Parallel homotopy algorithm for symmetric large sparse eigenproblems

In this paper, the homotopy continuation method is applied to solve the eigenproblem Ax = λx, λ ∈ R, x ∈ Rn⧹ {0} for a symmetric large sparse matrix A. A one-parameter family of matrices A(t) = tA + (1 − t)D is introduced and the eigenproblem A(t)x(t) = λ(t)x(t) is considered for t ∈ [0,1]. We discuss the problem of choosing an optimal starting matrix A(0) = D and consider the regularity and bifurcation problem of λ(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.