""" Implementation of optimized einsum. """ from __future__ import division, absolute_import, print_function from numpy.core.multiarray import c_einsum from numpy.core.numeric import asarray, asanyarray, result_type __all__ = ['einsum', 'einsum_path'] einsum_symbols = 'abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ' einsum_symbols_set = set(einsum_symbols) def _compute_size_by_dict(indices, idx_dict): """ 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 """ ret = 1 for i in indices: ret *= idx_dict[i] return ret def _find_contraction(positions, input_sets, output_set): """ 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'}) """ idx_contract = set() idx_remain = output_set.copy() remaining = [] for ind, value in enumerate(input_sets): if ind in positions: idx_contract |= value else: remaining.append(value) idx_remain |= value new_result = idx_remain & idx_contract idx_removed = (idx_contract - new_result) remaining.append(new_result) return (new_result, remaining, idx_removed, idx_contract) def _optimal_path(input_sets, output_set, idx_dict, memory_limit): """ 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)] """ full_results = [(0, [], input_sets)] for iteration in range(len(input_sets) - 1): iter_results = [] # Compute all unique pairs comb_iter = [] for x in range(len(input_sets) - iteration): for y in range(x + 1, len(input_sets) - iteration): comb_iter.append((x, y)) for curr in full_results: cost, positions, remaining = curr for con in comb_iter: # Find the contraction cont = _find_contraction(con, remaining, output_set) new_result, new_input_sets, idx_removed, idx_contract = cont # Sieve the results based on memory_limit new_size = _compute_size_by_dict(new_result, idx_dict) if new_size > memory_limit: continue # Find cost new_cost = _compute_size_by_dict(idx_contract, idx_dict) if idx_removed: new_cost *= 2 # Build (total_cost, positions, indices_remaining) new_cost += cost new_pos = positions + [con] iter_results.append((new_cost, new_pos, new_input_sets)) # Update list to iterate over full_results = iter_results # If we have not found anything return single einsum contraction if len(full_results) == 0: return [tuple(range(len(input_sets)))] path = min(full_results, key=lambda x: x[0])[1] return path def _greedy_path(input_sets, output_set, idx_dict, memory_limit): """ 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)] """ if len(input_sets) == 1: return [(0,)] path = [] for iteration in range(len(input_sets) - 1): iteration_results = [] comb_iter = [] # Compute all unique pairs for x in range(len(input_sets)): for y in range(x + 1, len(input_sets)): comb_iter.append((x, y)) for positions in comb_iter: # Find the contraction contract = _find_contraction(positions, input_sets, output_set) idx_result, new_input_sets, idx_removed, idx_contract = contract # Sieve the results based on memory_limit if _compute_size_by_dict(idx_result, idx_dict) > memory_limit: continue # Build sort tuple removed_size = _compute_size_by_dict(idx_removed, idx_dict) cost = _compute_size_by_dict(idx_contract, idx_dict) sort = (-removed_size, cost) # Add contraction to possible choices iteration_results.append([sort, positions, new_input_sets]) # If we did not find a new contraction contract remaining if len(iteration_results) == 0: path.append(tuple(range(len(input_sets)))) break # Sort based on first index best = min(iteration_results, key=lambda x: x[0]) path.append(best[1]) input_sets = best[2] return path def _parse_einsum_input(operands): """ 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]) """ if len(operands) == 0: raise ValueError("No input operands") if isinstance(operands[0], str): subscripts = operands[0].replace(" ", "") operands = [asanyarray(v) for v in operands[1:]] # Ensure all characters are valid for s in subscripts: if s in '.,->': continue if s not in einsum_symbols: raise ValueError("Character %s is not a valid symbol." % s) else: tmp_operands = list(operands) operand_list = [] subscript_list = [] for p in range(len(operands) // 2): operand_list.append(tmp_operands.pop(0)) subscript_list.append(tmp_operands.pop(0)) output_list = tmp_operands[-1] if len(tmp_operands) else None operands = [asanyarray(v) for v in operand_list] subscripts = "" last = len(subscript_list) - 1 for num, sub in enumerate(subscript_list): for