/
_ellip_harm.py
207 lines (159 loc) · 5.12 KB
/
_ellip_harm.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
import numpy as np
from ._ufuncs import _ellip_harm
from ._ellip_harm_2 import _ellipsoid, _ellipsoid_norm
def ellip_harm(h2, k2, n, p, s, signm=1, signn=1):
r"""
Ellipsoidal harmonic functions E^p_n(l)
These are also known as Lame functions of the first kind, and are
solutions to the Lame equation:
.. math:: (s^2 - h^2)(s^2 - k^2)E''(s) + s(2s^2 - h^2 - k^2)E'(s) + (a - q s^2)E(s) = 0
where :math:`q = (n+1)n` and :math:`a` is the eigenvalue (not
returned) corresponding to the solutions.
Parameters
----------
h2 : float
``h**2``
k2 : float
``k**2``; should be larger than ``h**2``
n : int
Degree
s : float
Coordinate
p : int
Order, can range between [1,2n+1]
signm : {1, -1}, optional
Sign of prefactor of functions. Can be +/-1. See Notes.
signn : {1, -1}, optional
Sign of prefactor of functions. Can be +/-1. See Notes.
Returns
-------
E : float
the harmonic :math:`E^p_n(s)`
See Also
--------
ellip_harm_2, ellip_normal
Notes
-----
The geometric interpretation of the ellipsoidal functions is
explained in [2]_, [3]_, [4]_. The `signm` and `signn` arguments control the
sign of prefactors for functions according to their type::
K : +1
L : signm
M : signn
N : signm*signn
.. versionadded:: 0.15.0
References
----------
.. [1] Digital Library of Mathematical Functions 29.12
https://dlmf.nist.gov/29.12
.. [2] Bardhan and Knepley, "Computational science and
re-discovery: open-source implementations of
ellipsoidal harmonics for problems in potential theory",
Comput. Sci. Disc. 5, 014006 (2012)
:doi:`10.1088/1749-4699/5/1/014006`.
.. [3] David J.and Dechambre P, "Computation of Ellipsoidal
Gravity Field Harmonics for small solar system bodies"
pp. 30-36, 2000
.. [4] George Dassios, "Ellipsoidal Harmonics: Theory and Applications"
pp. 418, 2012
Examples
--------
>>> from scipy.special import ellip_harm
>>> w = ellip_harm(5,8,1,1,2.5)
>>> w
2.5
Check that the functions indeed are solutions to the Lame equation:
>>> from scipy.interpolate import UnivariateSpline
>>> def eigenvalue(f, df, ddf):
... r = ((s**2 - h**2)*(s**2 - k**2)*ddf + s*(2*s**2 - h**2 - k**2)*df - n*(n+1)*s**2*f)/f
... return -r.mean(), r.std()
>>> s = np.linspace(0.1, 10, 200)
>>> k, h, n, p = 8.0, 2.2, 3, 2
>>> E = ellip_harm(h**2, k**2, n, p, s)
>>> E_spl = UnivariateSpline(s, E)
>>> a, a_err = eigenvalue(E_spl(s), E_spl(s,1), E_spl(s,2))
>>> a, a_err
(583.44366156701483, 6.4580890640310646e-11)
"""
return _ellip_harm(h2, k2, n, p, s, signm, signn)
_ellip_harm_2_vec = np.vectorize(_ellipsoid, otypes='d')
def ellip_harm_2(h2, k2, n, p, s):
r"""
Ellipsoidal harmonic functions F^p_n(l)
These are also known as Lame functions of the second kind, and are
solutions to the Lame equation:
.. math:: (s^2 - h^2)(s^2 - k^2)F''(s) + s(2s^2 - h^2 - k^2)F'(s) + (a - q s^2)F(s) = 0
where :math:`q = (n+1)n` and :math:`a` is the eigenvalue (not
returned) corresponding to the solutions.
Parameters
----------
h2 : float
``h**2``
k2 : float
``k**2``; should be larger than ``h**2``
n : int
Degree.
p : int
Order, can range between [1,2n+1].
s : float
Coordinate
Returns
-------
F : float
The harmonic :math:`F^p_n(s)`
Notes
-----
Lame functions of the second kind are related to the functions of the first kind:
.. math::
F^p_n(s)=(2n + 1)E^p_n(s)\int_{0}^{1/s}\frac{du}{(E^p_n(1/u))^2\sqrt{(1-u^2k^2)(1-u^2h^2)}}
.. versionadded:: 0.15.0
See Also
--------
ellip_harm, ellip_normal
Examples
--------
>>> from scipy.special import ellip_harm_2
>>> w = ellip_harm_2(5,8,2,1,10)
>>> w
0.00108056853382
"""
with np.errstate(all='ignore'):
return _ellip_harm_2_vec(h2, k2, n, p, s)
def _ellip_normal_vec(h2, k2, n, p):
return _ellipsoid_norm(h2, k2, n, p)
_ellip_normal_vec = np.vectorize(_ellip_normal_vec, otypes='d')
def ellip_normal(h2, k2, n, p):
r"""
Ellipsoidal harmonic normalization constants gamma^p_n
The normalization constant is defined as
.. math::
\gamma^p_n=8\int_{0}^{h}dx\int_{h}^{k}dy\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)}
Parameters
----------
h2 : float
``h**2``
k2 : float
``k**2``; should be larger than ``h**2``
n : int
Degree.
p : int
Order, can range between [1,2n+1].
Returns
-------
gamma : float
The normalization constant :math:`\gamma^p_n`
See Also
--------
ellip_harm, ellip_harm_2
Notes
-----
.. versionadded:: 0.15.0
Examples
--------
>>> from scipy.special import ellip_normal
>>> w = ellip_normal(5,8,3,7)
>>> w
1723.38796997
"""
with np.errstate(all='ignore'):
return _ellip_normal_vec(h2, k2, n, p)