N-dim array

N-dim array module.

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 Diofant 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 diofant.tensor.array import Array
>>> a1 = Array([[1, 2], [3, 4], [5, 6]])
>>> a1
[[1, 2], [3, 4], [5, 6]]
>>> a1.shape
(3, 2)
>>> a1.rank()
2
>>> 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:

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

Multiplication with other Diofant 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]]]

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 diofant.tensor.array import Array, tensorproduct
>>> 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:

>>> 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 diofant.tensor.array 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:

>>> 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]])

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 diofant.tensor.array import Array, tensorcontraction, derive_by_array

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
[[E**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))
E**x + z*cos(y*z)

Classes

class diofant.tensor.array.ImmutableDenseNDimArray[source]
class diofant.tensor.array.ImmutableSparseNDimArray[source]
class diofant.tensor.array.MutableDenseNDimArray[source]
class diofant.tensor.array.MutableSparseNDimArray[source]

Functions

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

Derivative by arrays. Supports both arrays and scalars.

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

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

Permutes the indices of an array.

Parameter specifies the permutation of the indices.

Examples

>>> from diofant.tensor.array import Array
>>> 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:

>>> 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:

>>> permutedims(b, Permutation([1, 2, 0]))
[[[1, 5], [2, 6]], [[3, 7], [4, 8]]]
diofant.tensor.array.tensorcontraction(array, *contraction_axes)[source]

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

Examples

>>> from diofant.tensor.array import Array
>>> 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 diofant.abc import 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]])
diofant.tensor.array.tensorproduct(*args)[source]

Tensor product among scalars or array-like objects.

Examples

>>> from diofant.tensor.array import Array
>>> 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.

>>> 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]]]]