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