  In this paper, we first establish the convergence criteria of the residual
iteration method for solving quadratic eigenvalue problem- s. We analyze the
impact of shift point and the subspace expansion on the convergence of this
method. 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, we propose an inner and outer iteration version of
the residual iteration method. The new method uses the iterative method to
solve the equations and uses the approximate solution to expand the subspace.
We analyze the relationship between inner and outer iterations and provide a
quantita- tive criterion for the inner iteration which can ensure the
convergence of the outer iteration. Finally, our numerical experiments provide
proof of our analysis and demonstrate the effectiveness of the inexact residual
iteration method.

Numerical Analysis