Lagrange interpolation in python

melpomene / lagrange.py

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import numpy as np

import matplotlib . pyplot as plt

import sys

def main ():

if len ( sys . argv ) == 1 or «-h» in sys . argv or «—help» in sys . argv :

print «python lagrange.py .. «

print «Example:»

print «python lagrange.py 0.1 2.4 4.5 3.2»

exit ()

points = []

for i in xrange ( len ( sys . argv )):

if i != 0 :

points . append (( int ( sys . argv [ i ]. split ( «.» )[ 0 ]), int ( sys . argv [ i ]. split ( «.» )[ 1 ])))

#points =[(0,0),(25,30),(50,10), (57,0)]

P = lagrange ( points )

nr = 2

print «(» + str ( points [ nr ][ 0 ]) + «, » + str ( points [ nr ][ 1 ]) + «) P(» + str ( points [ nr ][ 0 ]) + «)= » + str ( P ( points [ nr ][ 0 ]))

plot ( P , points )

def plot ( f , points ):

x = range ( — 10 , 100 )

y = map ( f , x )

print y

plt . plot ( x , y , linewidth = 2.0 )

x_list = []

y_list = []

for x_p , y_p in points :

x_list . append ( x_p )

y_list . append ( y_p )

print x_list

print y_list

plt . plot ( x_list , y_list , ‘ro’ )

plt . show ()

def lagrange ( points ):

def P ( x ):

total = 0

n = len ( points )

for i in xrange ( n ):

xi , yi = points [ i ]

def g ( i , n ):

tot_mul = 1

for j in xrange ( n ):

if i == j :

continue

xj , yj = points [ j ]

tot_mul *= ( x — xj ) / float ( xi — xj )

return tot_mul

total += yi * g ( i , n )

return total

return P

if __name__ == «__main__» :

main ()

Источник

scipy.interpolate.lagrange#

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.

Interpolate \(f(x) = x^3\) 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) 

Since there are only 3 points, Lagrange polynomial has degree 2. Explicitly, it is given by

>>> from numpy.polynomial.polynomial import Polynomial >>> Polynomial(poly.coef[::-1]).coef array([ 0., -2., 3.]) 

>>> import matplotlib.pyplot as plt >>> x_new = np.arange(0, 2.1, 0.1) >>> 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() >>> plt.show() 

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Pure-Python implementation of Lagrange interpolation over finite fields.

License

lapets/lagrange

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README.rst

Pure-Python implementation of Lagrange interpolation over finite fields.

This library provides a pure-Python implementation of the Lagrange interpolation algorithm over finite fields.

This library is available as a package on PyPI:

python -m pip install lagrange

The library can be imported in the usual way:

from lagrange import lagrange

Interpolation can be performed on collections of points represented in a variety of ways:

>>> lagrange(1: 15, 2: 9, 3: 3>, 17) 4 >>> lagrange([(1, 15), (2, 9), (3, 3)], 17) 4 >>> lagrange([15, 9, 3], 17) 4 >>> lagrange(\ 1: 119182, 2: 11988467, 3: 6052427, 4: 8694701,\ 5: 9050123, 6: 3676518, 7: 558333, 8: 12198248,\ 9: 7344866, 10: 10114014, 11: 2239291, 12: 2515398>,\ 15485867) 123 >>> lagrange(\ [119182, 11988467, 6052427, 8694701, 9050123, 3676518,\ 558333, 12198248, 7344866, 10114014, 2239291, 2515398],\ 15485867) 123

All installation and development dependencies are fully specified in pyproject.toml . The project.optional-dependencies object is used to specify optional requirements for various development tasks. This makes it possible to specify additional options (such as docs , lint , and so on) when performing installation using pip:

python -m pip install .[docs,lint]

The documentation can be generated automatically from the source files using Sphinx:

python -m pip install .[docs] cd docs sphinx-apidoc -f -E --templatedir=_templates -o _source .. && make html

All unit tests are executed and their coverage is measured when using pytest (see the pyproject.toml file for configuration details):

python -m pip install .[test] python -m pytest

Alternatively, all unit tests are included in the module itself and can be executed using doctest:

python src/lagrange/lagrange.py -v

Style conventions are enforced using Pylint:

python -m pip install .[lint] python -m pylint src/lagrange

In order to contribute to the source code, open an issue or submit a pull request on the GitHub page for this library.

