numpy.einsum

numpy.einsum(subscripts, *operands, out=None, dtype=None, order='K', casting='safe', optimize=False)

Evaluates the Einstein summation convention on the operands.

Using the Einstein summation convention, many common multi-dimensional array operations can be represented in a simple fashion. This function provides a way to compute such summations. The best way to understand this function is to try the examples below, which show how many common NumPy functions can be implemented as calls to einsum.

Parameters:

subscripts : str

Specifies the subscripts for summation.

operands : list of array_like

These are the arrays for the operation.

out : {ndarray, None}, optional

If provided, the calculation is done into this array.

dtype : {data-type, None}, optional

If provided, forces the calculation to use the data type specified. Note that you may have to also give a more liberal casting parameter to allow the conversions. Default is None.

order : {‘C’, ‘F’, ‘A’, ‘K’}, optional

Controls the memory layout of the output. ‘C’ means it should be C contiguous. ‘F’ means it should be Fortran contiguous, ‘A’ means it should be ‘F’ if the inputs are all ‘F’, ‘C’ otherwise. ‘K’ means it should be as close to the layout as the inputs as is possible, including arbitrarily permuted axes. Default is ‘K’.

casting : {‘no’, ‘equiv’, ‘safe’, ‘same_kind’, ‘unsafe’}, optional

Controls what kind of data casting may occur. Setting this to ‘unsafe’ is not recommended, as it can adversely affect accumulations.

• ‘no’ means the data types should not be cast at all.
• ‘equiv’ means only byte-order changes are allowed.
• ‘safe’ means only casts which can preserve values are allowed.
• ‘same_kind’ means only safe casts or casts within a kind, like float64 to float32, are allowed.
• ‘unsafe’ means any data conversions may be done.

Default is ‘safe’.

optimize : {False, True, ‘greedy’, ‘optimal’}, optional

Controls if intermediate optimization should occur. No optimization will occur if False and True will default to the ‘greedy’ algorithm. Also accepts an explicit contraction list from the np.einsum_path function. See np.einsum_path for more details. Defaults to False.

Returns:

output : ndarray

The calculation based on the Einstein summation convention.

See also

einsum_path, dot, inner, outer, tensordot, linalg.multi_dot

Notes

New in version 1.6.0.

The subscripts string is a comma-separated list of subscript labels, where each label refers to a dimension of the corresponding operand. Repeated subscripts labels in one operand take the diagonal. For example, np.einsum('ii', a) is equivalent to np.trace(a).

Whenever a label is repeated, it is summed, so np.einsum('i,i', a, b) is equivalent to np.inner(a,b). If a label appears only once, it is not summed, so np.einsum('i', a) produces a view of a with no changes.

The order of labels in the output is by default alphabetical. This means that np.einsum('ij', a) doesn’t affect a 2D array, while np.einsum('ji', a) takes its transpose.

The output can be controlled by specifying output subscript labels as well. This specifies the label order, and allows summing to be disallowed or forced when desired. The call np.einsum('i->', a) is like np.sum(a, axis=-1), and np.einsum('ii->i', a) is like np.diag(a). The difference is that einsum does not allow broadcasting by default.

To enable and control broadcasting, use an ellipsis. Default NumPy-style broadcasting is done by adding an ellipsis to the left of each term, like np.einsum('...ii->...i', a). To take the trace along the first and last axes, you can do np.einsum('i...i', a), or to do a matrix-matrix product with the left-most indices instead of rightmost, you can do np.einsum('ij...,jk...->ik...', a, b).

When there is only one operand, no axes are summed, and no output parameter is provided, a view into the operand is returned instead of a new array. Thus, taking the diagonal as np.einsum('ii->i', a) produces a view.

An alternative way to provide the subscripts and operands is as einsum(op0, sublist0, op1, sublist1, ..., [sublistout]). The examples below have corresponding einsum calls with the two parameter methods.

New in version 1.10.0.

Views returned from einsum are now writeable whenever the input array is writeable. For example, np.einsum('ijk...->kji...', a) will now have the same effect as np.swapaxes(a, 0, 2) and np.einsum('ii->i', a) will return a writeable view of the diagonal of a 2D array.

New in version 1.12.0.

Added the optimize argument which will optimize the contraction order of an einsum expression. For a contraction with three or more operands this can greatly increase the computational efficiency at the cost of a larger memory footprint during computation.

