def einsum(*operands, **kwargs):
"""
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. Default is False.
Returns
-------
output : ndarray
The calculation based on the Einstein summation convention.
See Also
--------
einsum_path, dot, inner, outer, tensordot, linalg.multi_dot
Notes
-----
.. versionadded:: 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.
.. versionadded:: 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.
.. versionadded:: 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.]])
"""
# Grab non-einsum kwargs
optimize_arg = kwargs.pop('optimize', False)
# If no optimization, run pure einsum
if optimize_arg is False:
return c_einsum(*operands, **kwargs)
valid_einsum_kwargs = ['out', 'dtype', 'order', 'casting']
einsum_kwargs = {
k: v
for (k, v) in kwargs.items() if k in valid_einsum_kwargs
}
# Make sure all keywords are valid
valid_contract_kwargs = ['optimize'] + valid_einsum_kwargs
unknown_kwargs = [
k for (k, v) in kwargs.items() if k not in valid_contract_kwargs
]
if len(unknown_kwargs):
raise TypeError("Did not understand the following kwargs: %s" %
unknown_kwargs)
# Special handeling if out is specified
specified_out = False
out_array = einsum_kwargs.pop('out', None)
if out_array is not None:
specified_out = True
# Build the contraction list and operand
operands, contraction_list = einsum_path(*operands,
optimize=optimize_arg,
einsum_call=True)
# Start contraction loop
for num, contraction in enumerate(contraction_list):
inds, idx_rm, einsum_str, remaining = contraction
tmp_operands = []
for x in inds:
tmp_operands.append(operands.pop(x))
# If out was specified
if specified_out and ((num + 1) == len(contraction_list)):
einsum_kwargs["out"] = out_array
# Do the contraction
new_view = c_einsum(einsum_str, *tmp_operands, **einsum_kwargs)
# Append new items and derefernce what we can
operands.append(new_view)
del tmp_operands, new_view
if specified_out:
return out_array
else:
return operands[0]
def einsum(*operands, **kwargs):
"""
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,
linear algebraic array operations can be represented in a simple fashion.
In *implicit* mode `einsum` computes these values.
In *explicit* mode, `einsum` provides further flexibility to compute
other array operations that might not be considered classical Einstein
summation operations, by disabling, or forcing summation over specified
subscript labels.
See the notes and examples for clarification.
Parameters
----------
subscripts : str
Specifies the subscripts for summation as comma separated list of
subscript labels. An implicit (classical Einstein summation)
calculation is performed unless the explicit indicator '->' is
included as well as subscript labels of the precise output form.
operands : list of array_like
These are the arrays for the operation.
out : ndarray, 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
-----
.. versionadded:: 1.6.0
The Einstein summation convention can be used to compute
many multi-dimensional, linear algebraic array operations. `einsum`
provides a succinct way of representing these.
A non-exhaustive list of these operations,
which can be computed by `einsum`, is shown below along with examples:
* Trace of an array, :py:func:`numpy.trace`.
* Return a diagonal, :py:func:`numpy.diag`.
* Array axis summations, :py:func:`numpy.sum`.
* Transpositions and permutations, :py:func:`numpy.transpose`.
* Matrix multiplication and dot product, :py:func:`numpy.matmul` :py:func:`numpy.dot`.
* Vector inner and outer products, :py:func:`numpy.inner` :py:func:`numpy.outer`.
* Broadcasting, element-wise and scalar multiplication, :py:func:`numpy.multiply`.
* Tensor contractions, :py:func:`numpy.tensordot`.
* Chained array operations, in efficient calculation order, :py:func:`numpy.einsum_path`.
The subscripts string is a comma-separated list of subscript labels,
where each label refers to a dimension of the corresponding operand.
Whenever a label is repeated it is summed, so ``np.einsum('i,i', a, b)``
is equivalent to :py:func:`np.inner(a,b) <numpy.inner>`. If a label
appears only once, it is not summed, so ``np.einsum('i', a)`` produces a
view of ``a`` with no changes. A further example ``np.einsum('ij,jk', a, b)``
describes traditional matrix multiplication and is equivalent to
:py:func:`np.matmul(a,b) <numpy.matmul>`. Repeated subscript labels in one
operand take the diagonal. For example, ``np.einsum('ii', a)`` is equivalent
to :py:func:`np.trace(a) <numpy.trace>`.
