An eigenvalue recursion for Toeplitz matrices

In this correspondence, a recursive algorithm for finding eigenvalues of a real symmetric Toeplitz matrix from the eigenvalues of nested Toeplitz submatrices is presented. Given the eigenvalues of two nested Toeplitz submatrices, the eigenvalues of the next larger Toeplitz matrix can be found simply by solving a Vandermonde set of equations and rooting two polynomials.