Algebraic properties of robust Padé approximants
Algebraic properties of robust Padé approximants
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DOI：
10.1016/j.jat.2014.05.018
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发表时间：
2013-10
期刊：
影响因子：
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通讯作者：
B. Beckermann;Ana C. Matos
中科院分区：
文献类型：
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作者：
B. Beckermann;Ana C. Matos
对于最近一种通过SVD技术计算所谓的鲁棒Padé逼近的新数值方法，作者给出了数值证据，证明这种逼近对数据中的扰动不敏感，并且没有所谓的伪极点，即具有接近零的极点或具有小残差的极点。消除伪极点的黑箱方法将对Padé逼近的收敛理论产生重大影响，因为已知容量收敛加上某个域D中极点的缺失意味着D中的局部一致收敛。在本文中，我们提供了一个向前稳定性（或鲁棒性）的证明，并显示了所谓的良好条件的Padé逼近的子类的虚假极点的情况下。给出了一类具有伪极点的鲁棒Padé逼近的数值例子，并讨论了相关问题。事实证明，这是不够的，只讨论基本的矩形Toeplitz矩阵的线性代数性质，因为在我们的结果中，其他矩阵，如西尔维斯特矩阵也出现。这些类型的矩阵已经在数值最大公约数计算中使用过。
For a recent new numerical method for computing so-called robust Padé approximants through SVD techniques, the authors gave numerical evidence that such approximants are insensitive to perturbations in the data and do not have so-called spurious poles, that is, poles with close-by zero or poles with small residuals. A black box procedure for eliminating spurious poles would have a major impact on the convergence theory of Padé approximants since it is known that convergence in capacity plus the absence of poles in some domain D implies locally uniform convergence in D. In the present paper we provide a proof for forward stability (or robustness) and show the absence of spurious poles for the subclass of so-called well-conditioned Padé approximants. We also give a numerical example of some robust Padé approximant which has spurious poles and discuss related questions. It turns out that it is not sufficient to discuss only linear algebra properties of the underlying rectangular Toeplitz matrix, since in our results other matrices like Sylvester matrices also occur. These types of matrices have been used before in numerical greatest common divisor computations.