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426 lines
11 KiB
426 lines
11 KiB
"""
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Discrete Fourier Transform, Number Theoretic Transform,
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Walsh Hadamard Transform, Mobius Transform
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"""
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from sympy.core import S, Symbol, sympify
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from sympy.core.function import expand_mul
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from sympy.core.numbers import pi, I
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from sympy.functions.elementary.trigonometric import sin, cos
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from sympy.ntheory import isprime, primitive_root
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from sympy.utilities.iterables import ibin, iterable
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from sympy.utilities.misc import as_int
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#----------------------------------------------------------------------------#
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# #
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# Discrete Fourier Transform #
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# #
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#----------------------------------------------------------------------------#
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def _fourier_transform(seq, dps, inverse=False):
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"""Utility function for the Discrete Fourier Transform"""
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if not iterable(seq):
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raise TypeError("Expected a sequence of numeric coefficients "
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"for Fourier Transform")
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a = [sympify(arg) for arg in seq]
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if any(x.has(Symbol) for x in a):
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raise ValueError("Expected non-symbolic coefficients")
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n = len(a)
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if n < 2:
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return a
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b = n.bit_length() - 1
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if n&(n - 1): # not a power of 2
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b += 1
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n = 2**b
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a += [S.Zero]*(n - len(a))
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for i in range(1, n):
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j = int(ibin(i, b, str=True)[::-1], 2)
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if i < j:
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a[i], a[j] = a[j], a[i]
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ang = -2*pi/n if inverse else 2*pi/n
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if dps is not None:
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ang = ang.evalf(dps + 2)
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w = [cos(ang*i) + I*sin(ang*i) for i in range(n // 2)]
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h = 2
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while h <= n:
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hf, ut = h // 2, n // h
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for i in range(0, n, h):
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for j in range(hf):
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u, v = a[i + j], expand_mul(a[i + j + hf]*w[ut * j])
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a[i + j], a[i + j + hf] = u + v, u - v
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h *= 2
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if inverse:
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a = [(x/n).evalf(dps) for x in a] if dps is not None \
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else [x/n for x in a]
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return a
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def fft(seq, dps=None):
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r"""
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Performs the Discrete Fourier Transform (**DFT**) in the complex domain.
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The sequence is automatically padded to the right with zeros, as the
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*radix-2 FFT* requires the number of sample points to be a power of 2.
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This method should be used with default arguments only for short sequences
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as the complexity of expressions increases with the size of the sequence.
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Parameters
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==========
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seq : iterable
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The sequence on which **DFT** is to be applied.
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dps : Integer
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Specifies the number of decimal digits for precision.
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Examples
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========
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>>> from sympy import fft, ifft
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>>> fft([1, 2, 3, 4])
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[10, -2 - 2*I, -2, -2 + 2*I]
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>>> ifft(_)
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[1, 2, 3, 4]
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>>> ifft([1, 2, 3, 4])
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[5/2, -1/2 + I/2, -1/2, -1/2 - I/2]
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>>> fft(_)
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[1, 2, 3, 4]
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>>> ifft([1, 7, 3, 4], dps=15)
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[3.75, -0.5 - 0.75*I, -1.75, -0.5 + 0.75*I]
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>>> fft(_)
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[1.0, 7.0, 3.0, 4.0]
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Cooley%E2%80%93Tukey_FFT_algorithm
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.. [2] https://mathworld.wolfram.com/FastFourierTransform.html
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"""
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return _fourier_transform(seq, dps=dps)
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def ifft(seq, dps=None):
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return _fourier_transform(seq, dps=dps, inverse=True)
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ifft.__doc__ = fft.__doc__
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#----------------------------------------------------------------------------#
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# #
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# Number Theoretic Transform #
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# #
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#----------------------------------------------------------------------------#
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def _number_theoretic_transform(seq, prime, inverse=False):
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"""Utility function for the Number Theoretic Transform"""
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if not iterable(seq):
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raise TypeError("Expected a sequence of integer coefficients "
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"for Number Theoretic Transform")
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p = as_int(prime)
