A divide and conquer algorithm on the double dimensional inverse eigenvalue problem for Jacobi matrices

摘要

This paper proposes a divide and conquer 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 Jacobi matrix Jn+1,2n′ and the first components of the unit eigenvectors of Jn+1,2n′, where Jn+1,2n′=Jn+1,2n-βne1e1T. The method needs not to reconstruct the leading principal submatrix Jn, and can avoid computing the coefficients of the characteristic polynomial for getting the eigenvalues of Jn.