Coverage for pygeodesy/auxilats/auxDST.py: 97%

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1 

2# -*- coding: utf-8 -*- 

3 

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+}. 

7 

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. 

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 

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

17 

18from pygeodesy.auxilats.auxily import _Dm 

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

20from pygeodesy.constants import PI_2, PI_4, isfinite, _0_0, _0_5, _naninf 

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

22from 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 

30class 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 

266def _len_N(F, *N): 

267 # Adjusted C{len(B{F})}. 

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

269 

270 

271def _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.