Levinson and Schur algorithms for Toeplitz matrices with singular minors

A simple general method for overcoming singularities in the Levinson (and Schur) recursions for Toeplitz systems is presented. It is based on a generalization of the conventional three-term recursion for polynomials orthogonal on the unit circle: the scalar coefficients of the conventional three-term recursion are replaced by polynomial coefficients whose degree is determined by the depth of singularity. The depth of the singularity is related to the number of additional zero elements that occur in the Schur recursion. The authors´ method also makes it possible to recursively determine the inertia of a Hermitian Toeplitz matrix