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489 lines
14 KiB
489 lines
14 KiB
5 months ago
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"""
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Convolution (using **FFT**, **NTT**, **FWHT**), Subset Convolution,
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Covering Product, Intersecting Product
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"""
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from sympy.core import S, sympify
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from sympy.core.function import expand_mul
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from sympy.discrete.transforms import (
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fft, ifft, ntt, intt, fwht, ifwht,
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mobius_transform, inverse_mobius_transform)
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from sympy.utilities.iterables import iterable
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from sympy.utilities.misc import as_int
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def convolution(a, b, cycle=0, dps=None, prime=None, dyadic=None, subset=None):
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"""
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Performs convolution by determining the type of desired
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convolution using hints.
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Exactly one of ``dps``, ``prime``, ``dyadic``, ``subset`` arguments
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should be specified explicitly for identifying the type of convolution,
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and the argument ``cycle`` can be specified optionally.
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For the default arguments, linear convolution is performed using **FFT**.
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Parameters
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==========
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a, b : iterables
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The sequences for which convolution is performed.
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cycle : Integer
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Specifies the length for doing cyclic convolution.
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dps : Integer
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Specifies the number of decimal digits for precision for
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performing **FFT** on the sequence.
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prime : Integer
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Prime modulus of the form `(m 2^k + 1)` to be used for
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performing **NTT** on the sequence.
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dyadic : bool
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Identifies the convolution type as dyadic (*bitwise-XOR*)
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convolution, which is performed using **FWHT**.
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subset : bool
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Identifies the convolution type as subset convolution.
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Examples
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========
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>>> from sympy import convolution, symbols, S, I
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>>> u, v, w, x, y, z = symbols('u v w x y z')
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>>> convolution([1 + 2*I, 4 + 3*I], [S(5)/4, 6], dps=3)
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[1.25 + 2.5*I, 11.0 + 15.8*I, 24.0 + 18.0*I]
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>>> convolution([1, 2, 3], [4, 5, 6], cycle=3)
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[31, 31, 28]
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>>> convolution([111, 777], [888, 444], prime=19*2**10 + 1)
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[1283, 19351, 14219]
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>>> convolution([111, 777], [888, 444], prime=19*2**10 + 1, cycle=2)
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[15502, 19351]
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>>> convolution([u, v], [x, y, z], dyadic=True)
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[u*x + v*y, u*y + v*x, u*z, v*z]
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>>> convolution([u, v], [x, y, z], dyadic=True, cycle=2)
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[u*x + u*z + v*y, u*y + v*x + v*z]
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>>> convolution([u, v, w], [x, y, z], subset=True)
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[u*x, u*y + v*x, u*z + w*x, v*z + w*y]
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>>> convolution([u, v, w], [x, y, z], subset=True, cycle=3)
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[u*x + v*z + w*y, u*y + v*x, u*z + w*x]
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"""
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c = as_int(cycle)
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if c < 0:
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raise ValueError("The length for cyclic convolution "
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"must be non-negative")
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dyadic = True if dyadic else None
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subset = True if subset else None
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if sum(x is not None for x in (prime, dps, dyadic, subset)) > 1:
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raise TypeError("Ambiguity in determining the type of convolution")
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if prime is not None:
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ls = convolution_ntt(a, b, prime=prime)
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return ls if not c else [sum(ls[i::c]) % prime for i in range(c)]
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if dyadic:
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ls = convolution_fwht(a, b)
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elif subset:
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ls = convolution_subset(a, b)
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else:
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ls = convolution_fft(a, b, dps=dps)
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return ls if not c else [sum(ls[i::c]) for i in range(c)]
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#----------------------------------------------------------------------------#
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# #
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# Convolution for Complex domain #
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# #
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#----------------------------------------------------------------------------#
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def convolution_fft(a, b, dps=None):
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"""
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Performs linear convolution using Fast Fourier Transform.
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Parameters
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==========
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a, b : iterables
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The sequences for which convolution is performed.
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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 S, I
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>>> from sympy.discrete.convolutions import convolution_fft
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>>> convolution_fft([2, 3], [4, 5])
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[8, 22, 15]
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>>> convolution_fft([2, 5], [6, 7, 3])
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[12, 44, 41, 15]
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>>> convolution_fft([1 + 2*I, 4 + 3*I], [S(5)/4, 6])
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[5/4 + 5*I/2, 11 + 63*I/4, 24 + 18*I]
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Convolution_theorem
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.. [2] https://en.wikipedia.org/wiki/Discrete_Fourier_transform_(general%29
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"""
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a, b = a[:], b[:]
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n = m = len(a) + len(b) - 1 # convolution size
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if n > 0 and n&(n - 1): # not a power of 2
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n = 2**n.bit_length()
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# padding with zeros
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a += [S.Zero]*(n - len(a))
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b += [S.Zero]*(n - len(b))
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a, b = fft(a, dps), fft(b, dps)
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a = [expand_mul(x*y) for x, y in zip(a, b)]
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a = ifft(a, dps)[:m]
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return a
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#----------------------------------------------------------------------------#
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# #
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# Convolution for GF(p) #
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# #
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#----------------------------------------------------------------------------#
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def convolution_ntt(a, b, prime):
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"""
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Performs linear convolution using Number Theoretic Transform.
