def convolution_fft(a, b, dps=None):
"""
Performs linear convolution using Fast Fourier Transform.
Parameters
==========
a, b : iterables
The sequences for which convolution is performed.
dps : Integer
Specifies the number of decimal digits for precision.
Examples
========
>>> from sympy import S, I
>>> from sympy.discrete.convolution import convolution_fft
>>> convolution_fft([2, 3], [4, 5])
[8, 22, 15]
>>> convolution_fft([2, 5], [6, 7, 3])
[12, 44, 41, 15]
>>> convolution_fft([1 + 2*I, 4 + 3*I], [S(5)/4, 6])
[5/4 + 5*I/2, 11 + 63*I/4, 24 + 18*I]
References
==========
.. [1] https://en.wikipedia.org/wiki/Convolution_theorem
.. [1] https://en.wikipedia.org/wiki/Discrete_Fourier_transform_(general)
"""
a, b = a[:], b[:]
n = m = len(a) + len(b) - 1 # convolution size
if n > 0 and 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 = fft(a, dps), fft(b, dps)
for i in range(n):
a[i] = expand_mul(a[i] * b[i])
a = ifft(a, dps)[:m]
return a
def convolution_fft(a, b, dps=None):
"""
Performs linear convolution using Fast Fourier Transform.
Parameters
==========
a, b : iterables
The sequences for which convolution is performed.
dps : Integer
Specifies the number of decimal digits for precision.
Examples
========
>>> from sympy import S, I
>>> from sympy.discrete.convolutions import convolution_fft
>>> convolution_fft([2, 3], [4, 5])
[8, 22, 15]
>>> convolution_fft([2, 5], [6, 7, 3])
[12, 44, 41, 15]
>>> convolution_fft([1 + 2*I, 4 + 3*I], [S(5)/4, 6])
[5/4 + 5*I/2, 11 + 63*I/4, 24 + 18*I]
References
==========
.. [1] https://en.wikipedia.org/wiki/Convolution_theorem
.. [2] https://en.wikipedia.org/wiki/Discrete_Fourier_transform_(general%29
"""
a, b = a[:], b[:]
n = m = len(a) + len(b) - 1 # convolution size
if n > 0 and 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 = fft(a, dps), fft(b, dps)
a = [expand_mul(x*y) for x, y in zip(a, b)]
a = ifft(a, dps)[:m]
return a