Inverse Eigenvalue Problems For Jacobi Matrices And Other Inverse Problems

In this dissertation, we mainly discussed inverse eigenvalue problems for Jacobi matrices. The dissertation divided into six parts.In the first part, we introduced the classification, importance and applications of inverse eigenvalue problems. Then we discussed several classical inverse problems.In the second part, we gave several basic and essential knowledge of inverse eigenvalue problems for Jacobi matrices: such as the properties of tridiagonal matrices, Jacobi matrices, orthogonal polynomials, Gauss quadrature formula and (k) inverse eigenvalue problem for Jacobi matrices.In the third part, we first introduced a new solution of an inverse problem for "Fixed-Fixed" and "Fixed-Free" Spring-Mass Systems. We formulated these equations by the changes in spring length from the equilibriums instead of formulating by the displacements of masses from their equilibrium positions, so the problem is easier to compute. Then we gave direct relations and equivalence property of the two forms.The fourth part is the central of this dissertation. Firstly, we introduced research history and advances in the inverse eigenvalue problem for Double Dimensional Jacobi matrices. Then, we proposed three algorithms for this problem. These three algorithms can avoid to recompute the nth order head submatrix of the Jacobi matrix. Moreover, the last two algorithms are more stable, their precision can match well with the algorithm of Boley and Golub. The first method base on the solution of inverse eigenvalue problem for (k) Jacobi matrices, Newton interpolation, and bisection method. The second method base on the solution of inverse eigenvalue problem for (k) Jacobi matrices too. Boley and Golub used Gauss quadrature formula to solve the inverse problem, but I used other different methods and thoughts. I obtained a new decomposition form of orthogonal matrix consisted by the unit eigenvectors of Jacobi matrix and found the direct relations between the first row and the orther rows of this orthogonal matrix. So I obtained a new stable method that can avoid computing the characteristic polynomial for getting the eigenvalues of the tail submatrix. Based on this idea and the Divide and Conquer method, I obtained another new stable method. The numerical examples showed these methods are quite good.In the fifth part, we obtained a solution of the boundary function for the Laplace equation by knowing some of its interior point's values of five-point difference approximation. The method- is to use linear combination of four lin-ear functions on the every boundary line segments to approximate the unknown boundary function. When the boundary function is linear on boundary line segments, the combination is just the same of the boundary function. When the boundary function is nearly linear, the combination is very approximate to the boundary function. At last, we gave an upper bound of the difference of the combination and the boundary.In the sixth part, I put forwarded several new inverse problems in actual life, music, physics, biology, chemistry, sociology, psychology, etc. Wish these inverse problems could attract more experts and scholars on mathematics, physics, engineering, biology, chemistry, sociology, psychology, etc. and to solve the positive inverse problems for living, production, and the social development around us. And exploit new research fields that have Chinese characteristic, don't depend on foreigners.