# 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

R141

https://en.wikipedia.org/wiki/Cooley%E2%80%93Tukey_FFT_algorithm

R142

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

R143

https://en.wikipedia.org/wiki/Cooley%E2%80%93Tukey_FFT_algorithm

R144

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

R145

http://www.apfloat.org/ntt.html

R146

http://mathworld.wolfram.com/NumberTheoreticTransform.html

R147

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

R148

http://www.apfloat.org/ntt.html

R149

http://mathworld.wolfram.com/NumberTheoreticTransform.html

R150

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

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

R151

R152

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

R153

R154

### 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

R155

https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula

R156

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

R157

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

R158

https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula

R159

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

R160

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.

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

R161

https://en.wikipedia.org/wiki/Convolution_theorem

R162

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

R163

https://en.wikipedia.org/wiki/Convolution_theorem

R164

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

### Convolution using Fast Walsh Hadamard Transform¶

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

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

R165

R166

### 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, I
>>> 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, 21, 5, 0]


References

R167

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

R168

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

R169

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