s in sub: if s is Ellipsis: subscripts += "..." elif isinstance(s, int): subscripts += einsum_symbols[s] else: raise TypeError("For this input type lists must contain " "either int or Ellipsis") if num != last: subscripts += "," if output_list is not None: subscripts += "->" for s in output_list: if s is Ellipsis: subscripts += "..." elif isinstance(s, int): subscripts += einsum_symbols[s] else: raise TypeError("For this input type lists must contain " "either int or Ellipsis") # Check for proper "->" if ("-" in subscripts) or (">" in subscripts): invalid = (subscripts.count("-") > 1) or (subscripts.count(">") > 1) if invalid or (subscripts.count("->") != 1): raise ValueError("Subscripts can only contain one '->'.") # Parse ellipses if "." in subscripts: used = subscripts.replace(".", "").replace(",", "").replace("->", "") unused = list(einsum_symbols_set - set(used)) ellipse_inds = "".join(unused) longest = 0 if "->" in subscripts: input_tmp, output_sub = subscripts.split("->") split_subscripts = input_tmp.split(",") out_sub = True else: split_subscripts = subscripts.split(',') out_sub = False for num, sub in enumerate(split_subscripts): if "." in sub: if (sub.count(".") != 3) or (sub.count("...") != 1): raise ValueError("Invalid Ellipses.") # Take into account numerical values if operands[num].shape == (): ellipse_count = 0 else: ellipse_count = max(operands[num].ndim, 1) ellipse_count -= (len(sub) - 3) if ellipse_count > longest: longest = ellipse_count if ellipse_count < 0: raise ValueError("Ellipses lengths do not match.") elif ellipse_count == 0: split_subscripts[num] = sub.replace('...', '') else: rep_inds = ellipse_inds[-ellipse_count:] split_subscripts[num] = sub.replace('...', rep_inds) subscripts = ",".join(split_subscripts) if longest == 0: out_ellipse = "" else: out_ellipse = ellipse_inds[-longest:] if out_sub: subscripts += "->" + output_sub.replace("...", out_ellipse) else: # Special care for outputless ellipses output_subscript = "" tmp_subscripts = subscripts.replace(",", "") for s in sorted(set(tmp_subscripts)): if s not in (einsum_symbols): raise ValueError("Character %s is not a valid symbol." % s) if tmp_subscripts.count(s) == 1: output_subscript += s normal_inds = ''.join(sorted(set(output_subscript) - set(out_ellipse))) subscripts += "->" + out_ellipse + normal_inds # Build output string if does not exist if "->" in subscripts: input_subscripts, output_subscript = subscripts.split("->") else: input_subscripts = subscripts # Build output subscripts tmp_subscripts = subscripts.replace(",", "") output_subscript = "" for s in sorted(set(tmp_subscripts)): if s not in einsum_symbols: raise ValueError("Character %s is not a valid symbol." % s) if tmp_subscripts.count(s) == 1: output_subscript += s # Make sure output subscripts are in the input for char in output_subscript: if char not in input_subscripts: raise ValueError("Output character %s did not appear in the input" % char) # Make sure number operands is equivalent to the number of terms if len(input_subscripts.split(',')) != len(operands): raise ValueError("Number of einsum subscripts must be equal to the " "number of operands.") return (input_subscripts, output_subscript, operands) def einsum_path(*operands, **kwargs): """ 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 """ # Make sure all keywords are valid valid_contract_kwargs = ['optimize', 'einsum_call'] 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) # Figure out what the path really is path_type = kwargs.pop('optimize', False) if path_type is True: path_type = 'greedy' if path_type is None: path_type = False memory_limit = None # No optimization or a named path algorithm if (path_type is False) or isinstance(path_type, str): pass # Given an explicit path elif len(path_type) and (path_type[0] == 'einsum_path'): pass # Path tuple with memory limit elif ((len(path_type) == 2) and isinstance(path_type[0], str) and isinstance(path_type[1], (int, float))): memory_limit = int(path_type[1]) path_type = path_type[0] else: raise TypeError("Did not understand the path: %s" % str(path_type)) # Hidden option, only einsum should call this einsum_call_arg = kwargs.pop("einsum_call", False) # Python side parsing input_subscripts, output_subscript, operands = _parse_einsum_input(operands) subscripts = input_subscripts + '->' + output_subscript # Build a few useful list and sets input_list = input_subscripts.split(',') input_sets = [set(x) for x in input_list] output_set = set(output_subscript) indices = set(input_subscripts.replace(',', '')) # Get length of each unique dimension and ensure all dimensions are correct dimension_dict = {} for tnum, term in enumerate(input_list): sh = operands[tnum].shape if len(sh) != len(term): raise ValueError("Einstein sum subscript %s does not contain the " "correct number of indices for operand %d.", input_subscripts[tnum], tnum) for cnum, char in enumerate(term): dim = sh[cnum] if char in dimension_dict.keys(): if dimension_dict[char] != dim: raise ValueError("Size of label '%s' for operand %d does " "not match previous terms.", char, tnum) else: dimension_dict[char] = dim # Compute size of each input array plus the output array size_list = [] for term in input_list + [output_subscript]: size_list.append(_compute_size_by_dict(term, dimension_dict)) max_size = max(size_list) if memory_limit is None: memory_arg = max_size else: memory_arg = memory_limit # Compute naive cost # This isnt quite right, need to look into exactly how einsum does this naive_cost = _compute_size_by_dict(indices, dimension_dict) indices_in_input = input_subscripts.replace(',', '') mult = max(len(input_list) - 1, 1) if (len(indices_in_input) - len(set(indices_in_input))): mult *= 2 naive_cost *= mult # Compute the path if (path_type is False) or (len(input_list) in [1, 2]) or (indices == output_set): # Nothing to be optimized, leave it to einsum path = [tuple(range(len(input_list)))] elif path_type == "greedy": # Maximum memory should be at most out_size for this algorithm memory_arg = min(memory_arg, max_size) path = _greedy_path(input_sets, output_set, dimension_dict, memory_arg) elif path_type == "optimal": path = _optimal_path(input_sets, output_set, dimension_dict, memory_arg) elif path_type[0] == 'einsum_path': path = path_type[1:] else: raise KeyError("Path name %s not found", path_type) cost_list, scale_list, size_list, contraction_list = [], [], [], [] # Build contraction tuple (positions, gemm, einsum_str, remaining) for cnum, contract_inds in enumerate(path): # Make sure we remove inds from right to left contract_inds = tuple(sorted(list(contract_inds), reverse=True)) contract = _find_contraction(contract_inds, input_sets, output_set) out_inds, input_sets, idx_removed, idx_contract = contract cost = _compute_size_by_dict(idx_contract, dimension_dict) if idx_removed: cost *= 2 cost_list.append(cost) scale_list.append(len(idx_contract)) size_list.append(_compute_size_by_dict(out_inds, dimension_dict)) tmp_inputs = [] for x in contract_inds: tmp_inputs.append(input_list.pop(x)) # Last contraction if (cnum - len(path)) == -1: idx_result = output_subscript else: sort_result = [(dimension_dict[ind], ind) for ind in out_inds] idx_result = "".join([x[1] for x in sorted(sort_result)]) input_list.append(idx_result) einsum_str = ",".join(tmp_inputs) + "->" + idx_result contraction = (contract_inds, idx_removed, einsum_str, input_list[:]) contraction_list.append(contraction) opt_cost = sum(cost_list) + 1 if einsum_call_arg: return (operands, contraction_list) # Return the path along with a nice string representation overall_contraction = input_subscripts + "->" + output_subscript header = ("scaling", "current", "remaining") speedup = naive_cost / opt_cost max_i = max(size_list) path_print = " Complete contraction: %s\n" % overall_contraction path_print += " Naive scaling: %d\n" % len(indices) path_print += " Optimized scaling: %d\n" % max(scale_list) path_print += " Naive FLOP count: %.3e\n" % naive_cost path_print += " Optimized FLOP count: %.3e\n" % opt_cost path_print += " Theoretical speedup: %3.3f\n" % speedup path_print += " Largest intermediate: %.3e elements\n" % max_i path_print += "-" * 74 + "\n" path_print += "%6s %24s %40s\n" % header path_print += "-" * 74 for n, contraction in enumerate(contraction_list): inds, idx_rm, einsum_str, remaining = contraction remaining_str = ",".join(remaining) + "->" + output_subscript path_run = (scale_list[n], einsum_str, remaining_str) path_print += "\n%4d %24s %40s" % path_run path = ['einsum_path'] + path return (path, path_print) # Rewrite einsum to handle different cases 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]