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

This paper proposes a divide and conquer algorithm for reconstructing a 2n2nth order Jacobi matrix J2nJ2n with a given n  th order leading principal submatrix JnJn and with all eigenvalues of J2nJ2n. This algorithm needs to compute the eigenvalues of the n  th 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 JnJn, and can avoid computing the coefficients of the characteristic polynomial for getting the eigenvalues of JnJn.