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numpy.einsum_path(subscripts, *operands, optimize='greedy')[source]

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


Specifies the subscripts for summation.

*operandslist 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’.

pathlist of tuples

A list representation of the einsum path.


A printable representation of the einsum path.


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.


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