File:Fast Multipoles - Local Expansion.svg
Summary
| Description |
English: For the Fast Multipole Method in 1D, this illustrates the interpolation of a pole at x=3 with an order-5 Chebyshev polynomial around [-1, 1] |
| Date | |
| Source | Own work |
| Author | Summentier |
Generated from the following Python script:
import numpy as np
import numpy.polynomial.chebyshev as np_cheby
import matplotlib.pyplot as pl
def f(x):
x = np.asarray(x)
with np.errstate(divide='ignore'):
return 1 / (x - 3)
ftilde = np_cheby.Chebyshev.interpolate(f, 5)
tk, _ = np_cheby.chebgauss(5)
pl.figure(figsize=(5,4))
x = np.linspace(-2, 4, 121)
xint = np.linspace(-1, 1, 101)
pl.plot(x, f(x), '-r', label='$(y-3)^{-1}$')
pl.plot([3,3], [-10,10], ':r', lw=1)
pl.plot(xint, ftilde(xint), '-k', label=r'$P_5(y)$')
pl.plot(x, ftilde(x), '--k', lw=1.2)
pl.plot(tk, ftilde(tk), '+k', lw=1, label='Chebyshev nodes')
pl.legend()
pl.ylim(-5, 5)
pl.xlim(-1.5, 4)
pl.title(r"Local expansion of a pole at $x=3$")
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