# N-dim array¶

N-dim array module for SymPy.

Four classes are provided to handle N-dim arrays, given by the combinations dense/sparse (i.e. whether to store all elements or only the non-zero ones in memory) and mutable/immutable (immutable classes are SymPy objects, but cannot change after they have been created).

## Examples¶

The following examples show the usage of Array. This is an abbreviation for ImmutableDenseNDimArray, that is an immutable and dense N-dim array, the other classes are analogous. For mutable classes it is also possible to change element values after the object has been constructed.

Array construction can detect the shape of nested lists and tuples:

>>> from sympy import Array
>>> a1 = Array([[1, 2], [3, 4], [5, 6]])
>>> a1
[[1, 2], [3, 4], [5, 6]]
>>> a1.shape
(3, 2)
>>> a1.rank()
2
>>> from sympy.abc import x, y, z
>>> a2 = Array([[[x, y], [z, x*z]], [[1, x*y], [1/x, x/y]]])
>>> a2
[[[x, y], [z, x*z]], [[1, x*y], [1/x, x/y]]]
>>> a2.shape
(2, 2, 2)
>>> a2.rank()
3


Otherwise one could pass a 1-dim array followed by a shape tuple:

>>> m1 = Array(range(12), (3, 4))
>>> m1
[[0, 1, 2, 3], [4, 5, 6, 7], [8, 9, 10, 11]]
>>> m2 = Array(range(12), (3, 2, 2))
>>> m2
[[[0, 1], [2, 3]], [[4, 5], [6, 7]], [[8, 9], [10, 11]]]
>>> m2[1,1,1]
7
>>> m2.reshape(4, 3)
[[0, 1, 2], [3, 4, 5], [6, 7, 8], [9, 10, 11]]


Slice support:

>>> m2[:, 1, 1]
[3, 7, 11]


Elementwise derivative:

>>> from sympy.abc import x, y, z
>>> m3 = Array([x**3, x*y, z])
>>> m3.diff(x)
[3*x**2, y, 0]
>>> m3.diff(z)
[0, 0, 1]


Multiplication with other SymPy expressions is applied elementwisely:

>>> (1+x)*m3
[x**3*(x + 1), x*y*(x + 1), z*(x + 1)]


To apply a function to each element of the N-dim array, use applyfunc:

>>> m3.applyfunc(lambda x: x/2)
[x**3/2, x*y/2, z/2]


N-dim arrays can be converted to nested lists by the tolist() method:

>>> m2.tolist()
[[[0, 1], [2, 3]], [[4, 5], [6, 7]], [[8, 9], [10, 11]]]
>>> isinstance(m2.tolist(), list)
True


If the rank is 2, it is possible to convert them to matrices with tomatrix():

>>> m1.tomatrix()
Matrix([
[0, 1,  2,  3],
[4, 5,  6,  7],
[8, 9, 10, 11]])


### Products and contractions¶

Tensor product between arrays $$A_{i_1,\ldots,i_n}$$ and $$B_{j_1,\ldots,j_m}$$ creates the combined array $$P = A \otimes B$$ defined as

$$P_{i_1,\ldots,i_n,j_1,\ldots,j_m} := A_{i_1,\ldots,i_n}\cdot B_{j_1,\ldots,j_m}.$$

It is available through tensorproduct(...):

>>> from sympy import Array, tensorproduct
>>> from sympy.abc import x,y,z,t
>>> A = Array([x, y, z, t])
>>> B = Array([1, 2, 3, 4])
>>> tensorproduct(A, B)
[[x, 2*x, 3*x, 4*x], [y, 2*y, 3*y, 4*y], [z, 2*z, 3*z, 4*z], [t, 2*t, 3*t, 4*t]]


Tensor product between a rank-1 array and a matrix creates a rank-3 array:

