1

2# -*- coding: utf-8 -*-

3

4 u'''Discrete Sine Transforms (AuxDST) in Python, transcoded from I{Karney}'s C++ class

5 U{DST<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1DST.html>}

6 in I{GeographicLib version 2.2+}.

7

8 Copyright (C) U{Charles Karney<mailto:Karney@Alum.MIT.edu>} (2022-2023) and licensed

9 under the MIT/X11 License. For more information, see the U{GeographicLib

10<https://GeographicLib.SourceForge.io>} documentation.

11

12@note: Class L{AuxDST} requires package U{numpy<https://PyPI.org/project/numpy>} to be

13 installed, version 1.16 or newer and needed for I{exact} area calculations.

14'''

15# make sure int/int division yields float quotient, see .basics

16 from __future__ import division as _; del _ # PYCHOK semicolon

17

18 from pygeodesy.basics import isodd, map2, neg, _reverange, _xnumpy

19 from pygeodesy.constants import PI_2, PI_4, isfinite, \

20 _0_0, _0_5, _1_0, _naninf

21# from pygeodesy.fsums import Fsum # from .karney

22 from pygeodesy.karney import _2cos2x, _ALL_DOCS, Fsum, property_RO

23# from pygeodesy.lazily import _ALL_DOCS # from .karney

24# from pygeodesy.props import property_RO # from .karney

25

26__all__ = ()

27__version__ = '23.08.20'

28

29

30 class AuxDST(object):

31 '''Discrete Sine Transforms (DST) for I{Auxiliary Latitudes}.

32

33 @see: I{Karney}'s C++ class U{DST

34 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1DST.html>}.

35 '''

36 _N = 0

37

38 def __init__(self, N):

39 '''New L{AuxDST} instance.

40

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

46

47 @staticmethod

48 def evaluate(sinx, cosx, F, *N):

49 '''Compute the Fourier sum given the sine and cosine of the angle,

50 using I{Clenshaw} summation C{sum(B{F}[i] * sin((2*i+1) * x))}

51 for C{i in range(min(len(B{F}), *B{N}))}.

52

53 @arg sinx: The sin(I{sigma}) (C{float}).

54 @arg cosx: The cos(I{sigma}) (C{float}).

55 @arg F: The Fourier coefficients (C{float}[]).

56 @arg N: Optional, (smaller) number of terms to evaluate (C{int}).

57

58 @return: Precison I{Clenshaw} sum (C{float}).

59

60 @see: Methods C{AuxDST.integral} and C{AuxDST.integral2}.

61 '''

62 a = -_2cos2x(cosx, sinx)

63 if isfinite(a):

64 Y0, Y1 = Fsum(), Fsum()

65 n = _len_N(F, *N)

66 Fn = list(F[:n])

67 _F = Fn.pop

68 if isodd(n):

69 Y0 -= _F()

70 while Fn: # Y0, Y1 negated

71 Y1 -= Y0 * a + _F()

72 Y0 -= Y1 * a + _F()

73 r = -float(_Ys(Y0, -Y1, sinx))

74 else:

75 r = _naninf(-a)

76 return r

77

78 @property_RO

79 def _fft_numpy(self):

80 '''(INTERNAL) Get the C{numpy.fft} module, I{once}.

81 '''

82 AuxDST._fft_numpy = fft = _xnumpy(AuxDST, 1, 16).fft # overwrite property_RO

83 return fft

84

85 def _fft_real(self, data):

86 '''(INTERNAL) NumPy's I{kissfft}-like C{transform_real} function,

87 taking C{float}[:N] B{C{data}} and returning C{complex}[:N*2].

88 '''

89 # <https://GitHub.com/mborgerding/kissfft/blob/master/test/testkiss.py>

90 return self._fft_numpy.rfftn(data)

91

92 def _ffts(self, data, cIV):

93 '''(INTERNAL) Compute the DST-III or DST-IV FFTransforms.

94

95 @arg data: Elements DST-III[0:N+1] or DST-IV[0:N] (C{float}[])

96 with DST_III[0] = 0.

