Homogeneous Jacobi-Davidson
Michiel E. Hochstenbach, Yvan Notay · TU/e Research Portal · 2006
Abstract. We study a homogeneous variant of the Jacobi–Davidson method for the generalized and polynomial eigenvalue problem. Special attention is given to the subspace expansion and the projection present in the correction equation. The resulting method can deal with both finite and infinite eigenvalues in a natural and unified way. We show relations with the multihomogeneous Newton method, Rayleigh quotient iteration, and (standard) Jacobi–Davidson for polynomial eigenproblems. Key words. Homogeneous form, quadratic eigenvalue problem, generalized eigenvalue problem, polynomial eigenvalue problem, infinite eigenvalues, correction equation, subspace method, subspace expansion, large sparse matrices, bihomogeneous Newton, multihomogeneous Newton, Rayleigh quotient iteration, Jacobi–Davidson. AMS subject classifications. 65F15, 65F50. 1. Introduction. We study a homogeneous Jacobi–Davidson variant for the polynomial eigenproblem (1.1) P (λ) x = (λ m Al + λ m−1 + · · · + A0) x = 0, where the matrices Ai are (possibly large sparse) n × n matrices with real or complex entries.