See np.einsum_path for more details.

Examples

>>> a = np.arange(25).reshape(5,5)
>>> b = np.arange(5)
>>> c = np.arange(6).reshape(2,3)

>>> np.einsum('ii', a)
60
>>> np.einsum(a, [0,0])
60
>>> np.trace(a)
60

>>> np.einsum('ii->i', a)
array([ 0,  6, 12, 18, 24])
>>> np.einsum(a, [0,0], [0])
array([ 0,  6, 12, 18, 24])
>>> np.diag(a)
array([ 0,  6, 12, 18, 24])

>>> np.einsum('ij,j', a, b)
array([ 30,  80, 130, 180, 230])
>>> np.einsum(a, [0,1], b, [1])
array([ 30,  80, 130, 180, 230])
>>> np.dot(a, b)
array([ 30,  80, 130, 180, 230])
>>> np.einsum('...j,j', a, b)
array([ 30,  80, 130, 180, 230])

>>> np.einsum('ji', c)
array([[0, 3],
       [1, 4],
       [2, 5]])
>>> np.einsum(c, [1,0])
array([[0, 3],
       [1, 4],
       [2, 5]])
>>> c.T
array([[0, 3],
       [1, 4],
       [2, 5]])

>>> np.einsum('..., ...', 3, c)
array([[ 0,  3,  6],
       [ 9, 12, 15]])
>>> np.einsum(',ij', 3, C)
array([[ 0,  3,  6],
       [ 9, 12, 15]])
>>> np.einsum(3, [Ellipsis], c, [Ellipsis])
array([[ 0,  3,  6],
       [ 9, 12, 15]])
>>> np.multiply(3, c)
array([[ 0,  3,  6],
       [ 9, 12, 15]])

>>> np.einsum('i,i', b, b)
30
>>> np.einsum(b, [0], b, [0])
30
>>> np.inner(b,b)
30

>>> np.einsum('i,j', np.arange(2)+1, b)
array([[0, 1, 2, 3, 4],
       [0, 2, 4, 6, 8]])
>>> np.einsum(np.arange(2)+1, [0], b, [1])
array([[0, 1, 2, 3, 4],
       [0, 2, 4, 6, 8]])
>>> np.outer(np.arange(2)+1, b)
array([[0, 1, 2, 3, 4],
       [0, 2, 4, 6, 8]])

>>> np.einsum('i...->...', a)
array([50, 55, 60, 65, 70])
>>> np.einsum(a, [0,Ellipsis], [Ellipsis])
array([50, 55, 60, 65, 70])
>>> np.sum(a, axis=0)
array([50, 55, 60, 65, 70])

>>> a = np.arange(60.).reshape(3,4,5)
>>> b = np.arange(24.).reshape(4,3,2)
>>> np.einsum('ijk,jil->kl', a, b)
array([[ 4400.,  4730.],
       [ 4532.,  4874.],
       [ 4664.,  5018.],
       [ 4796.,  5162.],
       [ 4928.,  5306.]])
>>> np.einsum(a, [0,1,2], b, [1,0,3], [2,3])
array([[ 4400.,  4730.],
       [ 4532.,  4874.],
       [ 4664.,  5018.],
       [ 4796.,  5162.],
       [ 4928.,  5306.]])
>>> np.tensordot(a,b, axes=([1,0],[0,1]))
array([[ 4400.,  4730.],
       [ 4532.,  4874.],
       [ 4664.,  5018.],
       [ 4796.,  5162.],
       [ 4928.,  5306.]])

>>> a = np.arange(6).reshape((3,2))
>>> b = np.arange(12).reshape((4,3))
>>> np.einsum('ki,jk->ij', a, b)
array([[10, 28, 46, 64],
       [13, 40, 67, 94]])
>>> np.einsum('ki,...k->i...', a, b)
array([[10, 28, 46, 64],
       [13, 40, 67, 94]])
>>> np.einsum('k...,jk', a, b)
array([[10, 28, 46, 64],
       [13, 40, 67, 94]])

>>> # since version 1.10.0
>>> a = np.zeros((3, 3))
>>> np.einsum('ii->i', a)[:] = 1
>>> a
array([[ 1.,  0.,  0.],
       [ 0.,  1.,  0.],
       [ 0.,  0.,  1.]])