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if not isprime(p):
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raise ValueError("Expected prime modulus for "
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"Number Theoretic Transform")
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a = [as_int(x) % p for x in seq]
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n = len(a)
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if n < 1:
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return a
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b = n.bit_length() - 1
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if n&(n - 1):
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b += 1
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n = 2**b
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if (p - 1) % n:
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raise ValueError("Expected prime modulus of the form (m*2**k + 1)")
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a += [0]*(n - len(a))
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for i in range(1, n):
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j = int(ibin(i, b, str=True)[::-1], 2)
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if i < j:
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a[i], a[j] = a[j], a[i]
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pr = primitive_root(p)
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rt = pow(pr, (p - 1) // n, p)
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if inverse:
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rt = pow(rt, p - 2, p)
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w = [1]*(n // 2)
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for i in range(1, n // 2):
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w[i] = w[i - 1]*rt % p
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h = 2
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while h <= n:
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hf, ut = h // 2, n // h
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for i in range(0, n, h):
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for j in range(hf):
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u, v = a[i + j], a[i + j + hf]*w[ut * j]
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a[i + j], a[i + j + hf] = (u + v) % p, (u - v) % p
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h *= 2
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if inverse:
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rv = pow(n, p - 2, p)
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a = [x*rv % p for x in a]
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return a
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def ntt(seq, prime):
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r"""
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Performs the Number Theoretic Transform (**NTT**), which specializes the
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Discrete Fourier Transform (**DFT**) over quotient ring `Z/pZ` for prime
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`p` instead of complex numbers `C`.
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The sequence is automatically padded to the right with zeros, as the
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*radix-2 NTT* requires the number of sample points to be a power of 2.
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Parameters
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==========
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seq : iterable
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The sequence on which **DFT** is to be applied.
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prime : Integer
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Prime modulus of the form `(m 2^k + 1)` to be used for performing
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**NTT** on the sequence.
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Examples
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========
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>>> from sympy import ntt, intt
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>>> ntt([1, 2, 3, 4], prime=3*2**8 + 1)
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[10, 643, 767, 122]
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>>> intt(_, 3*2**8 + 1)
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[1, 2, 3, 4]
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>>> intt([1, 2, 3, 4], prime=3*2**8 + 1)
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[387, 415, 384, 353]
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>>> ntt(_, prime=3*2**8 + 1)
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[1, 2, 3, 4]
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References
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==========
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.. [1] http://www.apfloat.org/ntt.html
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.. [2] https://mathworld.wolfram.com/NumberTheoreticTransform.html
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.. [3] https://en.wikipedia.org/wiki/Discrete_Fourier_transform_(general%29
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"""
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return _number_theoretic_transform(seq, prime=prime)
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def intt(seq, prime):
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return _number_theoretic_transform(seq, prime=prime, inverse=True)
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intt.__doc__ = ntt.__doc__
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#----------------------------------------------------------------------------#
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# #
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# Walsh Hadamard Transform #
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# #
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#----------------------------------------------------------------------------#
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def _walsh_hadamard_transform(seq, inverse=False):
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"""Utility function for the Walsh Hadamard Transform"""
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if not iterable(seq):
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raise TypeError("Expected a sequence of coefficients "
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"for Walsh Hadamard Transform")
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a = [sympify(arg) for arg in seq]
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n = len(a)
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if n < 2:
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return a
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if n&(n - 1):
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n = 2**n.bit_length()
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a += [S.Zero]*(n - len(a))
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h = 2
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while h <= n:
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hf = h // 2
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for i in range(0, n, h):
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for j in range(hf):
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u, v = a[i + j], a[i + j + hf]
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a[i + j], a[i + j + hf] = u + v, u - v
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h *= 2
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if inverse:
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a = [x/n for x in a]
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return a
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def fwht(seq):
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r"""
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Performs the Walsh Hadamard Transform (**WHT**), and uses Hadamard
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ordering for the sequence.
