Counting Zeros of Real Polynomials within the Unit Disk

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In this paper a scheme is presented to numerically evaluate the argument principle for a real polynomial on the unit circle. This scheme may then be used to determine the exact number of zeros possessed by a real polynomial in any open disk centered at the origin. After casting the argument principle in terms of Cauchy indices, a Sturm sequence of polynomials in Chebyshev form, of length at most half the degree of the given polynomial, is constructed to determine these indices. For real polynomials, the test of this paper requires about half the arithmetic operations used in the Schur–Cohn test.