| An inverse eigenvalue problem is to determine a structured matrix from a given spectral data. Inverse eigenvalue problems arise in many applications, including control design, system identification, seismic tomography, principal component analysis, exploration and remote sensing, antenna array processing, geophysics, molecular spectroscopy, particle physics, structure analysis, circuit theory, Hopfield neural networks, mechanical system simulation, and so on. In this thesis, we first note that Method III, originally proposed by Friedland, Nocedal, and Overton for solving inverse eigenvalue problems, is a Newton-type method. When the inverse problem is large, one can solve the 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 (outer) iterations. To overcome the shortcoming of Method III, we provide an inexact method, called inexact Cayley transform method, for solving inverse eigenvalue problems. Our inexact Cayley transform method can minimize the oversolving problem and improve the efficiency. Then we consider the solvability of the inverse eigenproblems for two special classes of matrices. The sufficient and necessary conditions are obtained. Also, we discuss the best approximation problems for the two special inverse eigenproblems. We show that the best approximations are unique and provide explicit expressions for the optimal solution. Moreover, we respectively propose the algorithms for computing the optimal solutions to the two best approximation problems. (Abstract shortened by UMI.)... |
