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���g���@s�dZddlZddlZddlmZddlmZmZddlm	Z	ddgZ
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Implementation of optimized einsum.

�N)�c_einsum)�
asanyarray�	tensordot)�array_function_dispatch�einsum�einsum_path�4abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZcCs,t||�}td|d�}|r|d7}||S)a�
    Computes the number of FLOPS in the contraction.

    Parameters
    ----------
    idx_contraction : iterable
        The indices involved in the contraction
    inner : bool
        Does this contraction require an inner product?
    num_terms : int
        The number of terms in a contraction
    size_dictionary : dict
        The size of each of the indices in idx_contraction

    Returns
    -------
    flop_count : int
        The total number of FLOPS required for the contraction.

    Examples
    --------

    >>> _flop_count('abc', False, 1, {'a': 2, 'b':3, 'c':5})
    30

    >>> _flop_count('abc', True, 2, {'a': 2, 'b':3, 'c':5})
    60

    �)�_compute_size_by_dict�max)�idx_contraction�inner�	num_terms�size_dictionary�overall_size�	op_factor�r�H/home/arjun/.local/lib/python3.10/site-packages/numpy/core/einsumfunc.py�_flop_counts

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    Computes the product of the elements in indices based on the dictionary
    idx_dict.

    Parameters
    ----------
    indices : iterable
        Indices to base the product on.
    idx_dict : dictionary
        Dictionary of index sizes

    Returns
    -------
    ret : int
        The resulting product.

    Examples
    --------
    >>> _compute_size_by_dict('abbc', {'a': 2, 'b':3, 'c':5})
    90

    r	r)�indices�idx_dict�ret�irrrr
8sr
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Csnt�}|��}g}t|�D]\}}||vr||O}q
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||@}||}	|�|�|||	|fS)a

    Finds the contraction for a given set of input and output sets.

    Parameters
    ----------
    positions : iterable
        Integer positions of terms used in the contraction.
    input_sets : list
        List of sets that represent the lhs side of the einsum subscript
    output_set : set
        Set that represents the rhs side of the overall einsum subscript

    Returns
    -------
    new_result : set
        The indices of the resulting contraction
    remaining : list
        List of sets that have not been contracted, the new set is appended to
        the end of this list
    idx_removed : set
        Indices removed from the entire contraction
    idx_contraction : set
        The indices used in the current contraction

    Examples
    --------

    # A simple dot product test case
    >>> pos = (0, 1)
    >>> isets = [set('ab'), set('bc')]
    >>> oset = set('ac')
    >>> _find_contraction(pos, isets, oset)
    ({'a', 'c'}, [{'a', 'c'}], {'b'}, {'a', 'b', 'c'})

    # A more complex case with additional terms in the contraction
    >>> pos = (0, 2)
    >>> isets = [set('abd'), set('ac'), set('bdc')]
    >>> oset = set('ac')
    >>> _find_contraction(pos, isets, oset)
    ({'a', 'c'}, [{'a', 'c'}, {'a', 'c'}], {'b', 'd'}, {'a', 'b', 'c', 'd'})
    )�set�copy�	enumerate�append)
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    Computes all possible pair contractions, sieves the results based
    on ``memory_limit`` and returns the lowest cost path. This algorithm
    scales factorial with respect to the elements in the list ``input_sets``.

    Parameters
    ----------
    input_sets : list
        List of sets that represent the lhs side of the einsum subscript
    output_set : set
        Set that represents the rhs side of the overall einsum subscript
    idx_dict : dictionary
        Dictionary of index sizes
    memory_limit : int
        The maximum number of elements in a temporary array

    Returns
    -------
    path : list
        The optimal contraction order within the memory limit constraint.

    Examples
    --------
    >>> isets = [set('abd'), set('ac'), set('bdc')]
    >>> oset = set()
    >>> idx_sizes = {'a': 1, 'b':2, 'c':3, 'd':4}
    >>> _optimal_path(isets, oset, idx_sizes, 5000)
    [(0, 2), (0, 1)]
    rr	�cS�|dS�Nrr��xrrr�<lambda>��z_optimal_path.<locals>.<lambda>��keycSr)r*rr+rrrr-�r.)
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    performing the contraction specified by ``positions``.

    Parameters
    ----------
    positions : tuple of int
        The locations of the proposed tensors to contract.
    input_sets : list of sets
        The indices found on each tensors.
    output_set : set
        The output indices of the expression.
    idx_dict : dict
        Mapping of each index to its size.
    memory_limit : int
        The total allowed size for an intermediary tensor.
    path_cost : int
        The contraction cost so far.
    naive_cost : int
        The cost of the unoptimized expression.

