A new shift strategy for the implicitly restarted generalized second-order Arnoldi method

摘要

In this paper, a new shift strategy for the implicitly restarted generalizedsecond-order Arnoldi (GSOAR) method is proposed. In implicitly restartedprocesses, we can get a $k$-step GSOAR decomposition from a $m$-step GSOARdecomposition by performing $p = m-k$ implicit shifted QR iterations. Theproblem of the implicitly restarted GSOAR is the mismatch between the number ofshifts and the dimension of the subspace. There are $2p$ shifts for $p$ QRiterations. We use the shifts to filter out the unwanted information in thecurrent subspace; when more shifts are used, one obtains a better updatedsubspace. But, if we use more than $p$ shifts, the structure of the GSOARdecomposition will be destroyed. We propose a novel method which can use all$2p$ candidates and preserve the special structure. The new method vastlyenhances the overall efficiency of the algorithm. Numerical experimentsillustrate the efficiency of every restart process.