Coverage for pygeodesy/auxilats/auxDST.py: 97%
100 statements
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2# -*- coding: utf-8 -*-
4u'''Discrete Sine Transforms (AuxDST) in Python, transcoded from I{Karney}'s C++ class
5U{DST<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1DST.html>}
6in I{GeographicLib version 2.2+}.
8Copyright (C) U{Charles Karney<mailto:Karney@Alum.MIT.edu>} (2022-2023) and licensed
9under the MIT/X11 License. For more information, see the U{GeographicLib
10<https://GeographicLib.SourceForge.io>} documentation.
12@note: Class L{AuxDST} requires U{numpy<https://PyPI.org/project/numpy>} to be
13 installed, version 1.16 or newer.
14'''
15# make sure int/int division yields float quotient, see .basics
16from __future__ import division as _; del _ # PYCHOK semicolon
18from pygeodesy.auxilats.auxily import _Dm
19from pygeodesy.basics import isodd, neg, _reverange, _xnumpy
20from pygeodesy.constants import PI_2, PI_4, isfinite, _0_0, _0_5, _naninf
21from pygeodesy.fsums import Fsum, property_RO
22from pygeodesy.karney import _2cos2x, _ALL_DOCS
23# from pygeodesy.lazily import _ALL_DOCS # from .karney
24# from pygeodesy.props import property_RO # from .fsums
26__all__ = ()
27__version__ = '23.12.02'
30class AuxDST(object):
31 '''Discrete Sine Transforms (DST) for I{Auxiliary} latitudes.
33 @see: I{Karney}'s C++ class U{DST
34 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1DST.html>}.
35 '''
36 _N = 0
38 def __init__(self, N):
39 '''New L{AuxDST} instance.
41 @arg N: Size, number of points (C{int}).
42 '''
43 if N > 0:
44 self._N = int(N)
45 # kissfft(N, False) # size, inverse
47 @staticmethod
48 def evaluate(sinx, cosx, F, *N):
49 '''Evaluate the Fourier sum given the sine and cosine of the angle,
50 using precision I{Clenshaw} summation.
52 @arg sinx: The sin(I{sigma}) (C{float}).
53 @arg cosx: The cos(I{sigma}) (C{float}).
54 @arg F: The Fourier coefficients (C{float}[]).
55 @arg N: Optional, (smaller) number of terms to evaluate (C{int}).
57 @return: Precison I{Clenshaw} sum (C{float}).
59 @see: Methods C{AuxDST.integral} and C{AuxDST.integral2}.
60 '''
61 a = -_2cos2x(cosx, sinx)
62 if isfinite(a):
63 Y0, Y1 = Fsum(), Fsum()
64 n = _len_N(F, *N)
65 Fn = list(F[:n])
66 _F = Fn.pop
67 if isodd(n):
68 Y0 -= _F()
69 while Fn: # Y0, Y1 negated
70 Y1 -= Y0 * a + _F()
71 Y0 -= Y1 * a + _F()
72 r = float(_Dm(-Y0, Y1, sinx))
73 else:
74 r = _naninf(-a)
75 return r
77 @property_RO
78 def _fft_numpy(self):
79 '''(INTERNAL) Get the C{numpy.fft} module, I{once}.
80 '''
81 AuxDST._fft_numpy = fft = _xnumpy(AuxDST, 1, 16).fft # overwrite property_RO
82 return fft
84 def _fft_real(self, data):
85 '''(INTERNAL) NumPy's I{kissfft}-like C{transform_real} function,
86 taking C{float}[:N] B{C{data}} and returning C{complex}[:N*2].
87 '''
88 # <https://GitHub.com/mborgerding/kissfft/blob/master/test/testkiss.py>
89 return self._fft_numpy.rfftn(data)
91 def _ffts(self, data, cIV):
92 '''(INTERNAL) Compute the DST-III or DST-IV FFTransforms.
