Two-sided Quadratic Residual Iteration

Qudratic residual iteration(QRI) is an algorithm to find the approximate eigenvalue and the corresponding right appriximate eigenvector of the quadratic eigenvalue problem(QEP).QRI generate an orthogonal basis,and orthogonal projection is applied to the QEP to reduce to the QEP with matrix dimension of lower order.This paper introduce a variant of QRI called two-sided quadratic residual iteration(TQRI).We use TQRI to solve the approximate eigenvalue and the corresponding left and right approximate eigenvector(or Ritz vector).Given a pair of starting vectors,TQRI generate biorthonormal bases of two subspaces.And the oblique projection is considered.The left and right approximating eigenvector can be solved from the two spaces at the same time.And the convergence behaviors of both right and left appriximating eigenvectors are essentially the same.Numerical experiment is given to identify the method.The advantage of TQRI is mainly embodied in two aspects.First,the Ritz value can approach the exact value more exactly.Second,it can approach the left eigenvector directly, especially for the nonsymmetric eigenvalue problem.In chapter 1,we give an introduction of quadratic eigenvalue problem(QEP), and the methods to solve it.In chpter 2,we introduce the Residual inverse iteration and the convergence behaivors of the approximate eigenvector and the approximate eigenvalue.In chapter 3,we introduce quadratic residual iteration. In chpter 4,we first introduce the motivation of the two-sided quadratic residual iteration we prsent in the paper in section 4.1.In section 4.2,we present the two-sided quadratic residual iteration.Lastly,we present the superiority of our method using the numerical experiments.