>>> from sympy import eye
>>> p1 = tensorproduct(A, eye(4))
>>> p1
[[[x, 0, 0, 0], [0, x, 0, 0], [0, 0, x, 0], [0, 0, 0, x]], [[y, 0, 0, 0], [0, y, 0, 0], [0, 0, y, 0], [0, 0, 0, y]], [[z, 0, 0, 0], [0, z, 0, 0], [0, 0, z, 0], [0, 0, 0, z]], [[t, 0, 0, 0], [0, t, 0, 0], [0, 0, t, 0], [0, 0, 0, t]]]


Now, to get back $$A_0 \otimes \mathbf{1}$$ one can access $$p_{0,m,n}$$ by slicing:

>>> p1[0,:,:]
[[x, 0, 0, 0], [0, x, 0, 0], [0, 0, x, 0], [0, 0, 0, x]]


Tensor contraction sums over the specified axes, for example contracting positions $$a$$ and $$b$$ means

$$A_{i_1,\ldots,i_a,\ldots,i_b,\ldots,i_n} \implies \sum_k A_{i_1,\ldots,k,\ldots,k,\ldots,i_n}$$

Remember that Python indexing is zero starting, to contract the a-th and b-th axes it is therefore necessary to specify $$a-1$$ and $$b-1$$

>>> from sympy import tensorcontraction
>>> C = Array([[x, y], [z, t]])


The matrix trace is equivalent to the contraction of a rank-2 array:

$$A_{m,n} \implies \sum_k A_{k,k}$$

>>> tensorcontraction(C, (0, 1))
t + x


Matrix product is equivalent to a tensor product of two rank-2 arrays, followed by a contraction of the 2nd and 3rd axes (in Python indexing axes number 1, 2).

$$A_{m,n}\cdot B_{i,j} \implies \sum_k A_{m, k}\cdot B_{k, j}$$

>>> D = Array([[2, 1], [0, -1]])
>>> tensorcontraction(tensorproduct(C, D), (1, 2))
[[2*x, x - y], [2*z, -t + z]]


One may verify that the matrix product is equivalent:

>>> from sympy import Matrix
>>> Matrix([[x, y], [z, t]])*Matrix([[2, 1], [0, -1]])
Matrix([
[2*x,  x - y],
[2*z, -t + z]])


or equivalently

>>> C.tomatrix()*D.tomatrix()
Matrix([
[2*x,  x - y],
[2*z, -t + z]])


### Diagonal operator¶

The tensordiagonal function acts in a similar manner as tensorcontraction, but the joined indices are not summed over, for example diagonalizing positions $$a$$ and $$b$$ means

$$A_{i_1,\ldots,i_a,\ldots,i_b,\ldots,i_n} \implies A_{i_1,\ldots,k,\ldots,k,\ldots,i_n} \implies \tilde{A}_{i_1,\ldots,i_{a-1},i_{a+1},\ldots,i_{b-1},i_{b+1},\ldots,i_n,k}$$

where $$\tilde{A}$$ is the array equivalent to the diagonal of $$A$$ at positions $$a$$ and $$b$$ moved to the last index slot.

Compare the difference between contraction and diagonal operators:

>>> from sympy import tensordiagonal
>>> from sympy.abc import a, b, c, d
>>> m = Matrix([[a, b], [c, d]])
>>> tensorcontraction(m, [0, 1])
a + d
>>> tensordiagonal(m, [0, 1])
[a, d]


In short, no summation occurs with tensordiagonal.

### Derivatives by array¶

The usual derivative operation may be extended to support derivation with respect to arrays, provided that all elements in the that array are symbols or expressions suitable for derivations.

