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B ��Fd͊ � @ s� d Z ddlmZmZmZ ddlmZ ddlmZm Z m Z ddgZ dZ e e �Zdd � Zd d � Zd d � Zdd� Zdd� Zdd� Zdd� ZdS )z& Implementation of optimized einsum. � )�division�absolute_import�print_function)�c_einsum)�asarray� asanyarray� result_type�einsum� einsum_pathZ4abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZc C s" d}x| D ]}||| 9 }q W |S )a� 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 � � )�indices�idx_dict�ret�ir r �H/opt/alt/python37/lib64/python3.7/site-packages/numpy/core/einsumfunc.py�_compute_size_by_dict s r c C sr t � }|�� }g }x8t|�D ],\}}|| kr6||O }q |�|� ||O }q W ||@ }|| } |�|� ||| |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) � positions� input_sets� output_set� idx_contractZ idx_remain� remaining�ind�value� new_result� idx_removedr r r �_find_contraction- s + r c C s< dg | fg}x�t t| �d �D ]�}g }g }xFt t| �| �D ]2}x,t |d t| �| �D ]} |�|| f� qXW q<W x�|D ]�} | \} } } xp|D ]h}t|| |�}|\}}}}t||�}||kr�q�t||�}|r�|d9 }|| 7 }| |g }|�|||f� q�W qxW |}q W t|�dk�r$tt t| ���gS t|dd� d�d }|S )a� 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} >>> _path__optimal_path(isets, oset, idx_sizes, 5000) [(0, 2), (0, 1)] r r � c S s | d S )Nr r )�xr r r �<lambda>� � z _optimal_path.<locals>.<lambda>)�key)�range�lenr r r �tuple�min)r r r � memory_limitZ full_results� iterationZ iter_results� comb_iterr" �y�curr�costr r ZconZcontr �new_input_setsr r Znew_sizeZnew_cost�new_pos�pathr r r � _optimal_pathi s4 r3 c C s, t | �dkrdgS g }�xtt | �d �D ]�}g }g }x>tt | ��D ].}x(t|d t | ��D ]} |�|| f� q\W qDW xb|D ]Z} t| | |�} | \} } }}t| |�|kr�q|t||�}t||�}| |f}|�|| | g� q|W t |�dkr�|�ttt | ���� P t|dd� d�}|�|d � |d } q*W |S )a� 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_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} >>> _path__greedy_path(isets, oset, idx_sizes, 5000) [(0, 2), (0, 1)] r )r r c S s | d S )Nr r )r" r r r r# � r$ z _greedy_path.<locals>.<lambda>)r% r! )r' r& r r r r( r) )r r r r* r2 r+ Ziteration_resultsr, r" r- r �contract� idx_resultr0 r r Z removed_sizer/ �sort�bestr r r � _greedy_path� s2 # r8 c C s� t | �dkrtd��t| d t�rx| d �dd�}dd� | dd� D �} x*|D ]"}|d kr\qN|tkrNtd | ��qNW �n@t| �}g }g }x8tt | �d �D ]$}|�|� d�� |�|� d�� q�W t |�r�|d nd}d d� |D �} d}t |�d }xjt |�D ]^\} } xD| D ]<}|t k�r$|d7 }n"t|t ��r>|t| 7 }nt d���q W | |kr�|d7 }q�W |dk �r�|d7 }xD|D ]<}|t k�r�|d7 }n"t|t ��r�|t| 7 }nt d���qxW d|k�s�d|k�r|�d�dk�p�|�d�dk} | �s |�d�dk�rtd��d|k�r"|�dd��dd��dd�} ttt| � �} d�| �}d}d|k�rt|�d�\}}|�d�}d}n|�d�}d}x�t |�D ]�\} } d| k�r�| �d�dk�s�| �d�dk�r�td��| | jdk�r�d}n t| | jd�}|t | �d 8 }||k�r |}|dk �r td��n:|dk�r:| �dd�|| < n || d� }| �d|�|| < �q�W d�|�}|dk�rxd}n|| d� }|�r�|d|�d|� 7 }n�d}|�dd�}xDtt|��D ]4}|tk�r�td | ��|�|�dk�r�||7 }�q�W d�tt|�t|� ��}|d| | 7 }d|k�r<|�d�\}}nZ|}|�dd�}d}xDtt|��D ]4}|tk�rxtd | ��|�|�dk�r^||7 }�q^W x$|D ] }||k�r�td | ���q�W t |�d��t | �k�r�td ��||| fS ) aj A reproduction of einsum c side einsum parsing in python. Returns ------- input_strings : str Parsed input strings output_string : str Parsed output string operands : list of array_like The operands to use in the numpy contraction Examples -------- The operand list is simplified to reduce printing: >>> a = np.random.rand(4, 4) >>> b = np.random.rand(4, 4, 4) >>> __parse_einsum_input(('...a,...a->...', a, b)) ('za,xza', 'xz', [a, b]) >>> __parse_einsum_input((a, [Ellipsis, 0], b, [Ellipsis, 0])) ('za,xza', 'xz', [a, b]) r zNo input operands� � c S s g | ] }t |��qS r )r )�.0�vr r r � <listcomp>" s z'_parse_einsum_input.<locals>.<listcomp>r Nz.,->z#Character %s is not a valid symbol.r! ���c S s g | ] }t |��qS r )r )r; r< r r r r= 4 s z...z=For this input type lists must contain either int or Ellipsis�,z->�-�>z%Subscripts can only contain one '->'.