Fixed database typo and removed unnecessary class identifier.

venv/Lib/site-packages/scipy/special/_precompute/lambertw.py

@@ -0,0 +1,68 @@
+"""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()