| Jacobi matrices inverse problem and tridiagonal quadratic pencil inverse prob-lem have been widely applied in control theory,geophysics,image processing,vi-bration theory,system recognition,structural mechanics,particle physics and other fields.Research on Jacobi matrices inverse problem is of theoretical and practical significance.It has aroused high attention among the the vast majorit,y of scientific researchers.In this present paper,the uniqueness of Jacobi matrices reconstruction solu-tion and the specific reconstruction algorithm are studied by using the property of alternating zero.The mains work is done as the following:In the first chapter,we elaborate the research advances of Jacobi matrices inverse problem,inverse problem with missing feature information(the eigenvalues of the original matrix and the eigenvalues of the perturbation matrix have equal elements),and tridiagonal quadratic pencil inverse problem,and then introduce the main work of this paper.In the second chapter,we give a summary of the research foundation of Jacobi matrices inverse problem,inverse problem with missing feature information,and tridiagonal quadratic pencil inverse problem.In the third chapter,the reconstruction of Jacobi matrices under mixed spectral data is studied.We give the necessary and sufficient conditions for the existence of a unique solution in two cases and prove them:The first case is that the Jacobi matrix is reconstructed from the partial eigenvalues of the Jacobi matrix,the partial eigenvalues of the principal submatrix,and a sequential principal submatrix,and the second case is that the Jacobi matrix is reconstructed from some eigenvalues,some norming constants,and one of its principal submatrix.We give a specific numerical algorithm and numerical examples for the solution.In the fourth chapter,the reconstruction of Jacobi matrices in the absence of feature information is studied.We analyze the reasons why Jacobi matrices are not unique in both cases:The first case is that we consider the reconstruction of the Jacobi matrix when the elements bN and aN-1 of the Jacobi matrix J[1,N]are disturbed and the eigenvalues before and after the perturbation are equal,and the second case is that we consider the reconstruction of the Jacobi matrix by using the eigenvalue pair of the Jacobi matrix J[1,M]and two submatrices J[1,n-1]and J[n+1,n]that eigenvalues are equal.We discuss the properties of norming constants,then the uniqueness of the Jacobi matrix reconstruction is realized by using the norming constants to compensate for the characteristic information,and a specific numerical algorithm and numerical examples are given.In the fifth chapter,the reconstruction of tridiagonal quadratic pencil is studied.We reconstruct the tridiagonal quadratic pencil through two perturbation methods based on Ram’s research of the tridiagonal quadratic pencil using eigenvalues of the original matrix and the primary submatrix:The first case is that we perturb the elements(αN,γN)of the quadratic pencil matrix(C[1,N],K[1,N]),and reconstruct(C[1,N],K[1,N])according to the eigenvalues before and after the perturbation,and the second case is that we divide the quadratic pencil matrix(C[1,N],K[1,N)into two parts(C[1,N],K[1,N]and(C[n+1,N],K[n+1,N]),and perturb them,then we reconstruct(C[1,N],K[1,N])by using eigenvalues before and after the perturbation.We give specific numerical algorithms and numerical examples. |
