Discrete¶

The discrete module in SymPy implements methods to compute discrete transforms and convolutions of finite sequences.

This module contains functions which operate on discrete sequences.

Transforms - fft, ifft, ntt, intt, fwht, ifwht,: mobius_transform, inverse_mobius_transform
Convolutions - convolution, convolution_fft, convolution_ntt,: convolution_fwht, convolution_subset, covering_product, intersecting_product

Since the discrete transforms can be used to reduce the computational complexity of the discrete convolutions, the convolutions module makes use of the transforms module for efficient computation (notable for long input sequences).

Transforms¶

This section lists the methods which implement the basic transforms for discrete sequences.

Fast Fourier Transform¶

sympy.discrete.transforms.fft(seq, dps=None)[source]¶

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

R161: https://en.wikipedia.org/wiki/Cooley%E2%80%93Tukey_FFT_algorithm
R162: http://mathworld.wolfram.com/FastFourierTransform.html

sympy.discrete.transforms.ifft(seq, dps=None)[source]¶

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

R163: https://en.wikipedia.org/wiki/Cooley%E2%80%93Tukey_FFT_algorithm
R164: http://mathworld.wolfram.com/FastFourierTransform.html

Number Theoretic Transform¶

sympy.discrete.transforms.ntt(seq, prime)[source]¶

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

R165: http://www.apfloat.org/ntt.html
R166: http://mathworld.wolfram.com/NumberTheoreticTransform.html
R167: https://en.wikipedia.org/wiki/Discrete_Fourier_transform_(general%29

sympy.discrete.transforms.intt(seq, prime)[source]¶

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

R168: http://www.apfloat.org/ntt.html
R169: http://mathworld.wolfram.com/NumberTheoreticTransform.html
R170: https://en.wikipedia.org/wiki/Discrete_Fourier_transform_(general%29

Fast Walsh Hadamard Transform¶

sympy.discrete.transforms.fwht(seq)[source]¶

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

R171: https://en.wikipedia.org/wiki/Hadamard_transform
R172: https://en.wikipedia.org/wiki/Fast_Walsh%E2%80%93Hadamard_transform

sympy.discrete.transforms.ifwht(seq)[source]¶

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

R173: https://en.wikipedia.org/wiki/Hadamard_transform
R174: https://en.wikipedia.org/wiki/Fast_Walsh%E2%80%93Hadamard_transform

Möbius Transform¶

sympy.discrete.transforms.mobius_transform(seq, subset=True)[source]¶

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

R175: https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula
R176: https://people.csail.mit.edu/rrw/presentations/subset-conv.pdf
R177: https://arxiv.org/pdf/1211.0189.pdf

sympy.discrete.transforms.inverse_mobius_transform(seq, subset=True)[source]¶

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

R178: https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula
R179: https://people.csail.mit.edu/rrw/presentations/subset-conv.pdf
R180: https://arxiv.org/pdf/1211.0189.pdf

Convolutions¶

This section lists the methods which implement the basic convolutions for discrete sequences.

Convolution¶

This is a general method for calculating the convolution of discrete sequences, which internally calls one of the methods convolution_fft, convolution_ntt, convolution_fwht, or convolution_subset.

sympy.discrete.convolutions.convolution(a, b, cycle=0, dps=None, prime=None, dyadic=None, subset=None)[source]¶

Performs convolution by determining the type of desired convolution using hints.

Exactly one of dps, prime, dyadic, subset arguments should be specified explicitly for identifying the type of convolution, and the argument cycle can be specified optionally.

For the default arguments, linear convolution is performed using FFT.

Parameters

a, b : iterables

The sequences for which convolution is performed.

cycle : Integer

Specifies the length for doing cyclic convolution.

dps : Integer

Specifies the number of decimal digits for precision for performing FFT on the sequence.

prime : Integer

Prime modulus of the form \((m 2^k + 1)\) to be used for performing NTT on the sequence.

dyadic : bool

Identifies the convolution type as dyadic (bitwise-XOR) convolution, which is performed using FWHT.

subset : bool

Identifies the convolution type as subset convolution.