In *implicit mode*, the chosen subscripts are important
since the axes of the output are reordered alphabetically. This
means that ``np.einsum('ij', a)`` doesn't affect a 2D array, while
``np.einsum('ji', a)`` takes its transpose. Additionally,
``np.einsum('ij,jk', a, b)`` returns a matrix multiplication, while,
``np.einsum('ij,jh', a, b)`` returns the transpose of the
multiplication since subscript 'h' precedes subscript 'i'.
In *explicit mode* the output can be directly controlled by
specifying output subscript labels. This requires the
identifier '->' as well as the list of output subscript labels.
This feature increases the flexibility of the function since
summing can be disabled or forced when required. The call
``np.einsum('i->', a)`` is like :py:func:`np.sum(a, axis=-1) <numpy.sum>`,
and ``np.einsum('ii->i', a)`` is like :py:func:`np.diag(a) <numpy.diag>`.
The difference is that `einsum` does not allow broadcasting by default.
Additionally ``np.einsum('ij,jh->ih', a, b)`` directly specifies the
order of the output subscript labels and therefore returns matrix
multiplication, unlike the example above in implicit mode.
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, one 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 (changed in version 1.10.0).
`einsum` also provides an alternative way to provide the subscripts
and operands as ``einsum(op0, sublist0, op1, sublist1, ..., [sublistout])``.
If the output shape is not provided in this format `einsum` will be
calculated in implicit mode, otherwise it will be performed explicitly.
The examples below have corresponding `einsum` calls with the two
parameter methods.
.. versionadded:: 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 :py:func:`np.swapaxes(a, 0, 2) <numpy.swapaxes>`
and ``np.einsum('ii->i', a)`` will return a writeable view of the diagonal
of a 2D array.
.. versionadded:: 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.
Typically a 'greedy' algorithm is applied which empirical tests have shown
returns the optimal path in the majority of cases. In some cases 'optimal'
will return the superlative path through a more expensive, exhaustive search.
For iterative calculations it may be advisable to calculate the optimal path
once and reuse that path by supplying it as an argument. An example is given
below.
See :py:func:`numpy.einsum_path` for more details.
Examples
--------
>>> a = np.arange(25).reshape(5,5)
>>> b = np.arange(5)
>>> c = np.arange(6).reshape(2,3)
Trace of a matrix:
>>> np.einsum('ii', a)
60
>>> np.einsum(a, [0,0])
60
>>> np.trace(a)
60
Extract the diagonal (requires explicit form):
>>> 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])
Sum over an axis (requires explicit form):
>>> np.einsum('ij->i', a)
array([ 10, 35, 60, 85, 110])
>>> np.einsum(a, [0,1], [0])
array([ 10, 35, 60, 85, 110])
>>> np.sum(a, axis=1)
array([ 10, 35, 60, 85, 110])
For higher dimensional arrays summing a single axis can be done with ellipsis:
>>> np.einsum('...j->...', a)
array([ 10, 35, 60, 85, 110])
>>> np.einsum(a, [Ellipsis,1], [Ellipsis])
array([ 10, 35, 60, 85, 110])
Compute a matrix transpose, or reorder any number of axes:
>>> np.einsum('ji', c)
array([[0, 3],
[1, 4],
[2, 5]])
>>> np.einsum('ij->ji', c)
array([[0, 3],
[1, 4],
[2, 5]])
>>> np.einsum(c, [1,0])
array([[0, 3],
[1, 4],
[2, 5]])
>>> np.transpose(c)
array([[0, 3],
[1, 4],
[2, 5]])
Vector inner products:
>>> np.einsum('i,i', b, b)
30
>>> np.einsum(b, [0], b, [0])
30
>>> np.inner(b,b)
30
Matrix vector multiplication:
>>> 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])
Broadcasting and scalar multiplication:
>>> 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]])
Vector outer product:
>>> 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]])
Tensor contraction:
>>> 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.]])
Writeable returned arrays (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.]])
