A Variant Of The Iterated Shift-and-invert Arnoldi Method For Quadratic Eigenvalue Problems

The direct projection method is the most popular method for solving large scale quadratic eigenvalue problems(QEP).This kind of the method projects the large QEP to a well chosen low- dimension subspace in order to preserve the structure of the original QEP.The iterated shift-and-invert Arnoldi algorithm is a new projection method, it combines with a shift-and-invert transformation and employs the Rayleigh-Ritz procedure to form the approximate eigenpairs.But the further theory analysis prove that the method may converge irregularly:the Ritz vectors are more difficult to converge compared with the corresponding Ritz values.Based on a residual norm minimizing scheme,a variant of the iterated shift-and-invert Arnoldi method is proposed in this paper.The Ritz value is used as the approximate eigenvalue,while the approximate eigenvector is derived by satisfying a certain optimal property,and it can be computed by a small sized singular value problem.Comparisons are done by numerical experiments and it shows that the new method has better performance.This paper includes four parts.In the first part,related problems and background is introduced and the basic method to solve the large scale QEP is also included.In the second part,we briefly describe the iterated shift-and-invert Arnoldi method.In the third part,based on describing the refined vector,the main idea of a variant of the iterated shift-and-invert Arnoldi method is presented.The last part is the numerical experiment.We test a few problems and the results show that the new variant has better performance than the original algorithm.