94 @arg data: Elements DST-III[0:N+1] or DST-IV[0:N] (C{float}[])
95 with DST_III[0] = 0.
96 @arg cIV: If C{True} DST-IV, otherwise DST-III.
98 @return: FFTransforms (C{float}[0:N]).
99 '''
100 T, N = (), self.N
101 if N > 0:
102 N2 = N * 2
103 d = tuple(data)
104 # assert len(d) == N + (0 if cIV else 1)
106 if cIV: # DST-IV
107 from cmath import exp as _cexp
109 def _cF(c, j, r=-PI_4 / N):
110 return c * _cexp(complex(0, r * j))
112 i = 0
113 else: # DST-III
114 i = 1
115 # assert d[0] == _0_0
117 def _cF(c, *unused): # PYCHOK redef
118 return c
120 d += tuple(reversed(d[i:N])) # i == len(d) - N
121 d += tuple(map(neg, d[:N2]))
122 c = self._fft_real(d) # complex[0:N*2]
123 n2 = float(-N2)
124 T = tuple(_cF(c[j], j).imag / n2 for j in range(1, N2, 2))
125 return T
127 def _ffts2(self, data, F):
128 '''(INTERNAL) Doubled FFTransforms.
130 @arg data: Grid centers (C{float}[N]).
131 @arg F: The transforms (C{float}[N])
133 @return: Doubled FFTransforms (C{float}[N*2]).
134 '''
135 __2 = _0_5 # N = self._N
136 # copy DST-IV order N transform to D[0:N]
137 D = self._ffts(data, True)
138 # assert len(D) == N and len(F) >= N
139 # (DST-IV order N - DST-III order N) / 2
140 M = tuple((d - f) * __2 for d, f in zip(D, F)) # strict=False
141 # (DST-IV order N + DST-III order N) / 2
142 P = tuple((d + f) * __2 for d, f in zip(D, F)) # strict=False
143 # assert len(M) == len(P) == self._N
144 return P + tuple(reversed(M))
146 @staticmethod
147 def integral(sinx, cosx, F, *N):
148 '''Evaluate the integral of Fourier sum given the sine and
149 cosine of the angle, using precision I{Clenshaw} summation.
151 @arg sinx: The sin(I{sigma}) (C{float}).
152 @arg cosx: The cos(I{sigma}) (C{float}).
153 @arg F: The Fourier coefficients (C{float}[]).
154 @arg N: Optional, C{len(B{F})} or a (smaller) number of
155 terms to evaluate (C{int}).
157 @return: Precison I{Clenshaw} intergral (C{float}).
159 @see: Methods C{AuxDST.evaluate} and C{AuxDST.integral2}.
160 '''
161 a = _2cos2x(cosx - sinx, cosx + sinx)
162 if isfinite(a):
163 Y0, Y1 = Fsum(), Fsum()
164 for r in _reverscaled(F, *N):
165 Y1 -= Y0 * a + r
166 Y1, Y0 = Y0, -Y1
167 r = float(_Dm(Y1, Y0, cosx))
168 else:
169 r = _naninf(a)
170 return r
172 @staticmethod
173 def integral2(sinx, cosx, siny, cosy, F, *N): # PYCHOK no cover
174 '''Compute the definite integral of Fourier sum given the
175 sine and cosine of the angles at the end points, using
176 precision I{Clenshaw} summation.
178 @arg sinx: The sin(I{sigma1}) (C{float}).
179 @arg cosx: The cos(I{sigma1}) (C{float}).
180 @arg siny: The sin(I{sigma2}) (C{float}).
181 @arg cosy: The cos(I{sigma2}) (C{float}).
182 @arg F: The Fourier coefficients (C{float}[]).
183 @arg N: Optional, C{len(B{F})} or a (smaller) number of
184 terms to evaluate (C{int}).
186 @return: Precison I{Clenshaw} integral (C{float}).