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Parameters
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==========
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a, b : iterables
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The sequences for which convolution is performed.
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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.discrete.convolutions import convolution_ntt
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>>> convolution_ntt([2, 3], [4, 5], prime=19*2**10 + 1)
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[8, 22, 15]
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>>> convolution_ntt([2, 5], [6, 7, 3], prime=19*2**10 + 1)
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[12, 44, 41, 15]
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>>> convolution_ntt([333, 555], [222, 666], prime=19*2**10 + 1)
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[15555, 14219, 19404]
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Convolution_theorem
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.. [2] https://en.wikipedia.org/wiki/Discrete_Fourier_transform_(general%29
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"""
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a, b, p = a[:], b[:], as_int(prime)
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n = m = len(a) + len(b) - 1 # convolution size
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if n > 0 and n&(n - 1): # not a power of 2
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n = 2**n.bit_length()
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# padding with zeros
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a += [0]*(n - len(a))
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b += [0]*(n - len(b))
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a, b = ntt(a, p), ntt(b, p)
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a = [x*y % p for x, y in zip(a, b)]
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a = intt(a, p)[:m]
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return a
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#----------------------------------------------------------------------------#
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# #
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# Convolution for 2**n-group #
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# #
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#----------------------------------------------------------------------------#
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def convolution_fwht(a, b):
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"""
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Performs dyadic (*bitwise-XOR*) convolution using Fast Walsh Hadamard
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Transform.
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The convolution 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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a, b : iterables
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The sequences for which convolution is performed.
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Examples
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========
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>>> from sympy import symbols, S, I
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>>> from sympy.discrete.convolutions import convolution_fwht
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>>> u, v, x, y = symbols('u v x y')
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>>> convolution_fwht([u, v], [x, y])
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[u*x + v*y, u*y + v*x]
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>>> convolution_fwht([2, 3], [4, 5])
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[23, 22]
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>>> convolution_fwht([2, 5 + 4*I, 7], [6*I, 7, 3 + 4*I])
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[56 + 68*I, -10 + 30*I, 6 + 50*I, 48 + 32*I]
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>>> convolution_fwht([S(33)/7, S(55)/6, S(7)/4], [S(2)/3, 5])
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[2057/42, 1870/63, 7/6, 35/4]
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References
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==========
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.. [1] https://www.radioeng.cz/fulltexts/2002/02_03_40_42.pdf
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.. [2] https://en.wikipedia.org/wiki/Hadamard_transform
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"""
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if not a or not b:
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return []
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a, b = a[:], b[:]
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n = max(len(a), len(b))
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if n&(n - 1): # not a power of 2
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n = 2**n.bit_length()
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# padding with zeros
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a += [S.Zero]*(n - len(a))
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b += [S.Zero]*(n - len(b))
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a, b = fwht(a), fwht(b)
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a = [expand_mul(x*y) for x, y in zip(a, b)]
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a = ifwht(a)
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return a
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#----------------------------------------------------------------------------#
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# #
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# Subset Convolution #
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# #
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#----------------------------------------------------------------------------#
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def convolution_subset(a, b):
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"""
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Performs Subset Convolution of given sequences.
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The indices of each argument, considered as bit strings, correspond to
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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 based on bitmasks (indices) requires the size of
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sequence to be a power of 2.
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Parameters
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==========
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a, b : iterables
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The sequences for which convolution is performed.
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Examples
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========
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>>> from sympy import symbols, S
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>>> from sympy.discrete.convolutions import convolution_subset
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>>> u, v, x, y, z = symbols('u v x y z')
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>>> convolution_subset([u, v], [x, y])
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[u*x, u*y + v*x]
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>>> convolution_subset([u, v, x], [y, z])
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[u*y, u*z + v*y, x*y, x*z]
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>>> convolution_subset([1, S(2)/3], [3, 4])
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[3, 6]
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>>> convolution_subset([1, 3, S(5)/7], [7])
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[7, 21, 5, 0]
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References
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==========
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.. [1] https://people.csail.mit.edu/rrw/presentations/subset-conv.pdf
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"""
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if not a or not b:
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return []
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if not iterable(a) or not iterable(b):
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raise TypeError("Expected a sequence of coefficients for convolution")
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a = [sympify(arg) for arg in a]
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b = [sympify(arg) for arg in b]
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n = max(len(a), len(b))
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if n&(n - 1): # not a power of 2
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n = 2**n.bit_length()
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# padding with zeros
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a += [S.Zero]*(n - len(a))
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b += [S.Zero]*(n - len(b))
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c = [S.Zero]*n
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for mask in range(n):
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smask = mask
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while smask > 0:
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c[mask] += expand_mul(a[smask] * b[mask^smask])
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smask = (smask - 1)&mask
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c[mask] += expand_mul(a[smask] * b[mask^smask])
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return c
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#----------------------------------------------------------------------------#
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# #
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# Covering Product #
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# #
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#----------------------------------------------------------------------------#
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def covering_product(a, b):
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"""
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Returns the covering product of given sequences.
