The inexact residual iteration method for quadratic eigenvalue problem and the analysis of convergence

The residual iteration method is a kind of direct projection methods commonly used for solving the quadratic eigenvalue problem. The convergence criterion of the residual iteration method was established, and the impact of shift point and subspace expansion on the convergence of this method has been analyzed. In the process of expanding subspace, this method needs to solve a linear system at every step. For large scale problems in which the equations cannot be solved directly, an inner and outer iteration version of the residual iteration method was proposed. The new method uses the iterative method to solve the equations and uses the approximate solution to expand the subspace. Based on analyzing the relationship between inner and outer iterations, a quantitative criterion for the inner iteration was established which can ensure the convergence of the outer iteration. Finally, the numerical experiments confirm the theory.