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.auxilats.auxily import _Dm

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

20 from pygeodesy.constants import PI_2, PI_4, isfinite, _0_0, _0_5, _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__ = '23.09.14'

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 '''Evaluate the Fourier sum given the sine and cosine of the angle,

50 using precision I{Clenshaw} summation.

51

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}).

56

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

58

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

76

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

83

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)

90

91 def _ffts(self, data, cIV):

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

93

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.

97

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)

105

106 if cIV: # DST-IV

107 from cmath import exp as _cexp

108

109 def _cF(c, j, r=-PI_4 / N):

110 return c * _cexp(complex(0, r * j))

111

112 i = 0

113 else: # DST-III

114 i = 1

115 # assert d[0] == _0_0

116

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

118 return c

119

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

126

127 def _ffts2(self, data, F):

128 '''(INTERNAL) Doubled FFTransforms.

129

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

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

132

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

134 '''

135 __2 = _0_5

136

137 def _dmF_2(d, F):

138 return (d - F) * __2

139

140 def _dpF_2(d, F):

141 return (d + F) * __2

142

143 N = self._N

144 # copy DST-IV order N transform to d[0:N]

145 d = self._ffts(data, True)

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

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

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

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

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

151 return p + tuple(reversed(m))

152

153 @staticmethod

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

155 '''Evaluate the integral of Fourier sum given the sine and

156 cosine of the angle, using precision I{Clenshaw} summation.

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, C{len(B{F})} or a (smaller) number of

162 terms to evaluate (C{int}).

163

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

165

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

167 '''

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

169 if isfinite(a):

170 Y0, Y1 = Fsum(), Fsum()

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

172 Y1 -= Y0 * a + r

173 Y1, Y0 = Y0, -Y1

174 r = float(_Dm(Y1, Y0, cosx))

175 else:

176 r = _naninf(a)

177 return r

178

179 @staticmethod

180 def integral2(sinx, cosx, siny, cosy, F, *N): # PYCHOK no cover

181 '''Compute the definite integral of Fourier sum given the

182 sine and cosine of the angles at the end points, using

183 precision I{Clenshaw} summation.

184

185 @arg sinx: The sin(I{sigma1}) (C{float}).

186 @arg cosx: The cos(I{sigma1}) (C{float}).

187 @arg siny: The sin(I{sigma2}) (C{float}).

188 @arg cosy: The cos(I{sigma2}) (C{float}).

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

190 @arg N: Optional, C{len(B{F})} or a (smaller) number of

191 terms to evaluate (C{int}).

192

193 @return: Precison I{Clenshaw} integral (C{float}).

194

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

196 '''

197 # 2 * cos(y - x) * cos(y + x) -> 2 * cos(2 * x)

198 c = _2cos2x(cosy * cosx, siny * sinx)

199 # -2 * sin(y - x) * sin(y + x) -> 0

200 s = -_2cos2x(siny * cosx, cosy * sinx)

201 if isfinite(c) and isfinite(s):

202 Y0, Y1 = Fsum(), Fsum()

203 Z0, Z1 = Fsum(), Fsum()

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

205 Y1 -= Y0 * c + Z0 * s + r

206 Z1 -= Y0 * s + Z0 * c

207 Y1, Y0 = Y0, -Y1

208 Z1, Z0 = Z0, -Z1

209 r = float(_Dm(Y1, Y0, cosy - cosx) +

210 _Dm(Z1, Z0, cosy + cosx))

211 else:

212 r = _naninf(c, s)

213 return r

214

215 @property_RO

216 def N(self):

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

218 '''

219 return self._N

220

221 def refine(self, f, F):

222 '''Refine the Fourier series by doubling the sampled points.

223

224 @arg f: Single-argument callable (C{B{f}(sigma)}).

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

226

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

228 '''

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

230 if N > 0:

231 r = PI_4 / N

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

233 yield _f(r * j)

234

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

236

237 def reset(self, N):

238 '''Reset this DST.

239

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

241

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

243 '''

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

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

246 return N

247

248 def transform(self, f):

249 '''Determine C{N + 1} terms in the Fourier series.

250

251 @arg f: Single-argument callable (C{B{f}(sigma)}).

252

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

254 leading 0).

255 '''

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

257 yield _0_0 # data[0] = 0

258 if N > 0:

259 r = PI_2 / N

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

261 yield _f(r * i)

262

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

264

265

266 def _len_N(F, *N):

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

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

269

270

271 def _reverscaled(F, *N):

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

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

274 yield F[n] / float(n * 2 + 1)

275

276

277__all__ += _ALL_DOCS(AuxDST)

278

279# **) MIT License

280#

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

282#

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

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

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

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

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

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

289#

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

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

292#

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

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

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

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

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

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

299# OTHER DEALINGS IN THE SOFTWARE.