Spaces:
Sleeping
Sleeping
File size: 11,681 Bytes
6a86ad5 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 |
"""
Discrete Fourier Transform, Number Theoretic Transform,
Walsh Hadamard Transform, Mobius Transform
"""
from sympy.core import S, Symbol, sympify
from sympy.core.function import expand_mul
from sympy.core.numbers import pi, I
from sympy.functions.elementary.trigonometric import sin, cos
from sympy.ntheory import isprime, primitive_root
from sympy.utilities.iterables import ibin, iterable
from sympy.utilities.misc import as_int
#----------------------------------------------------------------------------#
# #
# Discrete Fourier Transform #
# #
#----------------------------------------------------------------------------#
def _fourier_transform(seq, dps, inverse=False):
"""Utility function for the Discrete Fourier Transform"""
if not iterable(seq):
raise TypeError("Expected a sequence of numeric coefficients "
"for Fourier Transform")
a = [sympify(arg) for arg in seq]
if any(x.has(Symbol) for x in a):
raise ValueError("Expected non-symbolic coefficients")
n = len(a)
if n < 2:
return a
b = n.bit_length() - 1
if n&(n - 1): # not a power of 2
b += 1
n = 2**b
a += [S.Zero]*(n - len(a))
for i in range(1, n):
j = int(ibin(i, b, str=True)[::-1], 2)
if i < j:
a[i], a[j] = a[j], a[i]
ang = -2*pi/n if inverse else 2*pi/n
if dps is not None:
ang = ang.evalf(dps + 2)
w = [cos(ang*i) + I*sin(ang*i) for i in range(n // 2)]
h = 2
while h <= n:
hf, ut = h // 2, n // h
for i in range(0, n, h):
for j in range(hf):
u, v = a[i + j], expand_mul(a[i + j + hf]*w[ut * j])
a[i + j], a[i + j + hf] = u + v, u - v
h *= 2
if inverse:
a = [(x/n).evalf(dps) for x in a] if dps is not None \
else [x/n for x in a]
return a
def fft(seq, dps=None):
r"""
Performs the Discrete Fourier Transform (**DFT**) in the complex domain.
The sequence is automatically padded to the right with zeros, as the
*radix-2 FFT* requires the number of sample points to be a power of 2.
This method should be used with default arguments only for short sequences
as the complexity of expressions increases with the size of the sequence.
Parameters
==========
seq : iterable
The sequence on which **DFT** is to be applied.
dps : Integer
Specifies the number of decimal digits for precision.
Examples
========
>>> from sympy import fft, ifft
>>> fft([1, 2, 3, 4])
[10, -2 - 2*I, -2, -2 + 2*I]
>>> ifft(_)
[1, 2, 3, 4]
>>> ifft([1, 2, 3, 4])
[5/2, -1/2 + I/2, -1/2, -1/2 - I/2]
>>> fft(_)
[1, 2, 3, 4]
>>> ifft([1, 7, 3, 4], dps=15)
[3.75, -0.5 - 0.75*I, -1.75, -0.5 + 0.75*I]
>>> fft(_)
[1.0, 7.0, 3.0, 4.0]
References
==========
.. [1] https://en.wikipedia.org/wiki/Cooley%E2%80%93Tukey_FFT_algorithm
.. [2] https://mathworld.wolfram.com/FastFourierTransform.html
"""
return _fourier_transform(seq, dps=dps)
def ifft(seq, dps=None):
return _fourier_transform(seq, dps=dps, inverse=True)
ifft.__doc__ = fft.__doc__
#----------------------------------------------------------------------------#
# #
# Number Theoretic Transform #
# #
#----------------------------------------------------------------------------#
def _number_theoretic_transform(seq, prime, inverse=False):
"""Utility function for the Number Theoretic Transform"""
if not iterable(seq):
raise TypeError("Expected a sequence of integer coefficients "
"for Number Theoretic Transform")
p = as_int(prime)
if not isprime(p):
raise ValueError("Expected prime modulus for "
"Number Theoretic Transform")
a = [as_int(x) % p for x in seq]
n = len(a)
if n < 1:
return a
b = n.bit_length() - 1
if n&(n - 1):
b += 1
n = 2**b
if (p - 1) % n:
raise ValueError("Expected prime modulus of the form (m*2**k + 1)")
a += [0]*(n - len(a))
for i in range(1, n):
j = int(ibin(i, b, str=True)[::-1], 2)
if i < j:
a[i], a[j] = a[j], a[i]
pr = primitive_root(p)
rt = pow(pr, (p - 1) // n, p)
if inverse:
rt = pow(rt, p - 2, p)
w = [1]*(n // 2)
for i in range(1, n // 2):
w[i] = w[i - 1]*rt % p
h = 2
while h <= n:
hf, ut = h // 2, n // h
for i in range(0, n, h):
for j in range(hf):
u, v = a[i + j], a[i + j + hf]*w[ut * j]
a[i + j], a[i + j + hf] = (u + v) % p, (u - v) % p
h *= 2
if inverse:
rv = pow(n, p - 2, p)
a = [x*rv % p for x in a]
return a
def ntt(seq, prime):
r"""
Performs the Number Theoretic Transform (**NTT**), which specializes the
Discrete Fourier Transform (**DFT**) over quotient ring `Z/pZ` for prime
`p` instead of complex numbers `C`.
The sequence is automatically padded to the right with zeros, as the
*radix-2 NTT* requires the number of sample points to be a power of 2.
