Interpolation

This module contains a host of models and functions often used for testing interpolation algorithms.


temfpy.interpolation.runge(x)[source]

Runge function.

\[f(x) = \frac{1}{1 + 25x^2}\]
Parameters

x (float) – Input number, which is usually evaluated on the interval \(x_i \in [-1, 1]\).

Returns

Output domain

Return type

float

Notes

Runge found that interpolating this function with a Polynomial \(p_n(x)\) of degree \(n\) on an equidistant grid with grid points \(x_i = \frac{2i}{n}-1\) results in an interpolation function that oscilliates close to the interval boundaries \(-1\) and \(1\).

_images/fig-runge.png

References

R1901

Runge, C. (1901). Über empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten. Zeitschrift für Mathematik und Physik, 46: 224-243.

Examples

>>> from temfpy.interpolation import runge
>>> import numpy as np
>>>
>>> x = 0
>>> y = runge(x)
>>> np.testing.assert_almost_equal(y, 1)