    Returns
    -------
    cost : (int, int)
        A tuple containing the size of any indices removed, and the flop cost.
    positions : tuple of int
        The locations of the proposed tensors to contract.
    new_input_sets : list of sets
        The resulting new list of indices if this proposed contraction is performed.

    Nc3s�|]
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rScCs�|d}|\}}g}|D]\\}\}}}	||vs||vrq|	|t||k�t||k�=|	|t||k�t||k�=|	�d|dd�|t||k�t||k�|t||k�t||k�f}
|�||
|	f�q|S)aUpdate the positions and provisional input_sets of ``results`` based on
    performing the contraction result ``best``. Remove any involving the tensors
    contracted.

    Parameters
    ----------
    results : list
        List of contraction results produced by ``_parse_possible_contraction``.
    best : list
        The best contraction of ``results`` i.e. the one that will be performed.

    Returns
    -------
    mod_results : list
        The list of modified results, updated with outcome of ``best`` contraction.
    r	���r()�int�insertr)�results�best�best_con�bx�by�mod_resultsr<r,�y�con_sets�mod_conrrr�_update_other_resultss8r`c
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tt|�d�D]�}|
D]#}||d�||d�rWqGt|||||||	�}|durj|�|�qGt|�dkr�t�tt|��d�D]}t|||||||	�}|dur�|�|�q{t|�dkr�|
�t	tt|����|
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d�t��D�}
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�|d�||dd7}qC|
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    Finds the path by contracting the best pair until the input list is
    exhausted. The best pair is found by minimizing the tuple
    ``(-prod(indices_removed), cost)``.  What this amounts to is prioritizing
    matrix multiplication or inner product operations, then Hadamard like
    operations, and finally outer operations. Outer products are limited by
    ``memory_limit``. This algorithm scales cubically with respect to the
    number of elements in the list ``input_sets``.

    Parameters
    ----------
    input_sets : list
        List of sets that represent the lhs side of the einsum subscript
    output_set : set
        Set that represents the rhs side of the overall einsum subscript
    idx_dict : dictionary
        Dictionary of index sizes
    memory_limit : int
        The maximum number of elements in a temporary array

    Returns
    -------
    path : list
        The greedy contraction order within the memory limit constraint.

    Examples
    --------
    >>> isets = [set('abd'), set('ac'), set('bdc')]
    >>> oset = set()
    >>> idx_sizes = {'a': 1, 'b':2, 'c':3, 'd':4}
    >>> _greedy_path(isets, oset, idx_sizes, 5000)
    [(0, 2), (0, 1)]
    r	)rr()rr	rNcSr)r*rr+rrrr-�r.z_greedy_path.<locals>.<lambda>r/c3s�|]}|�fVqdSrEr)rGr��new_tensor_posrrrJ���z_greedy_path.<locals>.<genexpr>)r2r'r1rr3r4�
isdisjointrSrr6r5r`)rrrr7rNrOr?r&r rM�	comb_iter�known_contractionsrLrCr9r�resultrXrrar�_greedy_path8sR$�
��
��
rhc
CsXt|�dkrdSt|�dkrdS|\}}t||�D]-}|�|�|�|�}}|dks5|dks5||dkr8dS||dt||v�krGdSqt|�}t|�}	||}
|	|}t|�}||krbdS||	krhdS||d�|d|�krwdS|d|�||d�kr�dS||d�||d�kr�dS|d|�|d|�kr�dS|
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    Checks if we can use BLAS (np.tensordot) call and its beneficial to do so.

    Parameters
    ----------
    inputs : list of str
        Specifies the subscripts for summation.
    result : str
        Resulting summation.
    idx_removed : set
        Indices that are removed in the summation


    Returns
    -------
    type : bool
        Returns true if BLAS should and can be used, else False

    Notes
    -----
    If the operations is BLAS level 1 or 2 and is not already aligned
    we default back to einsum as the memory movement to copy is more
    costly than the operation itself.


