Refined Arnoldi-Chebyshev Method For Large Unsymmetric Eigenproblems

Arnoldi method is very efficient for computing the extreme eigenpairs of large unsymmetric matrices. But it also has problems: it needs many iterative steps when it's convergent, so it's complicated in computing; meanwhile to some eigenproblems, the corresponding approximate eigenvectors always converge very slowly even sometimes don't converge when approximate eigenvalues have converged. Thereby pre-treatment accelerating technique and refined strategy is usually used in it. Many methods have presented, such as Arnoldi-Chebyshev method, Arnoldi-QR method and refined Arnoldi method.This paper researches on Arnoldi-Chebyshev method and refined Arnoldi method, then advances some new algorithms. Firstly, we research and compare the Arnoldi-Chebyshev method and refined Arnoldi method, analyse their convergence. Moreover we simplify or improve some complicated parts. Then we advance the refined Arnoldi-Chebyshev method by analysing the theories for acceleration of Arnoldi-Chebyshev method and refined Arnoldi method.By means of numerical experiments and theory analysis, simplified Arnoldi-Chebyshev method and refined Arnoldi method accelerate convergence in Arnoldi method well. And improved refined Arnoldi method and refined Arnoldi-Chebyshev method have better convergence than former accelerating methods.