PARALLEL HOMOTOPY ALGORITHM FOR SYMMETRICAL LARGE SPARSE EIGENPROBLEMS

In this paper, the homotopy continuation method is applied to solve the eigenproblem Ax = lambda x, lambda is an element of R, x is an element of R(n)\{0} or a symmetric large sparge matrix A. A one-parameter family of matrices A(t) = tA + (1 - t) D is introduced and the eigenproblem A(t)x(t) = lambda(t)x(t) is considered for t is an element of [0, 1]. We discuss the problem of choosing an optimal starting matrix A(0) = D and consider the regularity and bifurcation problem of lambda(t) and x(t). A homotopy continuation algorithm is constructed and implemented on both parallel and vector machines for several types of matrices. The numerical experiments show that our method is efficient and highly parallel.