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Solution of the Yule-Walker equations

The structured condition number of the solution of the Yule-Walker system of equations is given. It is found that there is little difference between this structured condition and the general condition number of a Toeplitz matrix. As a consequence, general purpose linear system solvers are stable for solving the Yule-Walker equations. By constructing appropriate examples it is shown that the Levinson algorithm is only weakly stable and is less trustworthy than the LDLT algorithm. Our round-off error analysis also suggests that for better accuracy Schur coefficients should be computed by the Schur algorithm and then used in the Levinson algorithm for computing the solution of Yule-Walker equations.