Coverage for pygeodesy/auxilats/auxDST.py: 96%

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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:Charles@Karney.com>} (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 _2cos2x 

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

20from pygeodesy.constants import PI_2, PI_4, isfinite, \ 

21 _0_0, _0_5, _1_0, _inf_nan 

22from pygeodesy.fsums import Fsum, property_RO 

23from pygeodesy.lazily import _ALL_DOCS 

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

25 

26__all__ = () 

27__version__ = '23.08.12' 

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 '''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 if isodd(n): 

67 n -= 1 

68 Y0 -= F[n] 

69 while n > 0: # Y0, Y1 negated 

70 n -= 1; Y1 -= Y0 * a + F[n] # PYCHOK semicolon 

71 n -= 1; Y0 -= Y1 * a + F[n] # PYCHOK semicolon 

72 r = float(_Ys(Y0, -Y1, -sinx)) 

73 else: 

74 r = _inf_nan(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 = list(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, d=-PI_4 / N): 

110 return c * _cexp(complex(0, d * 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 += list(reversed(d[i:N])) # i == len(d) - N 

121 d += list(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 def _dmF_2(d, F): 

136 return (d - F) * _0_5 

137 

138 def _dpF_2(d, F): 

139 return (d + F) * _0_5 

140 

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

142 d = self._ffts(data, True) 

143 N = self._N 

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

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

146 m = map2(_dmF_2, d[:N], F[:N]) 

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

148 p = map2(_dpF_2, d[:N], F[:N]) 

149 return p + tuple(reversed(m)) 

150 

151 @staticmethod 

152 def integral(sinx, cosx, F, *N): 

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

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

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

156 

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

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

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

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

161 

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

163 

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

165 ''' 

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

167 if isfinite(a): 

168 Y0, Y1 = Fsum(), Fsum() 

169 for r in _reverscaled(F, *N): 

170 Y1 -= Y0 * a + r 

171 Y1, Y0 = Y0, -Y1 

172 r = float(_Ys(Y1, Y0, cosx)) 

173 else: 

174 r = _inf_nan(a) 

175 return r 

176 

177 @staticmethod 

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

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

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

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

182 

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

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

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

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

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

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

189 

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

191 

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

193 ''' 

194 # 2 * cos(y - x)*cos(y + x) -> 2 * cos(2 * x) 

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

196 # -2 * sin(y - x)*sin(y + x) -> 0 

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

198 if isfinite(a) and isfinite(b): 

199 Y0, Y1 = Fsum(), Fsum() 

200 Z0, Z1 = Fsum(), Fsum() 

201 for r in _reverscaled(F, *N): 

202 Y1 -= Y0 * a + Z0 * b + r 

203 Z1 -= Y0 * b + Z0 * a 

204 Y1, Y0 = Y0, -Y1 

205 Z1, Z0 = Z0, -Z1 

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

207 _Ys(Z1, Z0, cos2 + cos1)) 

208 else: 

209 r = _inf_nan(a, b) 

210 return r 

211 

212 @property_RO 

213 def N(self): 

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

215 ''' 

216 return self._N 

217 

218 def refine(self, f, F): 

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

220 

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

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

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

224 

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

226 ''' 

227 def _data(_f, N): # [:N] 

228 if N > 0: 

229 d = PI_4 / N 

230 for j in range(1, N*2, 2): 

231 yield _f(d * j) 

232 

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

234 

235 def reset(self, N): 

236 '''Reset this DST. 

237 

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

239 

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

241 ''' 

242 self._N = N = max(0, N) 

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

244 return N 

245 

246 def transform(self, f): 

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

248 

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

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

251 

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

253 ''' 

254 def _data(_f, N): # [:N + 1] 

255 yield _0_0 # data[0] = 0 

256 if N > 0: 

257 d = PI_2 / N 

258 for i in range(1, N + 1): 

259 yield _f(d * i) 

260 

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

262 

263 

264def _len_N(F, *N): 

265 # Adjusted C{len(B{F})}. 

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

267 

268 

269def _reverscaled(F, *N): 

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

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

272 yield F[n] / (n * 2 + _1_0) 

273 

274 

275def _Ys(X, Y, s): 

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

277 X -= Y 

278 X *= s 

279 return X 

280 

281 

282__all__ += _ALL_DOCS(AuxDST) 

283 

284# **) MIT License 

285# 

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

287# 

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

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

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

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

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

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

294# 

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

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

297# 

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

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

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

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

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

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

304# OTHER DEALINGS IN THE SOFTWARE.