Abstract
An existing general sparse-elimination algorithm was benchmarked on the Cyber 205 and the Cray 1-S. Computation rates on the Cyber 205 reached 16. 7 million floating point operations per second (MFLOPS) and 9. 8 MFLOPS on the Cray 1-S. It is concluded that the existing algorithm does not exploit the full potential of the vector computer. Several schemes are under investigation to improve the performance of the algorithm.
Introduction
General-purpose sparse-elimination techniques have been used successfully in solving systems of linear equations in reservoir simulation. These techniques allow Gaussian elimination to be performed efficiently with any given matrix ordering. The implementation of these techniques on a scalar computer such as the IBM 370/158 and on an array processor such as the Floating Point Systems AP120B are described in Refs. 1 and 2. respectively. The purpose of this paper is to report the results of a benchmark test to determine the performance of a known sparse-elimination algorithm on the vector computers Cyber 205 and Cray 1-S. The ground rules for the benchmark were as follows.The sparse-elimination algorithm was to remain unchanged.The use of vectorizable FORTRAN or vector syntax FORTRAN was allowed.The use of assembly-language coding was not allowed unless the assembly-language code was part of a systems or mathematics library callable by FORTRAN.The alternate diagonal grid ordering was to be used.
Implementation on the Vector Computer
The algorithm NNF in the Yale sparse matrix packages was used for benchmarking. The matrix equation to be solved is
A is factored into the lower triangular matrix, L, and the upper triangular matrix, U. The elements of A, L, and U are stored in a compressed storage mode as described by Gustavson. The solution vector x is obtained by forward substitution by using L, and backward substitution by using U. The Yale algorithm uses the row operations approach, which consists of two steps:multiply one equation by a nonzero number, andadd (or subtract) the equation from above to (or from) another equation.
Step 2 reduces the nonzeros in the lower triangular portion of A to zeros as factorization proceeds. portion of A to zeros as factorization proceeds. SPEJ
P. 743
Gaussian Elimination versus Greedy Methods for the Synthesis of Linear Reversible Circuits