�.TF� zInvalid Ellipses.r z Ellipses lengths do not match.z/Output character %s did not appear in the inputzDNumber of einsum subscripts must be equal to the number of operands.)r' � ValueError� isinstance�str�replace�einsum_symbols�listr& r �popr �Ellipsis�int� TypeError�count�einsum_symbols_setr �join�split�shape�max�ndim�sorted) �operands� subscripts�s� tmp_operandsZ operand_listZsubscript_list�pZ output_list�last�num�sub�invalid�usedZunusedZ ellipse_inds�longestZ input_tmpZ output_subZsplit_subscriptsZout_subZ ellipse_countZrep_indsZ out_ellipse�output_subscriptZtmp_subscriptsZ normal_inds�input_subscripts�charr r r �_parse_einsum_input s� rd c 5 s@ ddg��fdd�|� � D �}t|�r2td| ��|�dd�}|dkrJd}|d krVd}d }|dks�t|t�rnnht|�r�|d d kr�nRt|�d kr�t|d t�r�t|d ttf�r�t|d �}|d }ntdt|� ��|�dd�}t| �\}}} |d | }|� d�} dd� | D �} t |�} t |� dd��} i � x�t | �D ]�\} }| | j }t|�t|�k�rntd|| | ��xPt |�D ]D\}}|| }|� �� k�r�� | |k�r�td|| ��n|� |< �qxW �q:W g }x$| |g D ]}|�t|� �� �q�W t|�}|d k�r|}n|}t| � �}|� dd�}tt| �d d �}t|�tt |�� �rR|d 9 }||9 }|dk�s|t| �dk�s|| | k�r�ttt| ���g}nd|dk�r�t||�}t| | � |�}n@|dk�r�t| | � |�}n&|d d k�r�|d d � }n td|��g g g g f\}}}} �x$t |�D �]\}} ttt| �dd��} t| | | �} | \} } } }!t|!� �}"| �rb|"d 9 }"|�|"� |�t|!�� |�t| � �� g }#x | D ]}$|#�| �|$�� �q�W |t|� dk�r�|}%n*� fdd�| D �}&d� dd� t|&�D ��}%| �|%� d� |#�d |% }'| | |'| d d � f}(| �|(� �qW t |�d })|�rJ| | fS |d | }*d }+||) },t|�}-d |* }.|.d t| � 7 }.|.d t|� 7 }.|.d | 7 }.|.d!|) 7 }.|.d"|, 7 }.|.d#|- 7 }.|.d$7 }.|.d%|+ 7 }.|.d&7 }.xNt | �D ]B\}/}(|(\}0}1}'}2d� |2�d | }3||/ |'|3f}4|.d'|4 7 }.�q�W d g| }||.fS )(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 choosest 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 reprsented 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. >>> 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 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 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 �optimize� einsum_callc s g | ]\}}|� kr|�qS r r )r; �kr< )�valid_contract_kwargsr r r= s z einsum_path.<locals>.<listcomp>z+Did not understand the following kwargs: %sFTZgreedyNr r r! r z Did not understand the path: %sz->r? c S s g | ] }t |��qS r )r )r; r" r r r r= C s r: zXEinstein sum subscript %s does not contain the correct number of indices for operand %d.z@Size of label '%s' for operand %d does not match previous terms.)r r! ZoptimalzPath name %s not found)�reverser> c s g | ]}� | |f�qS r r )r; r )�dimension_dictr r r= � s c S s g | ] }|d �qS )r r )r; r" r r r r= � s )Zscaling�currentr z Complete contraction: %s z Naive scaling: %d z Optimized scaling: %d z Naive FLOP count: %.3e z Optimized FLOP count: %.3e z Theoretical speedup: %3.3f z' Largest intermediate: %.3e elements zK-------------------------------------------------------------------------- z%6s %24s %40s zJ--------------------------------------------------------------------------z %4d %24s %40s) �itemsr' rM rJ rE rF rL �floatrd rQ r rG r rR rD �keysr r rS r( r& r) r8 r3 �KeyErrorrU rI r rP �sum)5rV �kwargs�unknown_kwargs� path_typer* Zeinsum_call_argrb ra rW Z input_listr r r ZtnumZterm�shZcnumrc ZdimZ size_list�max_sizeZ memory_argZ naive_costZindices_in_inputZmultr2 Z cost_listZ scale_list�contraction_listZ contract_indsr4 Zout_indsr r r/ Z tmp_inputsr" r5 Z sort_result� einsum_str� contractionZopt_costZoverall_contraction�headerZspeedupZmax_iZ path_print�n�inds�idx_rmr Z remaining_strZpath_runr )rj rh r r � s� n " c s< |� dd�}|dkr t| |�S ddddg��fdd�|�� D �}dg� � � fd d �|�� D �}t|�rttd | ��d}|� dd �}|d k r�d }t| |d d��\} }x�t|�D ]t\}} | \} } } } g }x | D ]}|�| � |�� q�W |�r|d t|�k�r||d< t| f|�|�}| �|� ~~q�W |�r0|S | d S d S )aM! 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.]]) re F�outZdtype�orderZcastingc s i | ]\}}|� kr||�qS r r )r; rg r< )�valid_einsum_kwargsr r � <dictcomp>� s zeinsum.<locals>.<dictcomp>c s g | ]\}}|� kr|�qS r r )r; rg r< )rh r r r= � s zeinsum.<locals>.<listcomp>z+Did not understand the following kwargs: %sNT)re rf r r )rJ r rl r' rM r r r )rV rq Z optimize_argZ einsum_kwargsrr Z specified_outZ out_arrayrv r\ rx r{ r| rw r rY r" Znew_viewr )rh r r r � s< r N)�__doc__� __future__r r r Znumpy.core.multiarrayr Znumpy.core.numericr r r �__all__rH r rO r r r3 r8 rd r r r r r r �<module> s <KO )
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