Condition and Accuracy of Algorithms for Computing Schur Coefficients of Toeplitz Matrices

Abstract

A formula for the condition number of Schur coefficients of a positive definite Toeplitz matrix is obtained and an efficient algorithm for computing the condition number is given. New bounds of backward roundoff errors in Schur and Levinson algorithms for computing Schur coefficients are presented. These bounds, together, with the condition number, provide a posteriors estimate of the error in computed Schur coefficients. Numerical comparison of Schur and Levinson algorithms with the $LDL^T $ algorithm also indicates their forward stability.