We consider here the problem of constructing a general recursive algorithm to interpolate a given set of data with a rational function. While many algorithms of this kind already exist, they are either providing non-minimal degree solutions (like the Schur algorithm) or exhibit jumps in the degree of the interpolants (or of the partial realization, as the problem is called when the interpolation is at infinity, see Rissanen (SIAM J Control 9(3):420–430, 1971) and Gragg and Lindquist (in: Linear systems and control (special issue), linear algebra and its applications, vol 50. pp 277–319, 1983)). By imbedding the solution into a larger set of interpolants, we show that the increase in the degree of this representation is proportional to the increase in the length of the data. We provide an algorithm to interpolate multivariable tangential sets of data with arbitrary nodes, generalizing in a fundamental manner the results of Kuijper (Syst Control Lett 31:225–233, 1997). We use this new approach to discuss a special scalar case in detail. When the interpolation data are obtained from the Taylor-series expansion of a given function, then the Euclidean-type algorithm plays an important role.

在这里，我们考虑构造一个通用的递归算法，用一个有理函数对一个给定的数据集进行插值的问题。尽管已经存在许多这类算法，但它们要么提供非最小程度的解（如Schur算法），要么表现出插值的阶跃（或部分实现，因为当插值在无穷大，请参见Rissanen（SIAM J Control 9（3）：420–430，1971）和Gragg和Lindquist（in：线性系统和控制（特刊），线性代数及其应用，第50卷，第277–319页，1983年） ））。通过将解决方案嵌入更大的一组插值中，我们证明了这种表示形式的增加与数据长度的增加成比例。我们提供了一种算法，可以用任意节点插值多变量切向数据集，从根本上概括了Kuijper的结果（Syst Control Lett 31：225-233，1997）。我们使用这种新方法来详细讨论特殊的标量情况。当从给定函数的泰勒级数展开获得插值数据时，则欧几里得类型算法起着重要作用。