Example of ellipsis use:
>>> 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]])
Chained array operations. For more complicated contractions, speed ups
might be achieved by repeatedly computing a 'greedy' path or pre-computing the
'optimal' path and repeatedly applying it, using an
`einsum_path` insertion (since version 1.12.0). Performance improvements can be
particularly significant with larger arrays:
>>> a = np.ones(64).reshape(2,4,8)
# Basic `einsum`: ~1520ms (benchmarked on 3.1GHz Intel i5.)
>>> for iteration in range(500):
... np.einsum('ijk,ilm,njm,nlk,abc->',a,a,a,a,a)
# Sub-optimal `einsum` (due to repeated path calculation time): ~330ms
>>> for iteration in range(500):
... np.einsum('ijk,ilm,njm,nlk,abc->',a,a,a,a,a, optimize='optimal')
# Greedy `einsum` (faster optimal path approximation): ~160ms
>>> for iteration in range(500):
... np.einsum('ijk,ilm,njm,nlk,abc->',a,a,a,a,a, optimize='greedy')
# Optimal `einsum` (best usage pattern in some use cases): ~110ms
>>> path = np.einsum_path('ijk,ilm,njm,nlk,abc->',a,a,a,a,a, optimize='optimal')[0]
>>> for iteration in range(500):
... np.einsum('ijk,ilm,njm,nlk,abc->',a,a,a,a,a, optimize=path)
"""
# Grab non-einsum kwargs; never optimize 2-argument case.
optimize_arg = kwargs.pop('optimize', len(operands) > 3)
# If no optimization, run pure einsum
if optimize_arg is False:
return c_einsum(*operands, **kwargs)
valid_einsum_kwargs = ['out', 'dtype', 'order', 'casting']
einsum_kwargs = {k: v for (k, v) in kwargs.items() if
k in valid_einsum_kwargs}
# Make sure all keywords are valid
valid_contract_kwargs = ['optimize'] + valid_einsum_kwargs
unknown_kwargs = [k for (k, v) in kwargs.items() if
k not in valid_contract_kwargs]
if len(unknown_kwargs):
raise TypeError("Did not understand the following kwargs: %s"
% unknown_kwargs)
# Special handeling if out is specified
specified_out = False
out_array = einsum_kwargs.pop('out', None)
if out_array is not None:
specified_out = True
# Build the contraction list and operand
operands, contraction_list = einsum_path(*operands, optimize=optimize_arg,
einsum_call=True)
handle_out = False
# Start contraction loop
for num, contraction in enumerate(contraction_list):
inds, idx_rm, einsum_str, remaining, blas = contraction
tmp_operands = []
for x in inds:
tmp_operands.append(operands.pop(x))
# Do we need to deal with the output?
if specified_out and ((num + 1) == len(contraction_list)):
handle_out = True
# Handle broadcasting vs BLAS cases
if blas:
# Checks have already been handled
input_str, results_index = einsum_str.split('->')
input_left, input_right = input_str.split(',')
if 1 in tmp_operands[0].shape or 1 in tmp_operands[1].shape:
left_dims = {dim: size for dim, size in
zip(input_left, tmp_operands[0].shape)}
right_dims = {dim: size for dim, size in
zip(input_right, tmp_operands[1].shape)}
# If dims do not match we are broadcasting, BLAS off
if any(left_dims[ind] != right_dims[ind] for ind in idx_rm):
blas = False
# Call tensordot if still possible
if blas:
tensor_result = input_left + input_right
for s in idx_rm:
tensor_result = tensor_result.replace(s, "")
# Find indices to contract over
left_pos, right_pos = [], []
for s in idx_rm:
left_pos.append(input_left.find(s))
right_pos.append(input_right.find(s))
# Contract!
new_view = tensordot(*tmp_operands, axes=(tuple(left_pos), tuple(right_pos)))
# Build a new view if needed
if (tensor_result != results_index) or handle_out:
if handle_out:
einsum_kwargs["out"] = out_array
new_view = c_einsum(tensor_result + '->' + results_index, new_view, **einsum_kwargs)
# Call einsum
else:
# If out was specified
if handle_out:
einsum_kwargs["out"] = out_array
# Do the contraction
new_view = c_einsum(einsum_str, *tmp_operands, **einsum_kwargs)
# Append new items and dereference what we can
operands.append(new_view)
del tmp_operands, new_view
if specified_out:
return out_array
else:
return operands[0]
def einsum(*operands, **kwargs):
"""
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,
linear algebraic array operations can be represented in a simple fashion.
