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

Abstract

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 ) {\mathcal O}(n) additional arithmetic operations, which is in contrast to existing methods, which rely on a recursion, requiring O ( n 2 ) {\mathcal O}(n^2) additional arithmetic operations.