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The indices of each argument, considered as bit strings, correspond to
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subsets of a finite set.
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The covering product of given sequences is a sequence which contains
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the sum of products of the elements of the given sequences grouped by
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the *bitwise-OR* of the corresponding indices.
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The sequence is automatically padded to the right with zeros, as the
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definition of subset based on bitmasks (indices) requires the size of
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sequence to be a power of 2.
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Parameters
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==========
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a, b : iterables
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The sequences for which covering product is to be obtained.
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Examples
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========
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>>> from sympy import symbols, S, I, covering_product
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>>> u, v, x, y, z = symbols('u v x y z')
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>>> covering_product([u, v], [x, y])
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[u*x, u*y + v*x + v*y]
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>>> covering_product([u, v, x], [y, z])
|
||
|
|
[u*y, u*z + v*y + v*z, x*y, x*z]
|
||
|
|
|
||
|
|
>>> covering_product([1, S(2)/3], [3, 4 + 5*I])
|
||
|
|
[3, 26/3 + 25*I/3]
|
||
|
|
>>> covering_product([1, 3, S(5)/7], [7, 8])
|
||
|
|
[7, 53, 5, 40/7]
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
|
||
|
|
.. [1] https://people.csail.mit.edu/rrw/presentations/subset-conv.pdf
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
if not a or not b:
|
||
|
|
return []
|
||
|
|
|
||
|
|
a, b = a[:], b[:]
|
||
|
|
n = max(len(a), len(b))
|
||
|
|
|
||
|
|
if n&(n - 1): # not a power of 2
|
||
|
|
n = 2**n.bit_length()
|
||
|
|
|
||
|
|
# padding with zeros
|
||
|
|
a += [S.Zero]*(n - len(a))
|
||
|
|
b += [S.Zero]*(n - len(b))
|
||
|
|
|
||
|
|
a, b = mobius_transform(a), mobius_transform(b)
|
||
|
|
a = [expand_mul(x*y) for x, y in zip(a, b)]
|
||
|
|
a = inverse_mobius_transform(a)
|
||
|
|
|
||
|
|
return a
|
||
|
|
|
||
|
|
|
||
|
|
#----------------------------------------------------------------------------#
|
||
|
|
# #
|
||
|
|
# Intersecting Product #
|
||
|
|
# #
|
||
|
|
#----------------------------------------------------------------------------#
|
||
|
|
|
||
|
|
def intersecting_product(a, b):
|
||
|
|
"""
|
||
|
|
Returns the intersecting product of given sequences.
|
||
|
|
|
||
|
|
The indices of each argument, considered as bit strings, correspond to
|
||
|
|
subsets of a finite set.
|
||
|
|
|
||
|
|
The intersecting product of given sequences is the sequence which
|
||
|
|
contains the sum of products of the elements of the given sequences
|
||
|
|
grouped by the *bitwise-AND* of the corresponding indices.
|
||
|
|
|
||
|
|
The sequence is automatically padded to the right with zeros, as the
|
||
|
|
definition of subset based on bitmasks (indices) requires the size of
|
||
|
|
sequence to be a power of 2.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
a, b : iterables
|
||
|
|
The sequences for which intersecting product is to be obtained.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import symbols, S, I, intersecting_product
|
||
|
|
>>> u, v, x, y, z = symbols('u v x y z')
|
||
|
|
|
||
|
|
>>> intersecting_product([u, v], [x, y])
|
||
|
|
[u*x + u*y + v*x, v*y]
|
||
|
|
>>> intersecting_product([u, v, x], [y, z])
|
||
|
|
[u*y + u*z + v*y + x*y + x*z, v*z, 0, 0]
|
||
|
|
|
||
|
|
>>> intersecting_product([1, S(2)/3], [3, 4 + 5*I])
|
||
|
|
[9 + 5*I, 8/3 + 10*I/3]
|
||
|
|
>>> intersecting_product([1, 3, S(5)/7], [7, 8])
|
||
|
|
[327/7, 24, 0, 0]
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
|
||
|
|
.. [1] https://people.csail.mit.edu/rrw/presentations/subset-conv.pdf
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
if not a or not b:
|
||
|
|
return []
|
||
|
|
|
||
|
|
a, b = a[:], b[:]
|
||
|
|
n = max(len(a), len(b))
|
||
|
|
|
||
|
|
if n&(n - 1): # not a power of 2
|
||
|
|
n = 2**n.bit_length()
|
||
|
|
|
||
|
|
# padding with zeros
|
||
|
|
a += [S.Zero]*(n - len(a))
|
||
|
|
b += [S.Zero]*(n - len(b))
|
||
|
|
|
||
|
|
a, b = mobius_transform(a, subset=False), mobius_transform(b, subset=False)
|
||
|
|
a = [expand_mul(x*y) for x, y in zip(a, b)]
|
||
|
|
a = inverse_mobius_transform(a, subset=False)
|
||
|
|
|
||
|
|
return a
|