse methods. Out-of-core versions of both methods are now available. Numerical re sults on the CRAY X-MP for several structural engineering problems demonstrate the impact of these improvements.