FakeQuadrature

FOSS Python software related to the Fake Nodes approach for getting a more stable quadrature without computing new samples.

Authors: S. De Marchi, G. Elefante, E. Perracchione, D. Poggiali, Università di Padova.

Quote this work

To use this work in any scientific report or publication, please cite:

• S. De Marchi, G. Elefante, E. Perracchione, D. Poggiali, Quadrature at fake nodes link, preprint.
• S. De Marchi, F. Marchetti, E. Perracchione, D. Poggiali, Polynomial interpolation via mapped bases without resampling link, J. Comput. Appl. Math., 364 (2020), 112347--12.
• J.P. Berrut, S. De Marchi, G. Elefante, F. Marchetti, Treating the Gibbs phenomenon in barycentric rational interpolation via the S-Gibbs algorithm, to appear on Appl. Math. Letters. 2019.

How to use this software

Clone this repo in your folder and import it

import numpy as np
from fakequadrature import quadrature_weights

then define the function you want to integrate, the mapping function S and the nodes of quadrature, for instance

from scipy.special import erf
f = lambda x: erf( x )
S = lambda x: -2 * np.cos( np.pi* (x+2)/2 )
x = np.linspace( -2, 2, 20 )

so you can now compute the integral of f(x) in the interval [-2,2] with S-FakeNodes

weights = quadrature_weights( x, (-2,2), mapping = S )
Integral = np.dot( f(x) , weights ) 

if you want to compute the approximate integral the usual way, it is sufficient not to specify a mapping S.

Fake Nodes for quadrature

This software extends the fake nodes approach to quadrature formula for mitigating both Runge and Gibbs phenomena when computing the quadrature wheigts.

Quadrature under Fake Nodes mitigate the Runge and Gibbs effect.