Cholesky Factorization of Semi-de nite Toeplitz Matrices

It can be shown directly from consideration of the Schur algorithm that any nn semi-deenite rank r Toeplitz matrix, T, has a factorization T = C r C T r with C r = C 11 C 12 0 0 where C 11 is r r and upper triangular. This paper explores the reliability of computing such a decomposition with O(nr) complexity using the Schur algorithm and truncating the Cholesky factor after computing the rst r rows. The theoretical conclusion is that, as with Gaussian elimination with partial pivoting, the algorithm is stable in a qualiied sense: the backward error bounds can be expressed purely in terms of n and r and are independent of the condition number of the leading r r submatrix of T, T 11 , but they grow exponentially in r. Thus, the algorithm is completely reliable when r is small, but assessing its reliability for larger r requires making a judgment as to how realistic the error bounds are in practice. Experimental results are presented which suggest that the bounds are pessimistic, and the algorithm is stable for many types of semi-deenite Toeplitz matrices. The analysis and conclusions are similar to those in a paper by Higham dealing with general semi-deenite matrices.