Examples

>>> from sympy import convolution, symbols, S, I
>>> u, v, w, x, y, z = symbols('u v w x y z')

>>> convolution([1 + 2*I, 4 + 3*I], [S(5)/4, 6], dps=3)
[1.25 + 2.5*I, 11.0 + 15.8*I, 24.0 + 18.0*I]
>>> convolution([1, 2, 3], [4, 5, 6], cycle=3)
[31, 31, 28]

>>> convolution([111, 777], [888, 444], prime=19*2**10 + 1)
[1283, 19351, 14219]
>>> convolution([111, 777], [888, 444], prime=19*2**10 + 1, cycle=2)
[15502, 19351]

>>> convolution([u, v], [x, y, z], dyadic=True)
[u*x + v*y, u*y + v*x, u*z, v*z]
>>> convolution([u, v], [x, y, z], dyadic=True, cycle=2)
[u*x + u*z + v*y, u*y + v*x + v*z]

>>> convolution([u, v, w], [x, y, z], subset=True)
[u*x, u*y + v*x, u*z + w*x, v*z + w*y]
>>> convolution([u, v, w], [x, y, z], subset=True, cycle=3)
[u*x + v*z + w*y, u*y + v*x, u*z + w*x]

Convolution using Fast Fourier Transform¶

sympy.discrete.convolutions.convolution_fft(a, b, dps=None)[source]¶

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

R181: https://en.wikipedia.org/wiki/Convolution_theorem
R182: https://en.wikipedia.org/wiki/Discrete_Fourier_transform_(general%29

Convolution using Number Theoretic Transform¶

sympy.discrete.convolutions.convolution_ntt(a, b, prime)[source]¶

Performs linear convolution using Number Theoretic Transform.

Parameters

a, b : iterables

The sequences for which convolution is performed.

prime : Integer

Prime modulus of the form \((m 2^k + 1)\) to be used for performing NTT on the sequence.

Examples

>>> from sympy.discrete.convolutions import convolution_ntt
>>> convolution_ntt([2, 3], [4, 5], prime=19*2**10 + 1)
[8, 22, 15]
>>> convolution_ntt([2, 5], [6, 7, 3], prime=19*2**10 + 1)
[12, 44, 41, 15]
>>> convolution_ntt([333, 555], [222, 666], prime=19*2**10 + 1)
[15555, 14219, 19404]

References

R183: https://en.wikipedia.org/wiki/Convolution_theorem
R184: https://en.wikipedia.org/wiki/Discrete_Fourier_transform_(general%29

Convolution using Fast Walsh Hadamard Transform¶

sympy.discrete.convolutions.convolution_fwht(a, b)[source]¶

Performs dyadic (bitwise-XOR) convolution using Fast Walsh Hadamard Transform.

The convolution 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: a, b : iterables

The sequences for which convolution is performed.

Examples

>>> from sympy import symbols, S, I
>>> from sympy.discrete.convolutions import convolution_fwht

>>> u, v, x, y = symbols('u v x y')
>>> convolution_fwht([u, v], [x, y])
[u*x + v*y, u*y + v*x]

>>> convolution_fwht([2, 3], [4, 5])
[23, 22]
>>> convolution_fwht([2, 5 + 4*I, 7], [6*I, 7, 3 + 4*I])
[56 + 68*I, -10 + 30*I, 6 + 50*I, 48 + 32*I]

>>> convolution_fwht([S(33)/7, S(55)/6, S(7)/4], [S(2)/3, 5])
[2057/42, 1870/63, 7/6, 35/4]

References

R185: https://www.radioeng.cz/fulltexts/2002/02_03_40_42.pdf
R186: https://en.wikipedia.org/wiki/Hadamard_transform

Subset Convolution¶

sympy.discrete.convolutions.convolution_subset(a, b)[source]¶

Performs Subset Convolution of given sequences.

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 based on bitmasks (indices) requires the size of sequence to be a power of 2.

Parameters: a, b : iterables

The sequences for which convolution is performed.

Examples

>>> from sympy import symbols, S
>>> from sympy.discrete.convolutions import convolution_subset
>>> u, v, x, y, z = symbols('u v x y z')

>>> convolution_subset([u, v], [x, y])
[u*x, u*y + v*x]
>>> convolution_subset([u, v, x], [y, z])
[u*y, u*z + v*y, x*y, x*z]

>>> convolution_subset([1, S(2)/3], [3, 4])
[3, 6]
>>> convolution_subset([1, 3, S(5)/7], [7])
[7, 21, 5, 0]

References

R187: https://people.csail.mit.edu/rrw/presentations/subset-conv.pdf

Covering Product¶

sympy.discrete.convolutions.covering_product(a, b)[source]¶

Returns the covering product of given sequences.

The indices of each argument, considered as bit strings, correspond to subsets of a finite set.

The covering product of given sequences is a sequence which contains the sum of products of the elements of the given sequences grouped by the bitwise-OR 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 covering product is to be obtained.

Examples

>>> from sympy import symbols, S, I, covering_product
>>> u, v, x, y, z = symbols('u v x y z')

>>> covering_product([u, v], [x, y])
[u*x, u*y + v*x + v*y]
>>> 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

R188: https://people.csail.mit.edu/rrw/presentations/subset-conv.pdf

Intersecting Product¶

sympy.discrete.convolutions.intersecting_product(a, b)[source]¶

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

R189: https://people.csail.mit.edu/rrw/presentations/subset-conv.pdf

Core

Numerical Evaluation