Inexact Newton methods for inverse eigenvalue problems

In this paper, we survey some of the latest development in using inexact Newton-like methods for solving inverse eigenvalue problems. These methods require the solutions of nonsymmetric and large linear systems. One can solve the approximate Jacobian equation by iterative methods. 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-like (outer) iterations. The inexact methods can reduce or minimize the oversolving problem and improve the efficiency. The convergence rates of the inexact methods are superlinear and a good tradeoff between the required inner and outer iterations can be obtained. AMS Subject Classifications. 65F18, 65F10, 65F15.