# 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_symbol():

>>> 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]])
>>> oe.contract(x, (0, 1))  # original matrix
array([[0, 1],
[2, 0]])
>>> oe.contract(x, (1, 0)) # the transpose
array([[0, 2],
[1, 0]])
>>> oe.contract(x, ('a', 'b'))  # original matrix, consistent behavior
array([[0, 1],
[2, 0]])
>>> oe.contract(x, ('b', 'a')) # the transpose, consistent behavior
array([[0, 2],
[1, 0]])
>>> oe.contract(x, (0, 'a')) # relative sequence undefined, can't determine output
TypeError: For this input type lists must contain either Ellipsis or hashable and comparable object (e.g. int, str)