94 lines
2.4 KiB
Python
94 lines
2.4 KiB
Python
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#!/usr/bin/env python3
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# Runge's Phenomenon
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#----------------------------------
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import numpy as np
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def func_to_interpolate(x):
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return 1/( 1 + 25 * x**2 )
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def equidistant_nodes(n):
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return 2*np.arange(0,n+1)/n -1
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def chebychev_nodes(n):
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return np.cos( (2*np.arange(1,n+1) -1) * np.pi / (2*n) )
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def random_nodes(n):
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return 2*np.random.rand(n) -1
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def main():
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import matplotlib.pyplot as plt
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from scipy.interpolate import lagrange as scipy_lagrange
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np.random.seed(0)
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n_special = 20
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n_cheby = n_special
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n_equi = n_special
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n_random = n_special
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n_func = 1e4
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x_func = np.linspace(-1, 1, n_func, endpoint = True )
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fig, ax = plt.subplots()
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ax.set_title("Lagrange Interpolations with different types of nodes")
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ax.grid()
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ax.set_xlabel("x")
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ax.set_ylabel("y")
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func = func_to_interpolate
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# Plot the formula
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ax.plot(x_func, func(x_func), '--', label="Function $1/(1+25x^2)$")
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# Using equidistant nodes
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x_equi = equidistant_nodes(n_equi)
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ax.plot(x_equi, scipy_lagrange(x_equi, func(x_equi))(x_equi), '^', label="Equidistant (n={})".format(n_equi))
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# Using chebychev nodes
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x_cheby = chebychev_nodes(n_cheby)
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ax.plot(x_cheby, scipy_lagrange(x_cheby, func(x_cheby))(x_cheby), '>',label="Chebyshev (n={})".format(n_cheby))
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# Using random nodes
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x_random = random_nodes(n_random)
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ax.plot(x_random, scipy_lagrange(x_random, func(x_random))(x_random), '.', label="Random (n={})".format(n_random))
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ax.legend()
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print("""
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Runge's Phenomenon
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------------------
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According to the docs [1], the scipy Lagrange
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interpolation is unstable and there should not be more
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than 20 points.
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The region where the equidistant nodes break is near
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x = -1, where the interpolation is fluctuating between
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large negative and large positive values. From n > 25,
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this starts to get visible.
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The chebyshev nodes break near x = 1 when n > 30, where
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it interpolates the function to be negative.
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Funnily enough, the equidistant nodes vary much more
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than the chebyshev nodes.
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For the random nodes, you just have be lucky that not
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too many points are close together at the extremes of
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the interval.
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[1] = https://docs.scipy.org/doc/scipy/reference/generated/scipy.interpolate.lagrange.html
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""")
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plt.show()
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if __name__ == "__main__":
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main()
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