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−τ I)⁻¹ in a shifted Krylov subspace (A−τ I)𝒦 _m (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.

Keywords:

• Arnoldi process
• Harmonic Ritz value
• Harmonic Ritz vector
• Ritz value
• Ritz vector

Acknowledgements

We would like to thank the referees for their valuable comments and suggestions on this paper. This work is supported by National Natural Science Foundation of China Nos. 10531080.