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The sequence is automatically padded to the right with zeros, as the
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*radix-2 FWHT* requires the number of sample points to be a power of 2.
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Parameters
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==========
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seq : iterable
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The sequence on which WHT is to be applied.
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Examples
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========
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>>> from sympy import fwht, ifwht
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>>> fwht([4, 2, 2, 0, 0, 2, -2, 0])
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[8, 0, 8, 0, 8, 8, 0, 0]
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>>> ifwht(_)
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[4, 2, 2, 0, 0, 2, -2, 0]
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>>> ifwht([19, -1, 11, -9, -7, 13, -15, 5])
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[2, 0, 4, 0, 3, 10, 0, 0]
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>>> fwht(_)
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[19, -1, 11, -9, -7, 13, -15, 5]
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Hadamard_transform
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.. [2] https://en.wikipedia.org/wiki/Fast_Walsh%E2%80%93Hadamard_transform
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"""
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return _walsh_hadamard_transform(seq)
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def ifwht(seq):
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return _walsh_hadamard_transform(seq, inverse=True)
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ifwht.__doc__ = fwht.__doc__
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#----------------------------------------------------------------------------#
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# #
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# Mobius Transform for Subset Lattice #
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# #
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#----------------------------------------------------------------------------#
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def _mobius_transform(seq, sgn, subset):
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r"""Utility function for performing Mobius Transform using
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Yate's Dynamic Programming method"""
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if not iterable(seq):
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raise TypeError("Expected a sequence of coefficients")
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a = [sympify(arg) for arg in seq]
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n = len(a)
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if n < 2:
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return a
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if n&(n - 1):
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n = 2**n.bit_length()
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a += [S.Zero]*(n - len(a))
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if subset:
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i = 1
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while i < n:
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for j in range(n):
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if j & i:
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a[j] += sgn*a[j ^ i]
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i *= 2
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else:
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i = 1
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while i < n:
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for j in range(n):
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if j & i:
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continue
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a[j] += sgn*a[j ^ i]
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i *= 2
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return a
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def mobius_transform(seq, subset=True):
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r"""
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Performs the Mobius Transform for subset lattice with indices of
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sequence as bitmasks.
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The indices of each argument, considered as bit strings, correspond
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to subsets of a finite set.
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The sequence is automatically padded to the right with zeros, as the
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definition of subset/superset based on bitmasks (indices) requires
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the size of sequence to be a power of 2.
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Parameters
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==========
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seq : iterable
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The sequence on which Mobius Transform is to be applied.
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subset : bool
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Specifies if Mobius Transform is applied by enumerating subsets
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or supersets of the given set.
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Examples
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========
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>>> from sympy import symbols
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>>> from sympy import mobius_transform, inverse_mobius_transform
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>>> x, y, z = symbols('x y z')
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>>> mobius_transform([x, y, z])
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[x, x + y, x + z, x + y + z]
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>>> inverse_mobius_transform(_)
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[x, y, z, 0]
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>>> mobius_transform([x, y, z], subset=False)
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[x + y + z, y, z, 0]
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>>> inverse_mobius_transform(_, subset=False)
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[x, y, z, 0]
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>>> mobius_transform([1, 2, 3, 4])
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[1, 3, 4, 10]
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>>> inverse_mobius_transform(_)
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[1, 2, 3, 4]
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>>> mobius_transform([1, 2, 3, 4], subset=False)
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[10, 6, 7, 4]
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>>> inverse_mobius_transform(_, subset=False)
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[1, 2, 3, 4]
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula
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.. [2] https://people.csail.mit.edu/rrw/presentations/subset-conv.pdf
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.. [3] https://arxiv.org/pdf/1211.0189.pdf
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"""
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return _mobius_transform(seq, sgn=+1, subset=subset)
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def inverse_mobius_transform(seq, subset=True):
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return _mobius_transform(seq, sgn=-1, subset=subset)
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inverse_mobius_transform.__doc__ = mobius_transform.__doc__
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