In *implicit* mode `einsum` computes these values.
In *explicit* mode, `einsum` provides further flexibility to compute
other array operations that might not be considered classical Einstein
summation operations, by disabling, or forcing summation over specified
subscript labels.
See the notes and examples for clarification.
Parameters
----------
subscripts : str
Specifies the subscripts for summation as comma separated list of
subscript labels. An implicit (classical Einstein summation)
calculation is performed unless the explicit indicator '->' is
included as well as subscript labels of the precise output form.
operands : list of array_like
These are the arrays for the operation.
out : ndarray, 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
-----
.. versionadded:: 1.6.0
The Einstein summation convention can be used to compute
many multi-dimensional, linear algebraic array operations. `einsum`
provides a succinct way of representing these.
A non-exhaustive list of these operations,
which can be computed by `einsum`, is shown below along with examples:
* Trace of an array, :py:func:`numpy.trace`.
* Return a diagonal, :py:func:`numpy.diag`.
* Array axis summations, :py:func:`numpy.sum`.
* Transpositions and permutations, :py:func:`numpy.transpose`.
* Matrix multiplication and dot product, :py:func:`numpy.matmul` :py:func:`numpy.dot`.
* Vector inner and outer products, :py:func:`numpy.inner` :py:func:`numpy.outer`.
* Broadcasting, element-wise and scalar multiplication, :py:func:`numpy.multiply`.
* Tensor contractions, :py:func:`numpy.tensordot`.
* Chained array operations, in efficient calculation order, :py:func:`numpy.einsum_path`.
The subscripts string is a comma-separated list of subscript labels,
where each label refers to a dimension of the corresponding operand.
Whenever a label is repeated it is summed, so ``np.einsum('i,i', a, b)``
is equivalent to :py:func:`np.inner(a,b) <numpy.inner>`. If a label
appears only once, it is not summed, so ``np.einsum('i', a)`` produces a
view of ``a`` with no changes. A further example ``np.einsum('ij,jk', a, b)``
describes traditional matrix multiplication and is equivalent to
:py:func:`np.matmul(a,b) <numpy.matmul>`. Repeated subscript labels in one
operand take the diagonal. For example, ``np.einsum('ii', a)`` is equivalent
to :py:func:`np.trace(a) <numpy.trace>`.
In *implicit mode*, the chosen subscripts are important
since the axes of the output are reordered alphabetically. This
means that ``np.einsum('ij', a)`` doesn't affect a 2D array, while
``np.einsum('ji', a)`` takes its transpose. Additionally,
``np.einsum('ij,jk', a, b)`` returns a matrix multiplication, while,
``np.einsum('ij,jh', a, b)`` returns the transpose of the
multiplication since subscript 'h' precedes subscript 'i'.
In *explicit mode* the output can be directly controlled by
specifying output subscript labels. This requires the
identifier '->' as well as the list of output subscript labels.
This feature increases the flexibility of the function since
summing can be disabled or forced when required. The call
``np.einsum('i->', a)`` is like :py:func:`np.sum(a, axis=-1) <numpy.sum>`,
and ``np.einsum('ii->i', a)`` is like :py:func:`np.diag(a) <numpy.diag>`.
The difference is that `einsum` does not allow broadcasting by default.
Additionally ``np.einsum('ij,jh->ih', a, b)`` directly specifies the
order of the output subscript labels and therefore returns matrix
multiplication, unlike the example above in implicit mode.
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, one 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 (changed in version 1.10.0).
`einsum` also provides an alternative way to provide the subscripts
and operands as ``einsum(op0, sublist0, op1, sublist1, ..., [sublistout])``.
If the output shape is not provided in this format `einsum` will be
calculated in implicit mode, otherwise it will be performed explicitly.
The examples below have corresponding `einsum` calls with the two
parameter methods.
.. versionadded:: 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 :py:func:`np.swapaxes(a, 0, 2) <numpy.swapaxes>`
and ``np.einsum('ii->i', a)`` will return a writeable view of the diagonal
of a 2D array.
