Improving Eigenvectors in Arnoldi's Method (Journal of Computational Mathematics, 2000)
The Ritz vectors obtained by Arnoldi's method may not be good approximations and even may not converge even if the corresponding Ritz values do. In order to improve the quality of Ritz vectors and enhance the efficiency of Arnoldi type algorithms, we propose a strategy that uses Ritz values obtained from an $m$-dimensional Krylov subspace but chooses modified approximate eigenvectors in an $(m+1)$-dimensional Krylov subspace. Residual norm of each new approximate eigenpair is minimal over the span of the Ritz vector and the $(m+1)$th basis vector, which is available when the $m$-step Arnoldi process is run. The resulting modified $m$-step Arnoldi method is better than the standard $m$-step one in theory and cheaper than the standard $(m+1)$-step one. Based on this strategy, we present a modified $m$-step restarted Arnoldi algorithm.Numerical examples show that the modified $m$-step restarted algorithm and its version with Chebyshev acceleration are often considerably more efficient than the standard $(m+1)$-step restarted ones.
Keywords: Large unsymmetric; The m-step Arnoldi process; The m-step Arnoldi method; Eigenvalue; Ritz vector; Modified