linear_algebra_functions.py

from ._types import Tuple, Union, Sequence, array


def matmul(x1: array, x2: array, /) -> array:
    """
    Computes the matrix product.

    .. note::
       The ``matmul`` function must implement the same semantics as the built-in ``@`` operator (see `PEP 465 <https://www.python.org/dev/peps/pep-0465>`_).

    Parameters
    ----------
    x1: array
        first input array. Should have a numeric data type. Must have at least one dimension. If ``x1`` is one-dimensional having shape ``(M,)`` and ``x2`` has more than one dimension, ``x1`` must be promoted to a two-dimensional array by prepending ``1`` to its dimensions (i.e., must have shape ``(1, M)``). After matrix multiplication, the prepended dimensions in the returned array must be removed. If ``x1`` has more than one dimension (including after vector-to-matrix promotion), ``shape(x1)[:-2]`` must be compatible with ``shape(x2)[:-2]`` (after vector-to-matrix promotion) (see :ref:`broadcasting`). If ``x1`` has shape ``(..., M, K)``, the innermost two dimensions form matrices on which to perform matrix multiplication.
    x2: array
        second input array. Should have a numeric data type. Must have at least one dimension. If ``x2`` is one-dimensional having shape ``(N,)`` and ``x1`` has more than one dimension, ``x2`` must be promoted to a two-dimensional array by appending ``1`` to its dimensions (i.e., must have shape ``(N, 1)``). After matrix multiplication, the appended dimensions in the returned array must be removed. If ``x2`` has more than one dimension (including after vector-to-matrix promotion), ``shape(x2)[:-2]`` must be compatible with ``shape(x1)[:-2]`` (after vector-to-matrix promotion) (see :ref:`broadcasting`). If ``x2`` has shape ``(..., K, N)``, the innermost two dimensions form matrices on which to perform matrix multiplication.


    .. note::
       If either ``x1`` or ``x2`` has a complex floating-point data type, neither argument must be complex-conjugated or transposed. If conjugation and/or transposition is desired, these operations should be explicitly performed prior to computing the matrix product.

    Returns
    -------
    out: array
        -   if both ``x1`` and ``x2`` are one-dimensional arrays having shape ``(N,)``, a zero-dimensional array containing the inner product as its only element.
        -   if ``x1`` is a two-dimensional array having shape ``(M, K)`` and ``x2`` is a two-dimensional array having shape ``(K, N)``, a two-dimensional array containing the `conventional matrix product <https://en.wikipedia.org/wiki/Matrix_multiplication>`_ and having shape ``(M, N)``.
        -   if ``x1`` is a one-dimensional array having shape ``(K,)`` and ``x2`` is an array having shape ``(..., K, N)``, an array having shape ``(..., N)`` (i.e., prepended dimensions during vector-to-matrix promotion must be removed) and containing the `conventional matrix product <https://en.wikipedia.org/wiki/Matrix_multiplication>`_.
        -   if ``x1`` is an array having shape ``(..., M, K)`` and ``x2`` is a one-dimensional array having shape ``(K,)``, an array having shape ``(..., M)`` (i.e., appended dimensions during vector-to-matrix promotion must be removed) and containing the `conventional matrix product <https://en.wikipedia.org/wiki/Matrix_multiplication>`_.
        -   if ``x1`` is a two-dimensional array having shape ``(M, K)`` and ``x2`` is an array having shape ``(..., K, N)``, an array having shape ``(..., M, N)`` and containing the `conventional matrix product <https://en.wikipedia.org/wiki/Matrix_multiplication>`_ for each stacked matrix.
        -   if ``x1`` is an array having shape ``(..., M, K)`` and ``x2`` is a two-dimensional array having shape ``(K, N)``, an array having shape ``(..., M, N)`` and containing the `conventional matrix product <https://en.wikipedia.org/wiki/Matrix_multiplication>`_ for each stacked matrix.
        -   if either ``x1`` or ``x2`` has more than two dimensions, an array having a shape determined by :ref:`broadcasting` ``shape(x1)[:-2]`` against ``shape(x2)[:-2]`` and containing the `conventional matrix product <https://en.wikipedia.org/wiki/Matrix_multiplication>`_ for each stacked matrix.

        The returned array must have a data type determined by :ref:`type-promotion`.

    Notes
    -----

    .. versionchanged:: 2022.12
       Added complex data type support.

    **Raises**

    -   if either ``x1`` or ``x2`` is a zero-dimensional array.
    -   if ``x1`` is a one-dimensional array having shape ``(K,)``, ``x2`` is a one-dimensional array having shape ``(L,)``, and ``K != L``.
    -   if ``x1`` is a one-dimensional array having shape ``(K,)``, ``x2`` is an array having shape ``(..., L, N)``, and ``K != L``.
    -   if ``x1`` is an array having shape ``(..., M, K)``, ``x2`` is a one-dimensional array having shape ``(L,)``, and ``K != L``.
    -   if ``x1`` is an array having shape ``(..., M, K)``, ``x2`` is an array having shape ``(..., L, N)``, and ``K != L``.

