A new algorithm on the inverse eigenvalue problem for double dimensional Jacobi matrices

In this paper, we investigate some properties of eigenvalues and eigenvectors of Jacobi matrices. We propose a new algorithm for reconstructing a 2nth order Jacobi matrix J2n with a given nth order leading principal submatrix Jn and with all eigenvalues of J2n. This algorithm needs to compute the eigenvalues of the nth order tailing principal submatrix Jn+1,2n and the first components of the unit eigenvectors of Jn+1,2n. Our method needs not to reconstruct Jn, and can avoid computing the coefficients of the characteristic polynomial for getting the eigenvalues of Jn+1,2n. We also present some numerical results.