ncku-talk2014-02-10

NCTS(South) Seminar

DATE 2014-02-10　16:00-17:00

PLACE R204, 2F, NCTS, NCKU

SPEAKER 李鐵香教授（東南大學數學學院）

TITLE A Semiorthogonal Generalized Arnoldi Method and Its Variations for Quadratic Eigenvalue Problems

ABSTRACT In this paper, we are concerned with the computation of a few eigenpairs with smallest eigenvalues in absolute value of quadratic eigenvalue problems. We first develop a semiorthogonal generalized Arnoldi method where the name comes from the application of a pseudo inner product in the construction of a generalized Arnoldi reduction for a generalized eigenvalue problem. The method applies the Rayleigh-Ritz orthogonal projection technique on the quadratic eigenvalue problem. Consequently it preserves the spectral properties of the original quadratic eigenvalue problem. Furthermore, we propose a refinement scheme to improve the accuracy of the Ritz vectors for the quadratic eigenvalue problem. Given shifts, we also show how to restart the method by implicitly updating the starting vector and constructing better projection subspace. We combine the ideas of the refinement and the restart by selecting shifts upon the information of refined Ritz vectors. Finally an implicitly restarted refined semiorthogonal generalized Arnoldi method is developed. Numerical examples demonstrate that the implicitly restarted semiorthogonal generalized Arnoldi method with or without refinement has superior convergence behaviors than the implicitly restarted Anoldi method applied to the linearized quadratic eigenvalue problem.

NCTS(South) Seminar

DATE	2014-02-10　16:00-17:00

PLACE	R204, 2F, NCTS, NCKU

SPEAKER	李鐵香教授（東南大學數學學院）

TITLE	A Semiorthogonal Generalized Arnoldi Method and Its Variations for Quadratic Eigenvalue Problems

ABSTRACT	In this paper, we are concerned with the computation of a few eigenpairs with smallest eigenvalues in absolute value of quadratic eigenvalue problems. We first develop a semiorthogonal generalized Arnoldi method where the name comes from the application of a pseudo inner product in the construction of a generalized Arnoldi reduction for a generalized eigenvalue problem. The method applies the Rayleigh-Ritz orthogonal projection technique on the quadratic eigenvalue problem. Consequently it preserves the spectral properties of the original quadratic eigenvalue problem. Furthermore, we propose a refinement scheme to improve the accuracy of the Ritz vectors for the quadratic eigenvalue problem. Given shifts, we also show how to restart the method by implicitly updating the starting vector and constructing better projection subspace. We combine the ideas of the refinement and the restart by selecting shifts upon the information of refined Ritz vectors. Finally an implicitly restarted refined semiorthogonal generalized Arnoldi method is developed. Numerical examples demonstrate that the implicitly restarted semiorthogonal generalized Arnoldi method with or without refinement has superior convergence behaviors than the implicitly restarted Anoldi method applied to the linearized quadratic eigenvalue problem.