69 lines
1.9 KiB
Python
69 lines
1.9 KiB
Python
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"""Compute a Pade approximation for the principal branch of the
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Lambert W function around 0 and compare it to various other
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approximations.
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"""
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import numpy as np
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try:
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import mpmath
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import matplotlib.pyplot as plt
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except ImportError:
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pass
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def lambertw_pade():
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derivs = [mpmath.diff(mpmath.lambertw, 0, n=n) for n in range(6)]
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p, q = mpmath.pade(derivs, 3, 2)
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return p, q
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def main():
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print(__doc__)
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with mpmath.workdps(50):
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p, q = lambertw_pade()
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p, q = p[::-1], q[::-1]
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print(f"p = {p}")
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print(f"q = {q}")
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x, y = np.linspace(-1.5, 1.5, 75), np.linspace(-1.5, 1.5, 75)
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x, y = np.meshgrid(x, y)
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z = x + 1j*y
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lambertw_std = []
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for z0 in z.flatten():
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lambertw_std.append(complex(mpmath.lambertw(z0)))
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lambertw_std = np.array(lambertw_std).reshape(x.shape)
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fig, axes = plt.subplots(nrows=3, ncols=1)
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# Compare Pade approximation to true result
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p = np.array([float(p0) for p0 in p])
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q = np.array([float(q0) for q0 in q])
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pade_approx = np.polyval(p, z)/np.polyval(q, z)
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pade_err = abs(pade_approx - lambertw_std)
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axes[0].pcolormesh(x, y, pade_err)
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# Compare two terms of asymptotic series to true result
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asy_approx = np.log(z) - np.log(np.log(z))
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asy_err = abs(asy_approx - lambertw_std)
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axes[1].pcolormesh(x, y, asy_err)
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# Compare two terms of the series around the branch point to the
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# true result
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p = np.sqrt(2*(np.exp(1)*z + 1))
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series_approx = -1 + p - p**2/3
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series_err = abs(series_approx - lambertw_std)
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im = axes[2].pcolormesh(x, y, series_err)
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fig.colorbar(im, ax=axes.ravel().tolist())
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plt.show()
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fig, ax = plt.subplots(nrows=1, ncols=1)
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pade_better = pade_err < asy_err
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im = ax.pcolormesh(x, y, pade_better)
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t = np.linspace(-0.3, 0.3)
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ax.plot(-2.5*abs(t) - 0.2, t, 'r')
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fig.colorbar(im, ax=ax)
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plt.show()
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if __name__ == '__main__':
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main()
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