.. versionadded:: 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.
Typically a 'greedy' algorithm is applied which empirical tests have shown
returns the optimal path in the majority of cases. In some cases 'optimal'
will return the superlative path through a more expensive, exhaustive search.
For iterative calculations it may be advisable to calculate the optimal path
once and reuse that path by supplying it as an argument. An example is given
below.
See :py:func:`numpy.einsum_path` for more details.
Examples
--------
>>> a = np.arange(25).reshape(5,5)
>>> b = np.arange(5)
>>> c = np.arange(6).reshape(2,3)
Trace of a matrix:
>>> np.einsum('ii', a)
60
>>> np.einsum(a, [0,0])
60
>>> np.trace(a)
60
Extract the diagonal (requires explicit form):
>>> 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])
Sum over an axis (requires explicit form):
>>> np.einsum('ij->i', a)
array([ 10, 35, 60, 85, 110])
>>> np.einsum(a, [0,1], [0])
array([ 10, 35, 60, 85, 110])
>>> np.sum(a, axis=1)
array([ 10, 35, 60, 85, 110])
For higher dimensional arrays summing a single axis can be done with ellipsis:
>>> np.einsum('...j->...', a)
array([ 10, 35, 60, 85, 110])
>>> np.einsum(a, [Ellipsis,1], [Ellipsis])
array([ 10, 35, 60, 85, 110])
Compute a matrix transpose, or reorder any number of axes:
>>> np.einsum('ji', c)
array([[0, 3],
[1, 4],
[2, 5]])
>>> np.einsum('ij->ji', c)
array([[0, 3],
[1, 4],
[2, 5]])
>>> np.einsum(c, [1,0])
array([[0, 3],
[1, 4],
[2, 5]])
>>> np.transpose(c)
array([[0, 3],
[1, 4],
[2, 5]])
Vector inner products:
>>> np.einsum('i,i', b, b)
30
>>> np.einsum(b, [0], b, [0])
30
>>> np.inner(b,b)
30
Matrix vector multiplication:
>>> 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])
Broadcasting and scalar multiplication:
>>> 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]])
Vector outer product:
>>> 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]])
Tensor contraction:
>>> 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.]])
Writeable returned arrays (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.]])
Example of ellipsis use:
>>> 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]])
Chained array operations. For more complicated contractions, speed ups
might be achieved by repeatedly computing a 'greedy' path or pre-computing the
'optimal' path and repeatedly applying it, using an
`einsum_path` insertion (since version 1.12.0). Performance improvements can be
particularly significant with larger arrays:
>>> a = np.ones(64).reshape(2,4,8)
# Basic `einsum`: ~1520ms (benchmarked on 3.1GHz Intel i5.)
>>> for iteration in range(500):
... np.einsum('ijk,ilm,njm,nlk,abc->',a,a,a,a,a)
# Sub-optimal `einsum` (due to repeated path calculation time): ~330ms
>>> for iteration in range(500):
... np.einsum('ijk,ilm,njm,nlk,abc->',a,a,a,a,a, optimize='optimal')
# Greedy `einsum` (faster optimal path approximation): ~160ms
>>> for iteration in range(500):
... np.einsum('ijk,ilm,njm,nlk,abc->',a,a,a,a,a, optimize='greedy')
# Optimal `einsum` (best usage pattern in some use cases): ~110ms
>>> path = np.einsum_path('ijk,ilm,njm,nlk,abc->',a,a,a,a,a, optimize='optimal')[0]
>>> for iteration in range(500):
... np.einsum('ijk,ilm,njm,nlk,abc->',a,a,a,a,a, optimize=path)
"""
