"""
Implementation of optimized einsum.
"""
from __future__ import division, absolute_import, print_function
from numpy.compat import basestring
from numpy.core.multiarray import c_einsum
from numpy.core.numeric import asarray, asanyarray, result_type, tensordot, dot
__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 combinatorial list, if we did not find anything return best
# path + remaining contractions
if iter_results:
full_results = iter_results
else:
path = min(full_results, key=lambda x: x[0])[1]
path += [tuple(range(len(input_sets) - iteration))]
return path
# 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 _can_dot(inputs, result, idx_removed):
"""
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
"""
# All `dot` calls remove indices
if len(idx_removed) == 0:
return False
# BLAS can only handle two operands
if len(inputs) != 2:
return False
# Build a few temporaries
input_left, input_right = inputs
set_left = set(input_left)
set_right = set(input_right)
keep_left = set_left - idx_removed
keep_right = set_right - idx_removed
rs = len(idx_removed)
# Indices must overlap between the two operands
if not len(set_left & set_right):
return False
# We cannot have duplicate indices ("ijj, jk -> ik")
if (len(set_left) != len(input_left)) or (len(set_right) != len(input_right)):
return False
# Cannot handle partial inner ("ij, ji -> i")
if len(keep_left & keep_right):
return False
# At this point we are a DOT, GEMV, or GEMM operation
# Handle inner products
# DDOT with aligned data
if input_left == input_right:
return True
# DDOT without aligned data (better to use einsum)
if set_left == set_right:
return False
# Handle the 4 possible (aligned) GEMV or GEMM cases
# GEMM or GEMV no transpose
if input_left[-rs:] == input_right[:rs]:
return True
# GEMM or GEMV transpose both
if input_left[:rs] == input_right[-rs:]:
return True
# GEMM or GEMV transpose right
if input_left[-rs:] == input_right[-rs:]:
return True
# GEMM or GEMV transpose left
if input_left[:rs] == input_right[:rs]:
return True
# Einsum is faster than GEMV if we have to copy data
if not keep_left or not keep_right:
return False
# We are a matrix-matrix product, but we need to copy data
return True
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], basestring):
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', True)
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, basestring):
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], basestring) 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():
# For broadcasting cases we always want the largest dim size
if dimension_dict[char] == 1:
dimension_dict[char] = dim
elif dim not in (1, dimension_dict[char]):
raise ValueError("Size of label '%s' for operand %d (%d) "
"does not match previous terms (%d)."
% (char, tnum, dimension_dict[char], dim))
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))
do_blas = _can_dot(tmp_inputs, out_inds, idx_removed)
# 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[:], do_blas)
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, blas = 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)
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] or 1 in tmp_operands[1]:
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 derefernce what we can
operands.append(new_view)
del tmp_operands, new_view
if specified_out:
return out_array
else:
return operands[0]