A common way of finding the poles of a meromorphic function f in a domain, where an explicit expression of f is unknown but f can be evaluated at any given z, is to interpolate f by a rational function $$\frac{p}{q}$$pq such that $$r(\gamma _i)=f(\gamma _i)$$r(γi)=f(γi) at prescribed sample points $$\{\gamma _i\}_{i=1}^L$${γi}i=1L, and then find the roots of q. This is a two-step process and the type of the rational interpolant needs to be specified by the user. Many other algorithms for polefinding and rational interpolation (or least-squares fitting) have been proposed, but their numerical stability has remained largely unexplored. In this work we describe an algorithm with the following three features: (1) it automatically finds an appropriate type for a rational approximant, thereby allowing the user to input just the function f, (2) it finds the poles via a generalized eigenvalue problem of matrices constructed directly from the sampled values $$f(\gamma _i)$$f(γi) in a one-step fashion, and (3) it computes rational approximants $$\hat{p},\hat{q}$$p^,q^ in a numerically stable manner, in that $$(\hat{p}+\Delta p)/(\hat{q}+\Delta q)=f$$(p^+Δp)/(q^+Δq)=f with small $$\Delta p,\Delta q$$Δp,Δq at the sample points, making it the first rational interpolation (or approximation) algorithm with guaranteed numerical stability. Our algorithm executes an implicit change of polynomial basis by the QR factorization, and allows for oversampling combined with least-squares fitting. Through experiments we illustrate the resulting accuracy and stability, which can significantly outperform existing algorithms.

中文翻译：

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通过特征值进行稳定的极点查找和有理最小二乘拟合

在域中找到亚纯函数 f 的极点的一种常用方法，其中 f 的显式表达式未知但可以在任何给定 z 处计算 f，是通过有理函数 $$\frac{p}{ q}$$pq 使得 $$r(\gamma _i)=f(\gamma _i)$$r(γi)=f(γi) 在指定的样本点 $$\{\gamma _i\}_{i= 1}^L$${γi}i=1L，然后求q的根。这是一个两步过程，有理插值的类型需要由用户指定。已经提出了许多其他用于极点查找和有理插值（或最小二乘拟合）的算法，但它们的数值稳定性在很大程度上仍未得到探索。在这项工作中，我们描述了一种具有以下三个特征的算法：（1）它自动为有理逼近找到合适的类型，从而允许用户只输入函数 f，(2) 它通过一个矩阵的广义特征值问题以一步的方式直接从采样值 $$f(\gamma_i)$$f(γi) 中找到极点，以及 (3) 它计算有理近似值 $ $\hat{p},\hat{q}$$p^,q^ 以数值稳定的方式，即 $$(\hat{p}+\Delta p)/(\hat{q}+\Delta q)=f$$(p^+Δp)/(q^+Δq)=f 在样本点处具有小 $$\Delta p,\Delta q$$Δp,Δq，使其成为第一个有理插值（或近似）算法，保证数值稳定性。我们的算法通过 QR 分解执行多项式基的隐式变化，并允许过采样与最小二乘拟合相结合。通过实验，我们说明了由此产生的准确性和稳定性，它可以显着优于现有算法。