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bundles / scipy latest / scipy / interpolate / _interpolate / lagrange

function

scipy.interpolate._interpolate:lagrange

source: build-install/usr/lib/python3.14/site-packages/scipy/interpolate/_interpolate.py:27

Signature

def lagrange(x, w)

Summary

Return a Lagrange interpolating polynomial.

Extended Summary

Given two 1-D arrays x and w, returns the Lagrange interpolating polynomial through the points (x, w).

Warning: This implementation is numerically unstable. Do not expect to be able to use more than about 20 points even if they are chosen optimally.

Parameters

x : array_like

x represents the x-coordinates of a set of datapoints.

w : array_like

w represents the y-coordinates of a set of datapoints, i.e., f(x).

Returns

lagrange : `numpy.poly1d` instance

The Lagrange interpolating polynomial.

Notes

The name of this function refers to the fact that the returned object represents a Lagrange polynomial, the unique polynomial of lowest degree that interpolates a given set of data [1]. It computes the polynomial using Newton's divided differences formula [2]; that is, it works with Newton basis polynomials rather than Lagrange basis polynomials. For numerical calculations, the barycentric form of Lagrange interpolation (scipy.interpolate.BarycentricInterpolator) is typically more appropriate.

Array API Standard Support

lagrange is not in-scope for support of Python Array API Standard compatible backends other than NumPy.

See dev-arrayapi for more information.

Examples

Interpolate :math:`f(x) = x^3` by 3 points. 
import numpy as np
from scipy.interpolate import lagrange
x = np.array([0, 1, 2])
y = x**3
poly = lagrange(x, y)
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Since there are only 3 points, the Lagrange polynomial has degree 2. Explicitly, it is given by .. math:: \begin{aligned} L(x) &= 1\times \frac{x (x - 2)}{-1} + 8\times \frac{x (x-1)}{2} \\ &= x (-2 + 3x) \end{aligned} 
from numpy.polynomial.polynomial import Polynomial
Polynomial(poly.coef[::-1]).coef
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array([ 0., -2.,  3.])

import matplotlib.pyplot as plt
x_new = np.arange(0, 2.1, 0.1)
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plt.scatter(x, y, label='data')
plt.plot(x_new, Polynomial(poly.coef[::-1])(x_new), label='Polynomial')
plt.plot(x_new, 3*x_new**2 - 2*x_new + 0*x_new,
         label=r"$3 x^2 - 2 x$", linestyle='-.')
plt.legend()
✗
<matplotlib.legend.Legend object at 0x7f3f60f05550>

plt.show()
✓
fig-8e72370a8e3a4b2c.png

Aliases

  • scipy.interpolate.lagrange

Referenced by

This package