    Examples
    --------

    # Standard GEMM operation
    >>> _can_dot(['ij', 'jk'], 'ik', set('j'))
    True

    # Can use the standard BLAS, but requires odd data movement
    >>> _can_dot(['ijj', 'jk'], 'ik', set('j'))
    False

    # DDOT where the memory is not aligned
    >>> _can_dot(['ijk', 'ikj'], '', set('ijk'))
    False

    rFr(r	TN)r2r�countrU)
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|�}Wnty�}ztd�|�d}~ww|t|7}q|	|kr�|d7}qy|dur�|d7}|D]*}|tur�|d7}q�zt�
|�}Wnty�}ztd�|�d}~ww|t|7}q�d|vs�d|v�r|�d�dkp�|�d�dk}|�s
|�d�dk�rtd��d|v�r|�dd��dd��dd�}
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|�D]g\}	}
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�d�dk�sh|
�d�dk�rltd��||	jdk�rwd}nt||	jd�}|t|
�d8}||k�r�|}|dk�r�td��|dk�r�|
�dd�||	<�qO||d�}|
�d|�||	<�qOd�|�}|dk�r�d}n||d�}|�r�|d|�d|�7}n>d}|�dd�}tt|��D]}|tv�r�td
|��|�|�dk�r||7}�q�d�tt|�t|���}|d||7}d|v�r$|�d�\}}n+|}|�dd�}d}tt|��D]}|tv�rAtd
|��|�|�dk�rM||7}�q4|D]}||v�r^td|���qQt|�d��t|�k�rptd��|||fS)a�
    A reproduction of einsum c side einsum parsing in python.

    Returns
    -------
    input_strings : str
        Parsed input strings
    output_string : str
        Parsed output string
    operands : list of array_like
        The operands to use in the numpy contraction

    Examples
    --------
    The operand list is simplified to reduce printing:

    >>> np.random.seed(123)
    >>> a = np.random.rand(4, 4)
    >>> b = np.random.rand(4, 4, 4)
    >>> _parse_einsum_input(('...a,...a->...', a, b))
    ('za,xza', 'xz', [a, b]) # may vary

    >>> _parse_einsum_input((a, [Ellipsis, 0], b, [Ellipsis, 0]))
    ('za,xza', 'xz', [a, b]) # may vary
    rzNo input operands� �cS�g|]}t|��qSr�r�rG�vrrr�
<listcomp>+�z'_parse_einsum_input.<locals>.<listcomp>r	Nz.,->z#Character %s is not a valid symbol.r(rTcSrxrryrzrrrr|=r}z...z=For this input type lists must contain either int or Ellipsis�,�->�-�>z%Subscripts can only contain one '->'.�.TF�zInvalid Ellipses.rzEllipses lengths do not match.z/Output character %s did not appear in the inputzDNumber of einsum subscripts must be equal to the number of operands.)r2�
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isinstance�str�replace�einsum_symbols�listr1r�popr�Ellipsis�operator�index�	TypeErrorri�einsum_symbols_setr�join�split�shaper�ndim�sorted)�operands�
subscripts�s�tmp_operands�operand_list�subscript_listrH�output_list�last�num�sub�e�invalid�used�unused�ellipse_inds�longest�	input_tmp�
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ellipse_count�rep_inds�out_ellipse�output_subscript�tmp_subscripts�normal_inds�input_subscripts�charrrr�_parse_einsum_inputs���
����
���





 


�


��

�
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|�}|\}}
} }!t|!| t|���}"|�|"�|�t|!��|�t|���t�}#g}$|D]}%|$�|	�|%��|#|
�|%�O}#�q�|#| }&t| |#@��s�t|$|| �}'nd}'|t|�dk�r�|}(n�fdd�|D�})d
�dd�t|)�D��}(|	�|(�|
�|&�d
�|$�d|(}*|| |*|	dd�|'f}+|�|+��qmt|�d},t|	�dk�r0td�t|	�d���|�r7||fS|d|}-d}.||,}/t|�}0d|-}1|1d t|�7}1|1d!t|�7}1|1d"|7}1|1d#|,7}1|1d$|/7}1|1d%|07}1|1d&7}1|1d'|.7}1|1d(7}1t|�D]"\}2}+|+\}3}4}*}5}6d
�|5�d|}7||2|*|7f}8|1d)|87}1�q�dg|}||1fS)*a�
    einsum_path(subscripts, *operands, optimize='greedy')

    Evaluates the lowest cost contraction order for an einsum expression by
    considering the creation of intermediate arrays.

    Parameters
    ----------
    subscripts : str
        Specifies the subscripts for summation.
    *operands : list of array_like
        These are the arrays for the operation.
    optimize : {bool, list, tuple, 'greedy', 'optimal'}
        Choose the type of path. If a tuple is provided, the second argument is
        assumed to be the maximum intermediate size created. If only a single
        argument is provided the largest input or output array size is used
        as a maximum intermediate size.

        * if a list is given that starts with ``einsum_path``, uses this as the
          contraction path
        * if False no optimization is taken
        * if True defaults to the 'greedy' algorithm
        * 'optimal' An algorithm that combinatorially explores all possible
          ways of contracting the listed tensors and chooses the least costly
          path. Scales exponentially with the number of terms in the
          contraction.
        * 'greedy' An algorithm that chooses the best pair contraction
          at each step. Effectively, this algorithm searches the largest inner,
          Hadamard, and then outer products at each step. Scales cubically with
          the number of terms in the contraction. Equivalent to the 'optimal'
          path for most contractions.

