On the structure of the matrix in Sturm algorithm for determining the number of zeros in the open unit disk for a real polynomial

In 1984, C.W Schelin presented a numerical method for calculating the number of zeros of a real polynomial inside the unit disk using the argument principle and Cauchy index on [-1, 1] inspired by sturm’s algorithm [3], this method is based on calculation of the generalized Sturm sequence in Chebyshev form, however in some cases this algorithm does not efficient for this reason, it has been modified in another works [1], [2]. There are other numerical algorithms for calculating the number of zeros of a polynomial in the unit disk like shur -Cohn and Marden-Cohn but the numerical method presented in this study [1] is more efficient in cost of elementary operations (o (n2)). In this paper, we propose to improve this algorithm [1], to do this we study the matrix structure involved in the calculation of chebychev polynomials, for there on we prove that is a Toeplitz matrix which is known for their important properties in the complexity of calculations and we calculate its last line which is necessary to define the whole matrix to solve the linear system for calculating polynomials which are necessary to obtain Chebyshev sequence, this result will reduce calculation steps in this method [1]. This numerical algorithm may operate to study the stability of dynamical systems [2].