关于Bareiss分解算法和相关Toeplitz分解算法的稳定性
On the stability of the Bareiss and related Toeplitz factorization algorithms
摘要：
本文对位移秩为2的对称正定矩阵的Cholesky分解的分解算法进行了数值稳定性分析。类中的算法可以表示为基本的downdating步骤的序列。因子分解算法的稳定性直接来自于实现初等降期运算的算法的数值性质。证明了分解对称正定Toeplitz矩阵的Bareiss算法属于该类，因此Bareiss算法是稳定的.通过数值实验比较了Bareiss算法和Levinson算法的性能。这些实验表明，在一般情况下（当反射系数不都是积极的），莱文森算法是不稳定的;当然，它可以给更大的残差比Bareiss算法。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