The definition of a derivative by an array is as follows: given the array $$A_{i_1, \ldots, i_N}$$ and the array $$X_{j_1, \ldots, j_M}$$ the derivative of arrays will return a new array $$B$$ defined by

$$B_{j_1,\ldots,j_M,i_1,\ldots,i_N} := \frac{\partial A_{i_1,\ldots,i_N}}{\partial X_{j_1,\ldots,j_M}}$$

The function derive_by_array performs such an operation:

>>> from sympy import derive_by_array
>>> from sympy.abc import x, y, z, t
>>> from sympy import sin, exp


With scalars, it behaves exactly as the ordinary derivative:

>>> derive_by_array(sin(x*y), x)
y*cos(x*y)


Scalar derived by an array basis:

>>> derive_by_array(sin(x*y), [x, y, z])
[y*cos(x*y), x*cos(x*y), 0]


Deriving array by an array basis: $$B^{nm} := \frac{\partial A^m}{\partial x^n}$$

>>> basis = [x, y, z]
>>> ax = derive_by_array([exp(x), sin(y*z), t], basis)
>>> ax
[[exp(x), 0, 0], [0, z*cos(y*z), 0], [0, y*cos(y*z), 0]]


Contraction of the resulting array: $$\sum_m \frac{\partial A^m}{\partial x^m}$$

>>> tensorcontraction(ax, (0, 1))
z*cos(y*z) + exp(x)


## Classes¶

class sympy.tensor.array.ImmutableDenseNDimArray(iterable, shape=None, **kwargs)[source]
class sympy.tensor.array.ImmutableSparseNDimArray(iterable=None, shape=None, **kwargs)[source]
class sympy.tensor.array.MutableDenseNDimArray(iterable=None, shape=None, **kwargs)[source]
class sympy.tensor.array.MutableSparseNDimArray(iterable=None, shape=None, **kwargs)[source]

## Functions¶

sympy.tensor.array.derive_by_array(expr, dx)[source]

Derivative by arrays. Supports both arrays and scalars.

Explanation

Given the array $$A_{i_1, \ldots, i_N}$$ and the array $$X_{j_1, \ldots, j_M}$$ this function will return a new array $$B$$ defined by

$$B_{j_1,\ldots,j_M,i_1,\ldots,i_N} := \frac{\partial A_{i_1,\ldots,i_N}}{\partial X_{j_1,\ldots,j_M}}$$

Examples

>>> from sympy import derive_by_array
>>> from sympy.abc import x, y, z, t
>>> from sympy import cos
>>> derive_by_array(cos(x*t), x)
-t*sin(t*x)
>>> derive_by_array(cos(x*t), [x, y, z, t])
[-t*sin(t*x), 0, 0, -x*sin(t*x)]
>>> derive_by_array([x, y**2*z], [[x, y], [z, t]])
[[[1, 0], [0, 2*y*z]], [[0, y**2], [0, 0]]]

sympy.tensor.array.permutedims(expr, perm)[source]

Permutes the indices of an array.

Parameter specifies the permutation of the indices.

Examples

>>> from sympy.abc import x, y, z, t
>>> from sympy import sin
>>> from sympy import Array, permutedims
>>> a = Array([[x, y, z], [t, sin(x), 0]])
>>> a
[[x, y, z], [t, sin(x), 0]]
>>> permutedims(a, (1, 0))
[[x, t], [y, sin(x)], [z, 0]]


If the array is of second order, transpose can be used:

>>> from sympy import transpose
>>> transpose(a)
[[x, t], [y, sin(x)], [z, 0]]


Examples on higher dimensions:

>>> b = Array([[[1, 2], [3, 4]], [[5, 6], [7, 8]]])
>>> permutedims(b, (2, 1, 0))
[[[1, 5], [3, 7]], [[2, 6], [4, 8]]]
>>> permutedims(b, (1, 2, 0))
[[[1, 5], [2, 6]], [[3, 7], [4, 8]]]


Permutation objects are also allowed:

>>> from sympy.combinatorics import Permutation
>>> permutedims(b, Permutation([1, 2, 0]))
[[[1, 5], [2, 6]], [[3, 7], [4, 8]]]

sympy.tensor.array.tensorcontraction(array, *contraction_axes)[source]

Contraction of an array-like object on the specified axes.

