Efficient computation of tr{TR-1} for Toeplitz matrices

An efficient algorithm for the computation of tr{TR/sup -1/}, where T and R are Toeplitz matrices and R is also symmetric positive definite, is presented. The method exploits the fact that the trace of TR/sup -1/ depends only on the sum of the diagonals of R/sup -1/, and not on the whole matrix R/sup -1/. To obtain this sum, a fast efficient technique, built upon the Trench (1964) algorithm for computing the inverse of a Toeplitz matrix, is developed. The complexity of the algorithm depends on the generation function of matrix R and is O(N ln N) for generic functions and O(p ln p) for AR(p) functions.