Computation of the Newton step for the even and odd characteristic polynomials of a symmetric positive definite Toeplitz matrix

We compute the Newton step for the characteristic polynomial and for the even and odd characteristic polynomials of a symmetric positive definite Toeplitz matrix as the reciprocal of the trace of an appropriate matrix. We show that, after the Yule–Walker equations are solved, this trace can be computed in O(n) additional arithmetic operations, which is in contrast to existing methods, which rely on a recursion, requiring O(n2) additional arithmetic operations.