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Module « numpy.linalg »

Fonction tensordot - module numpy.linalg

Signature de la fonction tensordot

def tensordot(x1, x2, /, *, axes=2) 

Description

help(numpy.linalg.tensordot)

Compute tensor dot product along specified axes.

Given two tensors, `a` and `b`, and an array_like object containing
two array_like objects, ``(a_axes, b_axes)``, sum the products of
`a`'s and `b`'s elements (components) over the axes specified by
``a_axes`` and ``b_axes``. The third argument can be a single non-negative
integer_like scalar, ``N``; if it is such, then the last ``N`` dimensions
of `a` and the first ``N`` dimensions of `b` are summed over.

Parameters
----------
a, b : array_like
    Tensors to "dot".

axes : int or (2,) array_like
    * integer_like
      If an int N, sum over the last N axes of `a` and the first N axes
      of `b` in order. The sizes of the corresponding axes must match.
    * (2,) array_like
      Or, a list of axes to be summed over, first sequence applying to `a`,
      second to `b`. Both elements array_like must be of the same length.

Returns
-------
output : ndarray
    The tensor dot product of the input.

See Also
--------
dot, einsum

Notes
-----
Three common use cases are:
    * ``axes = 0`` : tensor product :math:`a\otimes b`
    * ``axes = 1`` : tensor dot product :math:`a\cdot b`
    * ``axes = 2`` : (default) tensor double contraction :math:`a:b`

When `axes` is integer_like, the sequence of axes for evaluation
will be: from the -Nth axis to the -1th axis in `a`,
and from the 0th axis to (N-1)th axis in `b`.
For example, ``axes = 2`` is the equal to
``axes = [[-2, -1], [0, 1]]``.
When N-1 is smaller than 0, or when -N is larger than -1,
the element of `a` and `b` are defined as the `axes`.

When there is more than one axis to sum over - and they are not the last
(first) axes of `a` (`b`) - the argument `axes` should consist of
two sequences of the same length, with the first axis to sum over given
first in both sequences, the second axis second, and so forth.
The calculation can be referred to ``numpy.einsum``.

The shape of the result consists of the non-contracted axes of the
first tensor, followed by the non-contracted axes of the second.

Examples
-------- 
An example on integer_like:

>>> a_0 = np.array([[1, 2], [3, 4]])
>>> b_0 = np.array([[5, 6], [7, 8]])
>>> c_0 = np.tensordot(a_0, b_0, axes=0)
>>> c_0.shape
(2, 2, 2, 2)
>>> c_0
array([[[[ 5,  6],
         [ 7,  8]],
        [[10, 12],
         [14, 16]]],
       [[[15, 18],
         [21, 24]],
        [[20, 24],
         [28, 32]]]])

An example on array_like:

>>> a = np.arange(60.).reshape(3,4,5)
>>> b = np.arange(24.).reshape(4,3,2)
>>> c = np.tensordot(a,b, axes=([1,0],[0,1]))
>>> c.shape
(5, 2)
>>> c
array([[4400., 4730.],
       [4532., 4874.],
       [4664., 5018.],
       [4796., 5162.],
       [4928., 5306.]])
       
A slower but equivalent way of computing the same...

>>> d = np.zeros((5,2))
>>> for i in range(5):
...   for j in range(2):
...     for k in range(3):
...       for n in range(4):
...         d[i,j] += a[k,n,i] * b[n,k,j]
>>> c == d
array([[ True,  True],
       [ True,  True],
       [ True,  True],
       [ True,  True],
       [ True,  True]])

An extended example taking advantage of the overloading of + and \*:

>>> a = np.array(range(1, 9))
>>> a.shape = (2, 2, 2)
>>> A = np.array(('a', 'b', 'c', 'd'), dtype=object)
>>> A.shape = (2, 2)
>>> a; A
array([[[1, 2],
        [3, 4]],
       [[5, 6],
        [7, 8]]])
array([['a', 'b'],
       ['c', 'd']], dtype=object)

>>> np.tensordot(a, A) # third argument default is 2 for double-contraction
array(['abbcccdddd', 'aaaaabbbbbbcccccccdddddddd'], dtype=object)

>>> np.tensordot(a, A, 1)
array([[['acc', 'bdd'],
        ['aaacccc', 'bbbdddd']],
       [['aaaaacccccc', 'bbbbbdddddd'],
        ['aaaaaaacccccccc', 'bbbbbbbdddddddd']]], dtype=object)

>>> np.tensordot(a, A, 0) # tensor product (result too long to incl.)
array([[[[['a', 'b'],
          ['c', 'd']],
          ...

>>> np.tensordot(a, A, (0, 1))
array([[['abbbbb', 'cddddd'],
        ['aabbbbbb', 'ccdddddd']],
       [['aaabbbbbbb', 'cccddddddd'],
        ['aaaabbbbbbbb', 'ccccdddddddd']]], dtype=object)

>>> np.tensordot(a, A, (2, 1))
array([[['abb', 'cdd'],
        ['aaabbbb', 'cccdddd']],
       [['aaaaabbbbbb', 'cccccdddddd'],
        ['aaaaaaabbbbbbbb', 'cccccccdddddddd']]], dtype=object)

>>> np.tensordot(a, A, ((0, 1), (0, 1)))
array(['abbbcccccddddddd', 'aabbbbccccccdddddddd'], dtype=object)

>>> np.tensordot(a, A, ((2, 1), (1, 0)))
array(['acccbbdddd', 'aaaaacccccccbbbbbbdddddddd'], dtype=object)

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