Beginning with version 0.2.0, the version number format for this library and the changes to the library associated with version number increments conform with Semantic Versioning 2.0.0.

This library can be published as a package on PyPI by a package maintainer. First, install the dependencies required for packaging and publishing:

python -m pip install .[publish]

Ensure that the correct version number appears in pyproject.toml , and that any links in this README document to the Read the Docs documentation of this package (or its dependencies) have appropriate version numbers. Also ensure that the Read the Docs project for this library has an automation rule that activates and sets as the default all tagged versions. Create and push a tag for this version (replacing . with the version number):

Remove any old build/distribution files. Then, package the source into a distribution archive:

rm -rf build dist src/*.egg-info python -m build --sdist --wheel .

Finally, upload the package distribution archive to PyPI:

python -m twine upload dist/*

About

Pure-Python implementation of Lagrange interpolation over finite fields.

Источник

Python Program for Lagrange Interpolation Method (with Output)

In this Python program, x and y are two array for storing x data and y data respectively. Here we create these array using numpy library. xp is interpolation point given by user and output of Lagrange interpolation method is obtained in yp .

Python Source Code: Lagrange Interpolation

 # Lagrange Interpolation # Importing NumPy Library import numpy as np # Reading number of unknowns n = int(input('Enter number of data points: ')) # Making numpy array of n & n x n size and initializing # to zero for storing x and y value along with differences of y x = np.zeros((n)) y = np.zeros((n)) # Reading data points print('Enter data for x and y: ') for i in range(n): x[i] = float(input( 'x['+str(i)+']=')) y[i] = float(input( 'y['+str(i)+']=')) # Reading interpolation point xp = float(input('Enter interpolation point: ')) # Set interpolated value initially to zero yp = 0 # Implementing Lagrange Interpolation for i in range(n): p = 1 for j in range(n): if i != j: p = p * (xp - x[j])/(x[i] - x[j]) yp = yp + p * y[i] # Displaying output print('Interpolated value at %.3f is %.3f.' % (xp, yp)) 

Python Output: Language Interpolation

Enter number of data points: 5 Enter data for x and y: x[0]=5 y[0]=150 x[1]=7 y[1]=392 x[2]=11 y[2]=1452 x[3]=13 y[3]=2366 x[4]=17 y[4]=5202 Enter interpolation point: 9 Interpolated value at 9.000 is 810.000.

Recommended Readings

• Algorithm for Bisection Method
• Pseudocode for Bisection Method
• C Program for Bisection Method
• C++ Program for Bisection Method
• MATLAB Program for Bisection Method
• Python Program for Bisection Method
• Bisection Method Advantages
• Bisection Method Disadvantages
• Bisection Method Features
• Convergence of Bisection Method
• Bisection Method Online Calculator

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The Lagrange interpolation formula is a way to find a polynomial, called Lagrange polynomial, that takes on certain values at arbitrary points. Lagrange’s interpolation is an Nth degree polynomial approximation to f(x). This program implements Lagrange Interpolation Method for finding a polynomial in python programming language and it’s necessar…

motisoltani/LagrangeInterpolationMethod-Python

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README.md

Lagrange Interpolation Method-Python

The Lagrange interpolation formula is a way to find a polynomial, called Lagrange polynomial, that takes on certain values at arbitrary points. Lagrange’s interpolation is an Nth degree polynomial approximation to f(x). This program implements Lagrange Interpolation Method for finding a polynomial in python programming language and it’s necessary to enter X & Y.
This program shows you the graph related to the polynomial and the data. I hope you find it useful. Email me if you have any questions.

About

The Lagrange interpolation formula is a way to find a polynomial, called Lagrange polynomial, that takes on certain values at arbitrary points. Lagrange’s interpolation is an Nth degree polynomial approximation to f(x). This program implements Lagrange Interpolation Method for finding a polynomial in python programming language and it’s necessar…

Источник