This report contains a numerical stability analysis of factorization
algorithms for computing the Cholesky decomposition of symmetric positive
definite matrices of displacement rank 2. The algorithms in the class can be
expressed as sequences of elementary downdating steps. The stability of the
factorization algorithms follows directly from the numerical properties of
algorithms for realizing elementary downdating operations. It is shown that the
Bareiss algorithm for factorizing a symmetric positive definite Toeplitz matrix
is in the class and hence the Bareiss algorithm is stable. Some numerical
experiments that compare behavior of the Bareiss algorithm and the Levinson
algorithm are presented. These experiments indicate that in general (when the
reflection coefficients are not all positive) the Levinson algorithm is not
stable; certainly it can give much larger residuals than the Bareiss algorithm.

对位移秩为 2 的对称正定矩阵的 Cholesky 分解算法进行了数值稳定性分析，得出 Bareiss 算法稳定，而 Levinson 算法在反射系数不全为正时并不稳定，实验结果表明 Bareiss 算法的残差要比 Levinson 算法小。