188 @see: Methods C{AuxDST.evaluate} and C{AuxDST.integral}.
189 '''
190 # 2 * cos(y - x) * cos(y + x) -> 2 * cos(2 * x)
191 c = _2cos2x(cosy * cosx, siny * sinx)
192 # -2 * sin(y - x) * sin(y + x) -> 0
193 s = -_2cos2x(siny * cosx, cosy * sinx)
194 if isfinite(c) and isfinite(s):
195 Y0, Y1 = Fsum(), Fsum()
196 Z0, Z1 = Fsum(), Fsum()
197 for r in _reverscaled(F, *N):
198 Y1 -= Y0 * c + Z0 * s + r
199 Z1 -= Y0 * s + Z0 * c
200 Y1, Y0 = Y0, -Y1
201 Z1, Z0 = Z0, -Z1
202 r = float(_Dm(Y1, Y0, cosy - cosx) +
203 _Dm(Z1, Z0, cosy + cosx))
204 else:
205 r = _naninf(c, s)
206 return r
208 @property_RO
209 def N(self):
210 '''Get this DST's size, number of points (C{int}).
211 '''
212 return self._N
214 def refine(self, f, F, *sentinel):
215 '''Refine the Fourier series by doubling the sampled points.
217 @arg f: Single-argument callable (C{B{f}(sigma)}).
218 @arg F: Initial Fourier series coefficients (C{float}[:N]).
219 @arg sentinel: Optional coefficient(s) to append (C{float}(s)).
221 @return: Fourier series coefficients (C{float}[:N*2]).
223 @note: Any initial C{B{F}[N:]} sentinel coefficients are ignored.
224 '''
225 def _data(_f, N): # [:N]
226 if N > 0:
227 r = PI_4 / N
228 for j in range(1, N*2, 2):
229 yield _f(r * j)
231 # F = F[:self.N] handled by zip strict=False in ._ffts2 above
232 return self._ffts2(_data(f, self.N), F) + sentinel
234 def reset(self, N):
235 '''Reset this DST.
237 @arg N: Size, number of points (C{int}).
239 @return: The new size (C{int}, non-negative).
240 '''
241 self._N = N = max(0, N)
242 # kissfft.assign(N*2, False) # "reset" size, inverse
243 return N
245 def transform(self, f):
246 '''Determine C{[N + 1]} terms in the Fourier series.
248 @arg f: Single-argument callable (C{B{f}(sigma)}).
250 @return: Fourier series coefficients (C{float}[:N+1],
251 leading 0).
252 '''
253 def _data(_f, N): # [:N + 1]
254 yield _0_0 # data[0] = 0
255 if N > 0:
256 r = PI_2 / N
257 for i in range(1, N + 1):
258 yield _f(r * i)
260 return self._ffts(_data(f, self.N), False)
263def _len_N(F, *N):
264 # Adjusted C{len(B{F})}.
265 return min(len(F), *N) if N else len(F)
268def _reverscaled(F, *N):
269 # Yield F[:N], reversed and scaled
270 for n in _reverange(_len_N(F, *N)):
271 yield F[n] / float(n * 2 + 1)
274__all__ += _ALL_DOCS(AuxDST)
276# **) MIT License
277#
278# Copyright (C) 2023-2024 -- mrJean1 at Gmail -- All Rights Reserved.
279#
280# Permission is hereby granted, free of charge, to any person obtaining a
281# copy of this software and associated documentation files (the "Software"),
282# to deal in the Software without restriction, including without limitation
283# the rights to use, copy, modify, merge, publish, distribute, sublicense,
284# and/or sell copies of the Software, and to permit persons to whom the
285# Software is furnished to do so, subject to the following conditions:
286#
287# The above copyright notice and this permission notice shall be included
288# in all copies or substantial portions of the Software.
289#
290# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
291# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
292# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
293# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
294# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
295# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
296# OTHER DEALINGS IN THE SOFTWARE.