The Approximation Property Of Berrut’s Rational Interpolant At Affine Nodes

Barycentric rational interpolation is a hot spot in the field of new approaches for its reduced calculation and well numerical stability. Concerning the choice of the inter-polation nodes has an important influence on the interpolation properties, this paper constructs a kind of n new interpolation nodes——affine nodes and the approximation property of Berrut’s rational interpolation at affine nodes is implemented research. The main research results include the following three aspects:At first, the different interpolation nodes in the same function interpolation will produce different effects, so what kinds of interpolation nodes to choose is of importance. Based on the analysis and summary of several known nodes, we construct a class of new interpolation nodes——affine nodes, whose properties are presented, such as affine property, symmetry, and density when scaling factor q takes distinct values, etc. At the same time, drawing the corresponding figure intuitively show the basic properties. The research laid the foundation for below Berrut rational interpolation at affine nodes.Next, we study the Berrut’s rational interpolant at affine nodes. It is natural to study the condition of this numerical approximation method. This article mainly from the per-spective of Lebesgue constant to study the approximation properties of Berrut’s rational interpolant at affine nodes. Based on important limit theory and bounds for the partial sums of the Leibniz series and the harmonic series, proves On the Lebesgue constant of Berrut’s rational interpolant at affine nodes, illustrates Berrut’s rational interpolation at affine nodes has good numerical stability.In this paper, using Matlab software to perform a number of numerical experiments. Not only draw picture for the Berrut’s rational interpolation at affine nodes when choosing distinct n and q. At the same time, the Berrut’s rational interpolation at affine nodes and existing nodes sush as equidistant node, the first kind of chebyshev points, the second chebyshev points etc.were compared in this paper. It fully embodies the significance of this article.