A HOMOTOPY ALGORITHM FOR 2-PARAMETER EIGENVALUE PROBLEMS

We present a homotopy algorithm for a certain class of two-parameter e igenvalue problems[1,2] and give theoretical background of the method. We consider the following two-parameter eigenvalue problem: T(1)x(1) = lambda B(1)z(1) + kappa mu C(1)x(1) (1) T(2)x(2) = kappa lambda B(2) x(2) + mu C(2)x(2), (2) where matrices T-i (i = 1, 2) are n x n irredu cible real symmetric tridiagonal matrices, and matrices B-i, C-i (i = 1,2) are nonsingular diagonal matrices with diagonal elements of the s ame sign. kappa (kappa is an element of [0, 1]) is a given constant, l ambda,mu are eigenvalues to be computed, and x(1),x(2) are correspondi ng eigen vectors satisfying normalizing conditions \x(1)(t)B(1)x(1)\ = 1, \x(2)(t)C(2)x(2)\ = 1. We assume the following condition, which we call the 'definiteness condition': 1 - kappa(2)(x(1)(t)C(1)x(1))(x(2) (t)B(2)x(2)) > 0. (3) We show that if the definiteness condition is sa tisfied, we can trace homotopy curves starting from n(2) points which can be easily computed, and can reach eigenvalues of the original two- parameter eigenvalue problem. The method is particularly suitable for parallel computation.