Parameters
==========
seq : iterable
The sequence on which **DFT** is to be applied.
prime : Integer
Prime modulus of the form `(m 2^k + 1)` to be used for performing
**NTT** on the sequence.
Examples
========
>>> from sympy import ntt, intt
>>> ntt([1, 2, 3, 4], prime=3*2**8 + 1)
[10, 643, 767, 122]
>>> intt(_, 3*2**8 + 1)
[1, 2, 3, 4]
>>> intt([1, 2, 3, 4], prime=3*2**8 + 1)
[387, 415, 384, 353]
>>> ntt(_, prime=3*2**8 + 1)
[1, 2, 3, 4]
References
==========
.. [1] http://www.apfloat.org/ntt.html
.. [2] https://mathworld.wolfram.com/NumberTheoreticTransform.html
.. [3] https://en.wikipedia.org/wiki/Discrete_Fourier_transform_(general%29
"""
return _number_theoretic_transform(seq, prime=prime)
def intt(seq, prime):
return _number_theoretic_transform(seq, prime=prime, inverse=True)
intt.__doc__ = ntt.__doc__
#----------------------------------------------------------------------------#
# #
# Walsh Hadamard Transform #
# #
#----------------------------------------------------------------------------#
def _walsh_hadamard_transform(seq, inverse=False):
"""Utility function for the Walsh Hadamard Transform"""
if not iterable(seq):
raise TypeError("Expected a sequence of coefficients "
"for Walsh Hadamard Transform")
a = [sympify(arg) for arg in seq]
n = len(a)
if n < 2:
return a
if n&(n - 1):
n = 2**n.bit_length()
a += [S.Zero]*(n - len(a))
h = 2
while h <= n:
hf = h // 2
for i in range(0, n, h):
for j in range(hf):
u, v = a[i + j], a[i + j + hf]
a[i + j], a[i + j + hf] = u + v, u - v
h *= 2
if inverse:
a = [x/n for x in a]
return a
def fwht(seq):
r"""
Performs the Walsh Hadamard Transform (**WHT**), and uses Hadamard
ordering for the sequence.
The sequence is automatically padded to the right with zeros, as the
*radix-2 FWHT* requires the number of sample points to be a power of 2.
Parameters
==========
seq : iterable
The sequence on which WHT is to be applied.
Examples
========
>>> from sympy import fwht, ifwht
>>> fwht([4, 2, 2, 0, 0, 2, -2, 0])
[8, 0, 8, 0, 8, 8, 0, 0]
>>> ifwht(_)
[4, 2, 2, 0, 0, 2, -2, 0]
>>> ifwht([19, -1, 11, -9, -7, 13, -15, 5])
[2, 0, 4, 0, 3, 10, 0, 0]
>>> fwht(_)
[19, -1, 11, -9, -7, 13, -15, 5]
References
==========
.. [1] https://en.wikipedia.org/wiki/Hadamard_transform
.. [2] https://en.wikipedia.org/wiki/Fast_Walsh%E2%80%93Hadamard_transform
"""
return _walsh_hadamard_transform(seq)
def ifwht(seq):
return _walsh_hadamard_transform(seq, inverse=True)
ifwht.__doc__ = fwht.__doc__
#----------------------------------------------------------------------------#
# #
# Mobius Transform for Subset Lattice #
# #
#----------------------------------------------------------------------------#
def _mobius_transform(seq, sgn, subset):
r"""Utility function for performing Mobius Transform using
Yate's Dynamic Programming method"""
if not iterable(seq):
raise TypeError("Expected a sequence of coefficients")
a = [sympify(arg) for arg in seq]
n = len(a)
if n < 2:
return a
if n&(n - 1):
n = 2**n.bit_length()
a += [S.Zero]*(n - len(a))
if subset:
i = 1
while i < n:
for j in range(n):
if j & i:
a[j] += sgn*a[j ^ i]
i *= 2
else:
i = 1
while i < n:
for j in range(n):
if j & i:
continue
a[j] += sgn*a[j ^ i]
i *= 2
return a
def mobius_transform(seq, subset=True):
r"""
Performs the Mobius Transform for subset lattice with indices of
sequence as bitmasks.
The indices of each argument, considered as bit strings, correspond
to subsets of a finite set.
The sequence is automatically padded to the right with zeros, as the
definition of subset/superset based on bitmasks (indices) requires
the size of sequence to be a power of 2.
Parameters
==========
seq : iterable
The sequence on which Mobius Transform is to be applied.
subset : bool
Specifies if Mobius Transform is applied by enumerating subsets
or supersets of the given set.
Examples
========
>>> from sympy import symbols
>>> from sympy import mobius_transform, inverse_mobius_transform
>>> x, y, z = symbols('x y z')
>>> mobius_transform([x, y, z])
[x, x + y, x + z, x + y + z]
>>> inverse_mobius_transform(_)
[x, y, z, 0]
>>> mobius_transform([x, y, z], subset=False)
[x + y + z, y, z, 0]
>>> inverse_mobius_transform(_, subset=False)
[x, y, z, 0]
>>> mobius_transform([1, 2, 3, 4])
[1, 3, 4, 10]
>>> inverse_mobius_transform(_)
[1, 2, 3, 4]
>>> mobius_transform([1, 2, 3, 4], subset=False)
[10, 6, 7, 4]
>>> inverse_mobius_transform(_, subset=False)
[1, 2, 3, 4]
References
==========
.. [1] https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula
.. [2] https://people.csail.mit.edu/rrw/presentations/subset-conv.pdf
.. [3] https://arxiv.org/pdf/1211.0189.pdf
"""
return _mobius_transform(seq, sgn=+1, subset=subset)
def inverse_mobius_transform(seq, subset=True):
return _mobius_transform(seq, sgn=-1, subset=subset)
inverse_mobius_transform.__doc__ = mobius_transform.__doc__
|