The Runge Example for Interpolation and Wilkinson's Examples for Rootfinding

Based on the Euler–Maclaurin formula, the Romberg quadrature method extrapolates trapezoidal values to improve their accuracy when computing the integral of smooth functions from equispaced samples. It has been known at least since an article of Lyness in 1971 that the Euler–Maclaurin formula may be extended to accommodate functions with jumps. In the present work, we develop an extrapolation method, based on this extended formula, for the quadrature of such discontinuous functions. We illustrate the method with numerical examples, using one as well as several sample vectors.

show abstract

“…is the interpolant of S p (h). But, according to a well-known result about the perturbation of the polynomial interpolant [3,7],…”

Section: Extrapolation Via a System Of Equationsmentioning

confidence: 99%

Extrapolation quadrature from equispaced samples of functions with jumps

Berrut

Trummer

2022

Numer Algor

View full text Add to dashboard Cite

show abstract

“…This is a well-known effect, of course, discussed in many textbooks and educational papers; see e.g. [23], which discusses it as an effect of the ill-conditioning of the truncated Taylor series. The situation gets very much worse as |z| increases.…”

mentioning

confidence: 99%

Computation and applications of Mathieu functions: A historical perspective

Brimacombe¹,

Corless²,

Zamir³

2020

Preprint

Self Cite

View full text Add to dashboard Cite

Mathieu functions of period π or 2π, also called elliptic cylinder functions, were introduced in 1868 by Émile Mathieu together with so-called modified Mathieu functions, in order to help understand the vibrations of an elastic membrane set in a fixed elliptical hoop. These functions still occur frequently in applications today: our interest, for instance, was stimulated by a problem of pulsatile blood flow in a blood vessel compressed into an elliptical cross-section. This paper surveys and recapitulates the historical development of the theory and methods of computation for Mathieu functions and modified Mathieu functions and identifies some gaps in current software capability, particularly to do with double eigenvalues of the Mathieu equation. We demonstrate how to compute Puiseux expansions of the Mathieu eigenvalues about such double eigenvalues, and give methods to compute the generalized eigenfunctions that arise there. In examining Mathieu's original contribution, we bring out that his use of anti-secularity predates that of Lindstedt. For interest, we also provide short biographies of some of the major mathematical researchers involved in the history of the Mathieu functions:

show abstract

On the accuracy of interpolation based on single-layer artificial neural networks with a focus on defeating the Runge phenomenon

Auricchio,

Belardo,

Calabrò

et al. 2024

Soft Comput

View full text Add to dashboard Cite

Artificial Neural Networks (ANNs) are a tool in approximation theory widely used to solve interpolation problems. In fact, ANNs can be assimilated to functions since they take an input and return an output. The structure of the specifically adopted network determines the underlying approximation space, while the form of the function is selected by fixing the parameters of the network. In the present paper, we consider one-hidden layer ANNs with a feedforward architecture, also referred to as shallow or two-layer networks, so that the structure is determined by the number and types of neurons. The determination of the parameters that define the function, called training, is done via the resolution of the approximation problem, so by imposing the interpolation through a set of specific nodes. We present the case where the parameters are trained using a procedure that is referred to as Extreme Learning Machine (ELM) that leads to a linear interpolation problem. In such hypotheses, the existence of an ANN interpolating function is guaranteed. Given that the ANN is interpolating, the error incurred occurs outside the sampling interpolation nodes provided by the user. In this study, various choices of nodes are analyzed: equispaced, Chebychev, and randomly selected ones. Then, the focus is on regular target functions, for which it is known that interpolation can lead to spurious oscillations, a phenomenon that in the ANN literature is referred to as overfitting. We obtain good accuracy of the ANN interpolating function in all tested cases using these different types of interpolating nodes and different types of neurons. The following study is conducted starting from the well-known bell-shaped Runge example, which makes it clear that the construction of a global interpolating polynomial is accurate only if trained on suitably chosen nodes, ad example the Chebychev ones. In order to evaluate the behavior when the number of interpolation nodes increases, we increase the number of neurons in our network and compare it with the interpolating polynomial. We test using Runge’s function and other well-known examples with different regularities. As expected, the accuracy of the approximation with a global polynomial increases only if the Chebychev nodes are considered. Instead, the error for the ANN interpolating function always decays, and in most cases we observe that the convergence follows what is observed in the polynomial case on Chebychev nodes, despite the set of nodes used for training. Then we can conclude that the use of such an ANN defeats the Runge phenomenon. Our results show the power of ANNs to achieve excellent approximations when interpolating regular functions also starting from uniform and random nodes, particularly for Runge’s function.

show abstract

The Runge Example for Interpolation and Wilkinson's Examples for Rootfinding

Cited by 6 publications

References 7 publications

Extrapolation quadrature from equispaced samples of functions with jumps

Extrapolation quadrature from equispaced samples of functions with jumps

Computation and applications of Mathieu functions: A historical perspective

On the accuracy of interpolation based on single-layer artificial neural networks with a focus on defeating the Runge phenomenon

Contact Info

Product

Resources

About