LANCZOS-TYPE METHODS FOR CONTINUATION PROBLEMS

We study the Lanczos type methods for continuation problems. First we indicate how the symmetric Lanczos method may be used to solve both po sitive 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 eigenval ue, the minimum singular value, and the condition number of the partia l tridiagonalizations of the coefficient matrices. The Ritz vector thu s obtained can be applied to compute the tangent vector at the bifurca tion point for branch-switching. Next, we indicate that the block or b and 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 ei genvalue of a nearly symmetric matrix, and therefore to detect the sim ple bifurcation point as well. Some preconditioning techniques are dis cussed. Sample numerical results are reported. Our test problems inclu de second order semilinear elliptic eigenvalue problems. (C) 1997 by J ohn Whey & Sons, Ltd.