An invert-free Arnoldi method for computing interior eigenpairs of large matrices

Abstract

This paper presents an invert-free Arnoldi method for extracting a few interior eigenpairs of large sparse matrices. It is derived by implicitly applying the Arnoldi process with the shifted and inverted operator (A-tau I)(-1) in a shifted Krylov subspace (A-tI) Km( A, v(1)). Due to a subtle relationship between the Krylov subspace K-m(A, v(1)) and its shifted Krylov subspace, we avoid forming the shifted and inverted operator explicitly. Comparisons are drawn between the harmonic Arnoldi method and the invert-free Arnoldi method. Finally, numerical results are reported to show the efficiency of the new method.