69 lines
2.0 KiB
Python
69 lines
2.0 KiB
Python
|
"""Compute a Pade approximation for the principle branch of the
|
||
|
|
Lambert W function around 0 and compare it to various other
|
||
|
|
approximations.
|
||
|
|
|
||
|
|
"""
|
||
|
|
import numpy as np
|
||
|
|
|
||
|
|
try:
|
||
|
|
import mpmath # type: ignore[import]
|
||
|
|
import matplotlib.pyplot as plt # type: ignore[import]
|
||
|
|
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("p = {}".format(p))
|
||
|
|
print("q = {}".format(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()
|