# Input Format¶

The `opt_einsum`

package is 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:

```
>>> 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)
```