Systolic algorithms for solving a sparse system of linear equations in circuit simulation

Abstract

Systolic algorithms providing near ideal speedups have been developed for solving a dense system of linear equations. However, in applications like circuit simulation, the resulting matrix equation is quite sparse ( > 99% zeros) which makes the dense matrix solution approach inefficient. Although sparse matrix solvers for circuit simulation have been devised for parallel shared memory machines and vector processors, their performance in terms of speedup has been limited due to tightly coupled and unstructured sparse equations. This paper first develops a systolic sparse matrix algorithm by modifying an existing partitionable dense matrix solution algorithm. The sparse algorithm is then further improved to include parallel execution by multiple systolic processor modules. Reordering schemes based on a modified Markowitz algorithm are also developed that minimize the fill-in-during the matrix solution by our sparse algorithms thereby improving the execution time. Results on several randomly generated sparse matrices and those generated from ISCAS benchmark circuits are presented to show the efficiency of our sparse algorithm and its excellent speedup as the number of processors in the systolic array is increased.