Lanczos-type Methods for Continuation Problems

Abstract

We study the Lanczos type methods for continuation problems. First we indicate how the symmetric Lanczos method may be used to solve both positive definite and indefinite linear systems. Furthermore, it can be used to monitor the simple bifurcation points on the solution curve of the eigenvalue problems. This includes computing the minimum eigenvalue, the minimum singular value, and the condition number of the partial tridiagonalizations of the coefficient matrices. The Ritz vector thus obtained can be applied to compute the tangent vector at the bifurcation point for branch-switching. Next, we indicate that the block or band Lanczos method can be used to monitor the multiple bifurcations as well as to solve the multiple right hand sides. We also show that the unsymmetric Lanczos method can be exploited to compute the minimum eigenvalue of a nearly symmetric matrix, and therefore to detect the simple bifurcation point as well. Some preconditioning techniques are discussed. Sample numerical results are reported. Our test problems include second order semilinear elliptic eigenvalue problems. © 1997 by John Wiley & Sons, Ltd.