An efficient nonsymmetric Lanczos method on parallel vector computers

Abstract

In this paper, we introduce the s-step biorthogonal Lanczos method for finding a few eigenvalues of a large sparse nonsymmetric matrix, and we prove that the s-step method generates reduction matrices which are similar to reduction matrices generated by the standard method. We prove that the breakdown conditions of the s-step method are less stringent than the standard one. One iteration of the s-step biorthogonal Lanczos algorithm corresponds to s iterations of the standard biorthogonal Lanczos algorithm, and the s-step method has improved data locality and minimized global communication and superior parallel properties to the standard one on parallel machines (Chronopoulos and Gear (1989) and Kim and Chronopoulos (1991)). We implement the s-step biorthogonal Lanczos method on the CRAY-2 super computer and discuss the breakdown conditions and demonstrate the superior performance of the s-step method to the standard one