Inexact Newton methods for inverse eigenvalue problems

In this paper, we survey some of the latest development in using inexact Newton methods for solving inverse eigenvalue problems. These methods require the solutions of nonsymmetric and large linear systems, i.e. the large Jacobian equations. One can solve these systems by iterative methods (inner iterations). However, iterative methods usually oversolve the problem in the sense that they require far more (inner) iterations than is required for the convergence of the Newton (outer) iterations. The inexact methods can avoid the oversolving problem and hence improve the efficiency. The convergence rate of the inexact methods are superlinear and a good tradeoff between the required inner and outer iterations can be obtained.