Research On Some Algebraic Inverse Eigenvalue Problems

Algebraic inverse eigenvalue problems arise in a remarkable variety of applications, such as control theory, system identification, structure mechanics, geophysics, molecular spectroscopy, particle physics and so on. Studies on these problems have important theoretical significance and application value. This dissertation deals with some inverse eigenvalue problems for Jacobi matrices and parameterized inverse eigenvalue problems. The main contributions are as follows.Two kinds of inverse eigenvalue problems for Jacobi matrices are presented. Firstly, the problem of reconstructing a Jacobi matrix from its eigenvalues, its leading principal submatrix and part of the eigenvalues of its submatrix is considered. The necessary and sufficient conditions for the existence and uniqueness of the solution to the prblem are derived. Two numerical algorithms for solving the problem are given. Secondly, another problem of reconstructing a Jacobi matrix from part of its eigenvalues and its submatrix is studied. The necessary and sufficient conditions for the existence and uniqueness of solution to this problem are established. A numerical algorithm is also presented. Nnumerical results show that the proposed methods are efficient.A kind of inverse eigenvalue problems for Jacobi matrix, which arises from the inverse vibration problem of a fixed-free mass spring system, is considered. The solvability condition of this problem is obtained, and two numerical methods for solving the problem are proposed. Numerical examples show that these two methods are efficient and the second method is numerically stable and more suitable for large scale problem.An inverse eigenvalue problem for the finite element model of a longitudinally vibrating rod whose one end is fixed and the other end is supported on a spring is considered. It is shown that the cross section areas can be determined from the spectrum of the rod. The necessary and sufficient conditions for the construction of a physically realizable rod with positive cross section areas are established. A numerical method is presented and an illustrative example is given.Numerical methods for solving parameterized inverse eigenvalue problems are considered. In order to reduce the highly computational cost of forming Jacobian matrix in Newton’s method for solving the problems, we present Broyden’s method for solving the problems, and analyze its convergence. When the multiple eigenvalues are present, the Broyden’s method is modified, and its locally suplinear convergence is proved. A smooth LU-like decomposition of a matrix dependent on several parameters is proposed. Based on the developed LU-like decomposition with complete pivoting, the Gauss-Newton method for solving parameterized inverse eigenvalue problems is presented, and its locally quadratic convergenc is analyzed. Numerical results show that the Broyden’s method and its modification, the Gauss-Newton method are efficient for both distinct and multiple eigenvalues cases..