97 @arg cIV: If C{True} DST-IV, otherwise DST-III.

98

99 @return: FFTransforms (C{float}[0:N]).

100 '''

101 t, N = (), self.N

102 if N > 0:

103 N2 = N * 2

104 d = list(data)

105 # assert len(d) == N + (0 if cIV else 1)

106

107 if cIV: # DST-IV

108 from cmath import exp as _cexp

109

110 def _cF(c, j, d=-PI_4 / N):

111 return c * _cexp(complex(0, d * j))

112

113 i = 0

114 else: # DST-III

115 i = 1

116 # assert d[0] == _0_0

117

118 def _cF(c, unused): # PYCHOK redef

119 return c

120

121 d += list(reversed(d[i:N])) # i == len(d) - N

122 d += list(map(neg, d[:N2]))

123 c = self._fft_real(d) # complex[0:N*2]

124 n2 = float(-N2)

125 t = tuple(_cF(c[j], j).imag / n2 for j in range(1, N2, 2))

126 return t

127

128 def _ffts2(self, data, F):

129 '''(INTERNAL) Doubled FFTransforms.

130

131 @arg data: Grid centers (C{float}[N]).

132 @arg F: The transforms (C{float}[N])

133

134 @return: Doubled FFTransforms (C{float}[N*2]).

135 '''

136 def _dmF_2(d, F):

137 return (d - F) * _0_5

138

139 def _dpF_2(d, F):

140 return (d + F) * _0_5

141

142 # Copy DST-IV order N transform to d[0:N]

143 d = self._ffts(data, True)

144 N = self._N

145 # assert len(d) >= N and len(F) >= N

146 # (DST-IV order N - DST-III order N) / 2

147 m = map2(_dmF_2, d[:N], F[:N])

148 # (DST-IV order N + DST-III order N) / 2

149 p = map2(_dpF_2, d[:N], F[:N])

150 return p + tuple(reversed(m))

151

152 @staticmethod

153 def integral(sinx, cosx, F, *N):

154 '''Compute the integral of Fourier sum given the sine and cosine

155 of the angle using I{Clenshaw} summation C{-sum(B{F}[i] / (2*i+1) *

156 cos((2*i+1) * x))} for C{i in range(min(len(B{F}), *B{N}))}.

157

158 @arg sinx: The sin(I{sigma}) (C{float}).

159 @arg cosx: The cos(I{sigma}) (C{float}).

160 @arg F: The Fourier coefficients (C{float}[]).

161 @arg N: Optional, (smaller) number of terms to evaluate (C{int}).

162

163 @return: Precison I{Clenshaw} intergral (C{float}).

164

165 @see: Methods C{AuxDST.evaluate} and C{AuxDST.integral2}.

166 '''

167 a = _2cos2x(cosx - sinx, cosx + sinx)

168 if isfinite(a):

169 Y0, Y1 = Fsum(), Fsum()

170 for r in _reverscaled(F, *N):

171 Y1 -= Y0 * a + r

172 Y1, Y0 = Y0, -Y1

173 r = float(_Ys(Y1, Y0, cosx))

174 else:

175 r = _naninf(a)

176 return r

177

178 @staticmethod

179 def integral2(sin1, cos1, sin2, cos2, F, *N): # PYCHOK no cover

180 '''Compute the integral of Fourier sum given the sine and cosine

181 of the angles at the end points using I{Clenshaw} summation

182 C{integral(siny, cosy, F) - integral(sinx, cosx, F)}.

183

184 @arg sin1: The sin(I{sigma1}) (C{float}).

185 @arg cos1: The cos(I{sigma1}) (C{float}).

186 @arg sin2: The sin(I{sigma2}) (C{float}).

187 @arg cos2: The cos(I{sigma2}) (C{float}).

188 @arg F: The Fourier coefficients (C{float}[]).

189 @arg N: Optional, (smaller) number of terms to evaluate (C{int}).

190

191 @return: Precison I{Clenshaw} intergral (C{float}).

