# Input Format

The opt_einsum package was originally designed as a drop-in replacement for the np.einsum function and supports all input formats that np.einsum supports. There are two styles of input accepted, a basic introduction to which can be found in the documentation for numpy.einsum. In addition to this, opt_einsum extends the allowed index labels to unicode or arbitrary hashable, comparable objects in order to handle large contractions with many indices.

## 'Equation' Input

As with numpy.einsum, here you specify an equation as a string, followed by the array arguments:

import opt_einsum as oe
eq = 'ijk,jkl->li'
x, y = np.random.rand(2, 3, 4), np.random.rand(3, 4, 5)
z = oe.contract(eq, x, y)
z.shape
#> (5, 2)


However, in addition to the standard alphabet, opt_einsum also supports unicode characters:

eq = "αβγ,βγδ->δα"
oe.contract(eq, x, y).shape
#> (5, 2)


This enables access to thousands of possible index labels. One way to access these programmatically is through the function get_symbols:

oe.get_symbol(805)
#> 'α'


which maps an int to a unicode characater. Note that as with numpy.einsum if the output is not specified with -> it will default to the sorted order of all indices appearing once:

eq = "αβγ,βγδ"  # "->αδ" is implicit
oe.contract(eq, x, y).shape
#> (2, 5)


## 'Interleaved' Input

The other input format is to 'interleave' the array arguments with their index labels ('subscripts') in pairs, optionally specifying the output indices as a final argument. As with numpy.einsum, integers are allowed as these index labels:

oe.contract(x, [1, 2, 3], y, [2, 3, 4], [4, 1]).shape
#> (5, 2)


with the default output order again specified by the sorted order of indices appearing once. However, unlike numpy.einsum, in opt_einsum you can also put anything hashable and comparable such as str in the subscript list. A simple example of this syntax is:

x, y, z = np.ones((1, 2)), np.ones((2, 2)), np.ones((2, 1))
oe.contract(x, ('left', 'bond1'), y, ('bond1', 'bond2'), z, ('bond2', 'right'), ('left', 'right'))
#> array([[4.]])


The subscripts need to be hashable so that opt_einsum can efficiently process them, and they should also be comparable so as to allow a default sorted output. For example:

x = np.array([[0, 1], [2, 0]])

# original matrix
oe.contract(x, (0, 1))
#> array([[0, 1],
#>        [2, 0]])

# the transpose
oe.contract(x, (1, 0))
#> array([[0, 2],
#>        [1, 0]])

# original matrix, consistent behavior
oe.contract(x, ('a', 'b'))
#> array([[0, 1],
#>        [2, 0]])

# the transpose, consistent behavior
>>> oe.contract(x, ('b', 'a'))
#> array([[0, 2],
#>        [1, 0]])

# relative sequence undefined, can't determine output
>>> oe.contract(x, (0, 'a'))
#> TypeError: For this input type lists must contain either Ellipsis
#> or hashable and comparable object (e.g. int, str)