215 lines
5.3 KiB
Python
215 lines
5.3 KiB
Python
import numpy as np
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from ._ufuncs import _ellip_harm
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from ._ellip_harm_2 import _ellipsoid, _ellipsoid_norm
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def ellip_harm(h2, k2, n, p, s, signm=1, signn=1):
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r"""
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Ellipsoidal harmonic functions E^p_n(l)
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These are also known as Lame functions of the first kind, and are
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solutions to the Lame equation:
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.. math:: (s^2 - h^2)(s^2 - k^2)E''(s)
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+ s(2s^2 - h^2 - k^2)E'(s) + (a - q s^2)E(s) = 0
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where :math:`q = (n+1)n` and :math:`a` is the eigenvalue (not
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returned) corresponding to the solutions.
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Parameters
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----------
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h2 : float
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``h**2``
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k2 : float
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``k**2``; should be larger than ``h**2``
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n : int
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Degree
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s : float
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Coordinate
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p : int
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Order, can range between [1,2n+1]
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signm : {1, -1}, optional
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Sign of prefactor of functions. Can be +/-1. See Notes.
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signn : {1, -1}, optional
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Sign of prefactor of functions. Can be +/-1. See Notes.
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Returns
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-------
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E : float
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the harmonic :math:`E^p_n(s)`
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See Also
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--------
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ellip_harm_2, ellip_normal
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Notes
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-----
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The geometric interpretation of the ellipsoidal functions is
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explained in [2]_, [3]_, [4]_. The `signm` and `signn` arguments control the
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sign of prefactors for functions according to their type::
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K : +1
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L : signm
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M : signn
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N : signm*signn
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.. versionadded:: 0.15.0
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References
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----------
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.. [1] Digital Library of Mathematical Functions 29.12
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https://dlmf.nist.gov/29.12
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.. [2] Bardhan and Knepley, "Computational science and
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re-discovery: open-source implementations of
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ellipsoidal harmonics for problems in potential theory",
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Comput. Sci. Disc. 5, 014006 (2012)
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:doi:`10.1088/1749-4699/5/1/014006`.
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.. [3] David J.and Dechambre P, "Computation of Ellipsoidal
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Gravity Field Harmonics for small solar system bodies"
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pp. 30-36, 2000
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.. [4] George Dassios, "Ellipsoidal Harmonics: Theory and Applications"
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pp. 418, 2012
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Examples
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--------
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>>> from scipy.special import ellip_harm
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>>> w = ellip_harm(5,8,1,1,2.5)
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>>> w
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2.5
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Check that the functions indeed are solutions to the Lame equation:
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>>> import numpy as np
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>>> from scipy.interpolate import UnivariateSpline
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>>> def eigenvalue(f, df, ddf):
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... r = (((s**2 - h**2) * (s**2 - k**2) * ddf
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... + s * (2*s**2 - h**2 - k**2) * df
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... - n * (n + 1)*s**2*f) / f)
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... return -r.mean(), r.std()
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>>> s = np.linspace(0.1, 10, 200)
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>>> k, h, n, p = 8.0, 2.2, 3, 2
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>>> E = ellip_harm(h**2, k**2, n, p, s)
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>>> E_spl = UnivariateSpline(s, E)
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>>> a, a_err = eigenvalue(E_spl(s), E_spl(s,1), E_spl(s,2))
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>>> a, a_err
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(583.44366156701483, 6.4580890640310646e-11)
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""" # noqa: E501
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return _ellip_harm(h2, k2, n, p, s, signm, signn)
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_ellip_harm_2_vec = np.vectorize(_ellipsoid, otypes='d')
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def ellip_harm_2(h2, k2, n, p, s):
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r"""
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Ellipsoidal harmonic functions F^p_n(l)
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These are also known as Lame functions of the second kind, and are
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solutions to the Lame equation:
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.. math:: (s^2 - h^2)(s^2 - k^2)F''(s)
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+ s(2s^2 - h^2 - k^2)F'(s) + (a - q s^2)F(s) = 0
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where :math:`q = (n+1)n` and :math:`a` is the eigenvalue (not
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returned) corresponding to the solutions.
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Parameters
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----------
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h2 : float
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``h**2``
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k2 : float
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``k**2``; should be larger than ``h**2``
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n : int
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Degree.
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p : int
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Order, can range between [1,2n+1].
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s : float
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Coordinate
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Returns
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-------
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F : float
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The harmonic :math:`F^p_n(s)`
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See Also
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--------
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ellip_harm, ellip_normal
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Notes
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-----
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Lame functions of the second kind are related to the functions of the first kind:
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.. math::
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F^p_n(s)=(2n + 1)E^p_n(s)\int_{0}^{1/s}
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\frac{du}{(E^p_n(1/u))^2\sqrt{(1-u^2k^2)(1-u^2h^2)}}
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.. versionadded:: 0.15.0
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Examples
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--------
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>>> from scipy.special import ellip_harm_2
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>>> w = ellip_harm_2(5,8,2,1,10)
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>>> w
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0.00108056853382
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"""
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with np.errstate(all='ignore'):
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return _ellip_harm_2_vec(h2, k2, n, p, s)
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def _ellip_normal_vec(h2, k2, n, p):
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return _ellipsoid_norm(h2, k2, n, p)
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_ellip_normal_vec = np.vectorize(_ellip_normal_vec, otypes='d')
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def ellip_normal(h2, k2, n, p):
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r"""
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Ellipsoidal harmonic normalization constants gamma^p_n
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The normalization constant is defined as
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.. math::
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\gamma^p_n=8\int_{0}^{h}dx\int_{h}^{k}dy
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\frac{(y^2-x^2)(E^p_n(y)E^p_n(x))^2}{\sqrt((k^2-y^2)(y^2-h^2)(h^2-x^2)(k^2-x^2)}
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Parameters
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----------
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h2 : float
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``h**2``
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k2 : float
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``k**2``; should be larger than ``h**2``
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n : int
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Degree.
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p : int
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Order, can range between [1,2n+1].
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Returns
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-------
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gamma : float
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The normalization constant :math:`\gamma^p_n`
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See Also
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--------
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ellip_harm, ellip_harm_2
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Notes
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-----
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.. versionadded:: 0.15.0
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Examples
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--------
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>>> from scipy.special import ellip_normal
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>>> w = ellip_normal(5,8,3,7)
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>>> w
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1723.38796997
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"""
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with np.errstate(all='ignore'):
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return _ellip_normal_vec(h2, k2, n, p)
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