Euclid algorithm, orthogonal polynomials, and generalized Routh-Hurwitz algorithm

This paper is concerned with the relations between the Euclid algorithm, the theory of orthogonal polynomials, and the problem of locating the zeros of a complex polynomial with respect to the imaginary axis. In particular, a simple generalized Routh-Hurwitz algorithm is proposed, which allows one to determine, in any situation, the numbers of zeros of an arbitrary complex polynomial in the right half plane, on the imaginary axis, and hence in the left half plane; moreover, it turns out that this algorithm yields, as a side result, a well-defined factorization of the considered polynomial. Furthermore as a straightforward consequence of the adopted approach, two presumably original algorithms are put into light, which involve linear arithmetic operations only: a polynomial nonnegativity test on the real axis, and a characterization of positive real functions.