# Grab non-einsum kwargs; do not optimize by default.
optimize_arg = kwargs.pop('optimize', False)
# If no optimization, run pure einsum
if optimize_arg is False:
return c_einsum(*operands, **kwargs)
valid_einsum_kwargs = ['out', 'dtype', 'order', 'casting']
einsum_kwargs = {
k: v
for (k, v) in kwargs.items() if k in valid_einsum_kwargs
}
# Make sure all keywords are valid
valid_contract_kwargs = ['optimize'] + valid_einsum_kwargs
unknown_kwargs = [
k for (k, v) in kwargs.items() if k not in valid_contract_kwargs
]
if len(unknown_kwargs):
raise TypeError("Did not understand the following kwargs: %s" %
unknown_kwargs)
# Special handeling if out is specified
specified_out = False
out_array = einsum_kwargs.pop('out', None)
if out_array is not None:
specified_out = True
# Build the contraction list and operand
operands, contraction_list = einsum_path(*operands,
optimize=optimize_arg,
einsum_call=True)
handle_out = False
# Start contraction loop
for num, contraction in enumerate(contraction_list):
inds, idx_rm, einsum_str, remaining, blas = contraction
tmp_operands = []
for x in inds:
tmp_operands.append(operands.pop(x))
# Do we need to deal with the output?
handle_out = specified_out and ((num + 1) == len(contraction_list))
# Call tensordot if still possible
if blas:
# Checks have already been handled
input_str, results_index = einsum_str.split('->')
input_left, input_right = input_str.split(',')
tensor_result = input_left + input_right
for s in idx_rm:
tensor_result = tensor_result.replace(s, "")
# Find indices to contract over
left_pos, right_pos = [], []
for s in idx_rm:
left_pos.append(input_left.find(s))
right_pos.append(input_right.find(s))
# Contract!
new_view = tensordot(*tmp_operands,
axes=(tuple(left_pos), tuple(right_pos)))
# Build a new view if needed
if (tensor_result != results_index) or handle_out:
if handle_out:
einsum_kwargs["out"] = out_array
new_view = c_einsum(tensor_result + '->' + results_index,
new_view, **einsum_kwargs)
# Call einsum
else:
# If out was specified
if handle_out:
einsum_kwargs["out"] = out_array
# Do the contraction
new_view = c_einsum(einsum_str, *tmp_operands, **einsum_kwargs)
# Append new items and dereference what we can
operands.append(new_view)
del tmp_operands, new_view
if specified_out:
return out_array
else:
return operands[0]
def einsum(*operands, **kwargs):
"""
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. Default is False.
Returns
-------
output : ndarray
The calculation based on the Einstein summation convention.
See Also
--------
einsum_path, dot, inner, outer, tensordot, linalg.multi_dot
Notes
-----
.. versionadded:: 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.
.. versionadded:: 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.
.. versionadded:: 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(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.]])
"""
# Grab non-einsum kwargs
optimize_arg = kwargs.pop('optimize', False)
# If no optimization, run pure einsum
if optimize_arg is False:
return c_einsum(*operands, **kwargs)
valid_einsum_kwargs = ['out', 'dtype', 'order', 'casting']
einsum_kwargs = {k: v for (k, v) in kwargs.items() if
k in valid_einsum_kwargs}
# Make sure all keywords are valid
valid_contract_kwargs = ['optimize'] + valid_einsum_kwargs
unknown_kwargs = [k for (k, v) in kwargs.items() if
k not in valid_contract_kwargs]
if len(unknown_kwargs):
raise TypeError("Did not understand the following kwargs: %s"
% unknown_kwargs)
# Special handeling if out is specified
specified_out = False
out_array = einsum_kwargs.pop('out', None)
if out_array is not None:
specified_out = True
# Build the contraction list and operand
operands, contraction_list = einsum_path(*operands, optimize=optimize_arg,
einsum_call=True)
# Start contraction loop
for num, contraction in enumerate(contraction_list):
inds, idx_rm, einsum_str, remaining = contraction
tmp_operands = []
for x in inds:
tmp_operands.append(operands.pop(x))
# If out was specified
if specified_out and ((num + 1) == len(contraction_list)):
einsum_kwargs["out"] = out_array
# Do the contraction
new_view = c_einsum(einsum_str, *tmp_operands, **einsum_kwargs)
# Append new items and derefernce what we can
operands.append(new_view)
del tmp_operands, new_view
if specified_out:
return out_array
else:
return operands[0]
def _rl_construct(self, tensor):
# fine to use pure c_einsum here as it's used anyway
return c_einsum(self.rl_einstr, tensor,
*[flat_metric(len(tensor))] * self._rl_diff)