192

193 @see: Methods C{AuxDST.evaluate} and C{AuxDST.integral}.

194 '''

195 # 2 * cos(y - x)*cos(y + x) -> 2 * cos(2 * x)

196 a = _2cos2x(cos2 * cos1, sin2 * sin1)

197 # -2 * sin(y - x)*sin(y + x) -> 0

198 b = -_2cos2x(sin2 * cos1, cos2 * sin1)

199 if isfinite(a) and isfinite(b):

200 Y0, Y1 = Fsum(), Fsum()

201 Z0, Z1 = Fsum(), Fsum()

202 for r in _reverscaled(F, *N):

203 Y1 -= Y0 * a + Z0 * b + r

204 Z1 -= Y0 * b + Z0 * a

205 Y1, Y0 = Y0, -Y1

206 Z1, Z0 = Z0, -Z1

207 r = float(_Ys(Y1, Y0, cos2 - cos1) +

208 _Ys(Z1, Z0, cos2 + cos1))

209 else:

210 r = _naninf(a, b)

211 return r

212

213 @property_RO

214 def N(self):

215 '''Get this DST's size, number of points (C{int}).

216 '''

217 return self._N

218

219 def refine(self, f, F):

220 '''Double the number of sampled points on a Fourier series.

221

222 @arg f: Single-argument function (C{callable(sigma)} with

223 C{sigma = PI_4 * j / N for j in range(1, N*2, 2)}.

224 @arg F: The initial Fourier series coefficients (C{float}[:N]).

225

226 @return: Fourier series coefficients (C{float}[:N*2]).

227 '''

228 def _data(_f, N): # [:N]

229 if N > 0:

230 d = PI_4 / N

231 for j in range(1, N*2, 2):

232 yield _f(d * j)

233

234 return self._ffts2(_data(f, self.N), F)

235

236 def reset(self, N):

237 '''Reset this DST.

238

239 @arg N: Size, number of points (C{int}).

240

241 @return: The new size (C{int}, non-negative).

242 '''

243 self._N = N = max(0, N)

244 # kissfft.assign(N*2, False) # "reset" size, inverse

245 return N

246

247 def transform(self, f):

248 '''Compute C{N + 1} terms in the Fourier series.

249

250 @arg f: Single-argument function (C{callable(sigma)} with

251 C{sigma = PI_2 * i / N for i in range(1, N + 1)}.

252

253 @return: Fourier series coefficients (C{float}[:N + 1]).

254 '''

255 def _data(_f, N): # [:N + 1]

256 yield _0_0 # data[0] = 0

257 if N > 0:

258 d = PI_2 / N

259 for i in range(1, N + 1):

260 yield _f(d * i)

261

262 return self._ffts(_data(f, self.N), False)

263

264

265 def _len_N(F, *N):

266 # Adjusted C{len(B{F})}.

267 return min(len(F), *N) if N else len(F)

268

269

270 def _reverscaled(F, *N):

271 # Yield F[:N], reversed and scaled

272 for n in _reverange(_len_N(F, *N)):

273 yield F[n] / (n * 2 + _1_0)

274

275

276 def _Ys(X, Y, s):

277 # Return M{(X - Y) * s}, overwriting X

278 X -= Y

279 X *= s

280 return X

281

282

283__all__ += _ALL_DOCS(AuxDST)

284

285# **) MIT License

286#

287# Copyright (C) 2023-2023 -- mrJean1 at Gmail -- All Rights Reserved.

288#

289# Permission is hereby granted, free of charge, to any person obtaining a

290# copy of this software and associated documentation files (the "Software"),

291# to deal in the Software without restriction, including without limitation

292# the rights to use, copy, modify, merge, publish, distribute, sublicense,

293# and/or sell copies of the Software, and to permit persons to whom the

294# Software is furnished to do so, subject to the following conditions:

295#

296# The above copyright notice and this permission notice shall be included

297# in all copies or substantial portions of the Software.

298#

299# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

300# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

301# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

302# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

303# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

304# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

305# OTHER DEALINGS IN THE SOFTWARE.