Abstract |
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This paper is concerned with the polynomial zero location problem with respect to the unit circle (z = 1) for complex parareciprocal polynomials, i.e. polynomials of the form p(z) = Sigma(i=0)(n)p(i)z(i) = epsilon Sigma(i=0)(n)<(p)over bar (n-i)>z(i) where epsilon is a unit modulus complex number. It is shown, in particular, that the problem can be solved by repeatedly applying the standard Schur-Cohn algorithm to a degree decreasing sequence of polynomials, which are recursively defined from the given polynomial p(z). The resulting algorithm yields, as a side result, a well-defined factorization of p(z). Furthermore, two new algorithms based on this factorization property are proposed, which involve linear arithmetic operations only: a trigonometric polynomial nonnegativity test and a characterization of positive (real) functions. These results extend to the unit disk (z < 1) similar results originally obtained in the right half plane within the context of the generalized Routh-Hurwitz algorithm. |