Examples

>>> from sympy import Array, tensorcontraction
>>> from sympy import Matrix, eye
>>> tensorcontraction(eye(3), (0, 1))
3
>>> A = Array(range(18), (3, 2, 3))
>>> A
[[[0, 1, 2], [3, 4, 5]], [[6, 7, 8], [9, 10, 11]], [[12, 13, 14], [15, 16, 17]]]
>>> tensorcontraction(A, (0, 2))
[21, 30]


Matrix multiplication may be emulated with a proper combination of tensorcontraction and tensorproduct

>>> from sympy import tensorproduct
>>> from sympy.abc import a,b,c,d,e,f,g,h
>>> m1 = Matrix([[a, b], [c, d]])
>>> m2 = Matrix([[e, f], [g, h]])
>>> p = tensorproduct(m1, m2)
>>> p
[[[[a*e, a*f], [a*g, a*h]], [[b*e, b*f], [b*g, b*h]]], [[[c*e, c*f], [c*g, c*h]], [[d*e, d*f], [d*g, d*h]]]]
>>> tensorcontraction(p, (1, 2))
[[a*e + b*g, a*f + b*h], [c*e + d*g, c*f + d*h]]
>>> m1*m2
Matrix([
[a*e + b*g, a*f + b*h],
[c*e + d*g, c*f + d*h]])

sympy.tensor.array.tensorproduct(*args)[source]

Tensor product among scalars or array-like objects.

Examples

>>> from sympy.tensor.array import tensorproduct, Array
>>> from sympy.abc import x, y, z, t
>>> A = Array([[1, 2], [3, 4]])
>>> B = Array([x, y])
>>> tensorproduct(A, B)
[[[x, y], [2*x, 2*y]], [[3*x, 3*y], [4*x, 4*y]]]
>>> tensorproduct(A, x)
[[x, 2*x], [3*x, 4*x]]
>>> tensorproduct(A, B, B)
[[[[x**2, x*y], [x*y, y**2]], [[2*x**2, 2*x*y], [2*x*y, 2*y**2]]], [[[3*x**2, 3*x*y], [3*x*y, 3*y**2]], [[4*x**2, 4*x*y], [4*x*y, 4*y**2]]]]


Applying this function on two matrices will result in a rank 4 array.

>>> from sympy import Matrix, eye
>>> m = Matrix([[x, y], [z, t]])
>>> p = tensorproduct(eye(3), m)
>>> p
[[[[x, y], [z, t]], [[0, 0], [0, 0]], [[0, 0], [0, 0]]], [[[0, 0], [0, 0]], [[x, y], [z, t]], [[0, 0], [0, 0]]], [[[0, 0], [0, 0]], [[0, 0], [0, 0]], [[x, y], [z, t]]]]

sympy.tensor.array.tensordiagonal(array, *diagonal_axes)[source]

Diagonalization of an array-like object on the specified axes.

This is equivalent to multiplying the expression by Kronecker deltas uniting the axes.

The diagonal indices are put at the end of the axes.

Examples

tensordiagonal acting on a 2-dimensional array by axes 0 and 1 is equivalent to the diagonal of the matrix:

>>> from sympy import Array, tensordiagonal
>>> from sympy import Matrix, eye
>>> tensordiagonal(eye(3), (0, 1))
[1, 1, 1]

>>> from sympy.abc import a,b,c,d
>>> m1 = Matrix([[a, b], [c, d]])
>>> tensordiagonal(m1, [0, 1])
[a, d]


In case of higher dimensional arrays, the diagonalized out dimensions are appended removed and appended as a single dimension at the end:

>>> A = Array(range(18), (3, 2, 3))
>>> A
[[[0, 1, 2], [3, 4, 5]], [[6, 7, 8], [9, 10, 11]], [[12, 13, 14], [15, 16, 17]]]
>>> tensordiagonal(A, (0, 2))
[[0, 7, 14], [3, 10, 17]]
>>> from sympy import permutedims
>>> tensordiagonal(A, (0, 2)) == permutedims(Array([A[0, :, 0], A[1, :, 1], A[2, :, 2]]), [1, 0])
True