    """


def matrix_transpose(x: array, /) -> array:
    """
    Transposes a matrix (or a stack of matrices) ``x``.

    Parameters
    ----------
    x: array
        input array having shape ``(..., M, N)`` and whose innermost two dimensions form ``MxN`` matrices.

    Returns
    -------
    out: array
        an array containing the transpose for each matrix and having shape ``(..., N, M)``. The returned array must have the same data type as ``x``.
    """


def tensordot(
    x1: array,
    x2: array,
    /,
    *,
    axes: Union[int, Tuple[Sequence[int], Sequence[int]]] = 2,
) -> array:
    """
    Returns a tensor contraction of ``x1`` and ``x2`` over specific axes.

    .. note::
       The ``tensordot`` function corresponds to the generalized matrix product.

    Parameters
    ----------
    x1: array
        first input array. Should have a numeric data type.
    x2: array
        second input array. Should have a numeric data type. Corresponding contracted axes of ``x1`` and ``x2`` must be equal.

        .. note::
           Contracted axes (dimensions) must not be broadcasted.

    axes: Union[int, Tuple[Sequence[int], Sequence[int]]]
        number of axes (dimensions) to contract or explicit sequences of axis (dimension) indices for ``x1`` and ``x2``, respectively.

        If ``axes`` is an ``int`` equal to ``N``, then contraction must be performed over the last ``N`` axes of ``x1`` and the first ``N`` axes of ``x2`` in order. The size of each corresponding axis (dimension) must match. Must be nonnegative.

        -   If ``N`` equals ``0``, the result is the tensor (outer) product.
        -   If ``N`` equals ``1``, the result is the tensor dot product.
        -   If ``N`` equals ``2``, the result is the tensor double contraction (default).

        If ``axes`` is a tuple of two sequences ``(x1_axes, x2_axes)``, the first sequence must apply to ``x1`` and the second sequence to ``x2``. Both sequences must have the same length. Each axis (dimension) ``x1_axes[i]`` for ``x1`` must have the same size as the respective axis (dimension) ``x2_axes[i]`` for ``x2``. Each index referred to in a sequence must be unique. If ``x1`` has rank (i.e, number of dimensions) ``N``, a valid ``x1`` axis must reside on the half-open interval ``[-N, N)``. If ``x2`` has rank ``M``, a valid ``x2`` axis must reside on the half-open interval ``[-M, M)``.


    .. note::
       If either ``x1`` or ``x2`` has a complex floating-point data type, neither argument must be complex-conjugated or transposed. If conjugation and/or transposition is desired, these operations should be explicitly performed prior to computing the generalized matrix product.

    Returns
    -------
    out: array
        an array containing the tensor contraction whose shape consists of the non-contracted axes (dimensions) of the first array ``x1``, followed by the non-contracted axes (dimensions) of the second array ``x2``. The returned array must have a data type determined by :ref:`type-promotion`.

    Notes
    -----

    .. versionchanged:: 2022.12
       Added complex data type support.
    """


def vecdot(x1: array, x2: array, /, *, axis: int = -1) -> array:
    r"""
    Computes the (vector) dot product of two arrays.

    Let :math:`\mathbf{a}` be a vector in ``x1`` and :math:`\mathbf{b}` be a corresponding vector in ``x2``. The dot product is defined as

    .. math::
       \mathbf{a} \cdot \mathbf{b} = \sum_{i=0}^{n-1} a_i\overline{b_i}

    over the dimension specified by ``axis`` and where :math:`n` is the dimension size and :math:`\overline{b_i}` denotes the complex conjugate if :math:`b_i` is complex and the identity if :math:`b_i` is real-valued.

    Parameters
    ----------
    x1: array
        first input array. Should have a floating-point data type.
    x2: array
        second input array. Must be compatible with ``x1`` for all non-contracted axes (see :ref:`broadcasting`). The size of the axis over which to compute the dot product must be the same size as the respective axis in ``x1``. Should have a floating-point data type.

        .. note::
           The contracted axis (dimension) must not be broadcasted.

    axis: int
        axis over which to compute the dot product. Must be an integer on the interval ``[-N, N)``, where ``N`` is the rank (number of dimensions) of the shape determined according to :ref:`broadcasting`. If specified as a negative integer, the function must determine the axis along which to compute the dot product by counting backward from the last dimension (where ``-1`` refers to the last dimension). By default, the function must compute the dot product over the last axis. Default: ``-1``.

    Returns
    -------
    out: array
        if ``x1`` and ``x2`` are both one-dimensional arrays, a zero-dimensional containing the dot product; otherwise, a non-zero-dimensional array containing the dot products and having rank ``N-1``, where ``N`` is the rank (number of dimensions) of the shape determined according to :ref:`broadcasting` along the non-contracted axes. The returned array must have a data type determined by :ref:`type-promotion`.

    Notes
    -----

    .. versionchanged:: 2022.12
       Added complex data type support.

    **Raises**

    -   if the size of the axis over which to compute the dot product is not the same (before broadcasting) for both ``x1`` and ``x2``.
    """


__all__ = ["matmul", "matrix_transpose", "tensordot", "vecdot"]