Zero location of polynomials with respect to the unit-circle unhampered by nonessential singularities

A method to determine the distribution of the zeros of a polynomial with respect to the unit-circle, proposed by this author in the past, is revisited and refined. The revised procedure remains recursive and nonsingular for polynomials whose Schur-Cohn matrix is not singular. Other nonessential singularities that previously caused interruption of the recursion are assimilated into a more general regular form of the three-term recursion of symmetric polynomials that underlies the method. The new form of the procedure does not compromise the simplicity of the rules to extract the information on the distribution that are proved using a different and more direct proof, based on the evaluation of the Cauchy index along the unit-circle. The low count of operations of the original procedure (recognized as the least cost solution for the problem) is maintained and actually gets better by the elimination of nonessential singularities. The improved features make the revised procedure a better all-around unit-circle zero location method for any real or complex polynomial. Its wider range of regularity should also benefit a variety of related signal processing and algebraic problems including some that were already affected by the original formulation.