A Global Arnoldi Method for Large non-Hermitian Eigenproblems with Special Applications to Multiple Eigenproblems∗

Global projection methods have been used for solving numerous large matrix equations, but nothing has been known on if and how a global projection method can be proposed for solving large eigenproblems. In this paper, based on the global Arnoldi process that generates an Forthonormal basis of a matrix Krylov subspace, a global Arnold method is proposed for large eigenproblems. It computes certain F-Ritz pairs that are used to approximate some eigenpairs. The global Arnoldi method inherits convergence properties of the standard Arnoldi method applied to a larger matrix whose distinct eigenvalues are the eigenvalues of the original given matrix. Some properties of the F-Ritz vectors are presented. As an application, assuming that A is diagonalizable, we show that the global Arnoldi method is able to solve multiple eigenvalue problems both in theory and practice. To be practical, we develop an implicitly restarted global Arnoldi algorithm with certain F-shifts suggested. In particular, this algorithm can be adaptively used to solve multiple eigenvalue problems. Numerical experiments show that the algorithm is efficient for the eigenproblem and is reliable for quite ill-conditioned multiple eigenproblems.