Restarted GMRES augmented with harmonic Ritz vectors for shifted linear systems

Abstract

Frommer and Glassner [Frommer, A. and Glassner, U., 1998, Restarted GMRES for shifted linear systems, SIAM Journal on Scientific Computing, 19, 15–26.] develop a variant of the restarted GMRES method for shifted linear systems at the expense of only one matrix–vector multiplication per iteration. However, restarting slows down the convergence, even stagnation. We present a variant of the restarted GMRES augmented with some approximate eigenvectors for the shifted systems. The convergence can be much faster at little extra expense. Numerical experiments show its efficiency.

Keywords:

• Shifted linear systems
• Augmented GMRES method
• Harmonic Ritz vector
• Restarting

Acknowledgements

Supported by the National Natural Science Foundation of China (10271075) and the Science and Technology Developing Foundation of University in Shanghai of China (02AK41).