        Default is 'greedy'.

    Returns
    -------
    path : list of tuples
        A list representation of the einsum path.
    string_repr : str
        A printable representation of the einsum path.

    Notes
    -----
    The resulting path indicates which terms of the input contraction should be
    contracted first, the result of this contraction is then appended to the
    end of the contraction list. This list can then be iterated over until all
    intermediate contractions are complete.

    See Also
    --------
    einsum, linalg.multi_dot

    Examples
    --------

    We can begin with a chain dot example. In this case, it is optimal to
    contract the ``b`` and ``c`` tensors first as represented by the first
    element of the path ``(1, 2)``. The resulting tensor is added to the end
    of the contraction and the remaining contraction ``(0, 1)`` is then
    completed.

    >>> np.random.seed(123)
    >>> a = np.random.rand(2, 2)
    >>> b = np.random.rand(2, 5)
    >>> c = np.random.rand(5, 2)
    >>> path_info = np.einsum_path('ij,jk,kl->il', a, b, c, optimize='greedy')
    >>> print(path_info[0])
    ['einsum_path', (1, 2), (0, 1)]
    >>> print(path_info[1])
      Complete contraction:  ij,jk,kl->il # may vary
             Naive scaling:  4
         Optimized scaling:  3
          Naive FLOP count:  1.600e+02
      Optimized FLOP count:  5.600e+01
       Theoretical speedup:  2.857
      Largest intermediate:  4.000e+00 elements
    -------------------------------------------------------------------------
    scaling                  current                                remaining
    -------------------------------------------------------------------------
       3                   kl,jk->jl                                ij,jl->il
       3                   jl,ij->il                                   il->il


    A more complex index transformation example.

    >>> I = np.random.rand(10, 10, 10, 10)
    >>> C = np.random.rand(10, 10)
    >>> path_info = np.einsum_path('ea,fb,abcd,gc,hd->efgh', C, C, I, C, C,
    ...                            optimize='greedy')

    >>> print(path_info[0])
    ['einsum_path', (0, 2), (0, 3), (0, 2), (0, 1)]
    >>> print(path_info[1]) 
      Complete contraction:  ea,fb,abcd,gc,hd->efgh # may vary
             Naive scaling:  8
         Optimized scaling:  5
          Naive FLOP count:  8.000e+08
      Optimized FLOP count:  8.000e+05
       Theoretical speedup:  1000.000
      Largest intermediate:  1.000e+04 elements
    --------------------------------------------------------------------------
    scaling                  current                                remaining
    --------------------------------------------------------------------------
       5               abcd,ea->bcde                      fb,gc,hd,bcde->efgh
       5               bcde,fb->cdef                         gc,hd,cdef->efgh
       5               cdef,gc->defg                            hd,defg->efgh
       5               defg,hd->efgh                               efgh->efgh
    Tr�NFrrr(r	zDid not understand the path: %sr~cSrxr�r�rGr,rrrr|Tr}zeinsum_path.<locals>.<listcomp>rwcSsg|]}g�qSrrr�rrrr|ZszXEinstein sum subscript %s does not contain the correct number of indices for operand %d.zJSize of label '%s' for operand %d (%d) does not match previous terms (%d).cSrxrr�r�rrrr|tr}csg|]}t|���qSrrF)rG�term��dimension_dictrrr|ws�css�|]}t|�VqdSrE)r2r�rrrrJ�rczeinsum_path.<locals>.<genexpr>)r	r(�optimalzPath name %s not found)�reverserTcsg|]}�||f�qSrr)rGr#r�rrr|�scSsg|]}|d�qS)r	rr�rrrr|�r}rzHInvalid einsum_path is specified: {} more operands has to be contracted.)�scaling�currentr"z  Complete contraction:  %s
z         Naive scaling:  %d
z     Optimized scaling:  %d
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z  Optimized FLOP count:  %.3e
z   Theoretical speedup:  %3.3f
z'  Largest intermediate:  %.3e elements
zK--------------------------------------------------------------------------
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    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
    einops :
        similar verbose interface is provided by
        `einops <https://github.com/arogozhnikov/einops>`_ package to cover
        additional operations: transpose, reshape/flatten, repeat/tile,
        squeeze/unsqueeze and reductions.
    opt_einsum :
        `opt_einsum <https://optimized-einsum.readthedocs.io/en/stable/>`_
        optimizes contraction order for einsum-like expressions
        in backend-agnostic manner.

    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)

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