Module « numpy.linalg »

Fonction svd - module numpy.linalg

Signature de la fonction svd

def svd(a, full_matrices=True, compute_uv=True, hermitian=False) 

Description

svd.__doc__

    Singular Value Decomposition.

    When `a` is a 2D array, it is factorized as ``u @ np.diag(s) @ vh
    = (u * s) @ vh``, where `u` and `vh` are 2D unitary arrays and `s` is a 1D
    array of `a`'s singular values. When `a` is higher-dimensional, SVD is
    applied in stacked mode as explained below.

    Parameters
    ----------
    a : (..., M, N) array_like
        A real or complex array with ``a.ndim >= 2``.
    full_matrices : bool, optional
        If True (default), `u` and `vh` have the shapes ``(..., M, M)`` and
        ``(..., N, N)``, respectively.  Otherwise, the shapes are
        ``(..., M, K)`` and ``(..., K, N)``, respectively, where
        ``K = min(M, N)``.
    compute_uv : bool, optional
        Whether or not to compute `u` and `vh` in addition to `s`.  True
        by default.
    hermitian : bool, optional
        If True, `a` is assumed to be Hermitian (symmetric if real-valued),
        enabling a more efficient method for finding singular values.
        Defaults to False.

        .. versionadded:: 1.17.0

    Returns
    -------
    u : { (..., M, M), (..., M, K) } array
        Unitary array(s). The first ``a.ndim - 2`` dimensions have the same
        size as those of the input `a`. The size of the last two dimensions
        depends on the value of `full_matrices`. Only returned when
        `compute_uv` is True.
    s : (..., K) array
        Vector(s) with the singular values, within each vector sorted in
        descending order. The first ``a.ndim - 2`` dimensions have the same
        size as those of the input `a`.
    vh : { (..., N, N), (..., K, N) } array
        Unitary array(s). The first ``a.ndim - 2`` dimensions have the same
        size as those of the input `a`. The size of the last two dimensions
        depends on the value of `full_matrices`. Only returned when
        `compute_uv` is True.

    Raises
    ------
    LinAlgError
        If SVD computation does not converge.

    See Also
    --------
    scipy.linalg.svd : Similar function in SciPy.
    scipy.linalg.svdvals : Compute singular values of a matrix.

    Notes
    -----

    .. versionchanged:: 1.8.0
       Broadcasting rules apply, see the `numpy.linalg` documentation for
       details.

    The decomposition is performed using LAPACK routine ``_gesdd``.

    SVD is usually described for the factorization of a 2D matrix :math:`A`.
    The higher-dimensional case will be discussed below. In the 2D case, SVD is
    written as :math:`A = U S V^H`, where :math:`A = a`, :math:`U= u`,
    :math:`S= \mathtt{np.diag}(s)` and :math:`V^H = vh`. The 1D array `s`
    contains the singular values of `a` and `u` and `vh` are unitary. The rows
    of `vh` are the eigenvectors of :math:`A^H A` and the columns of `u` are
    the eigenvectors of :math:`A A^H`. In both cases the corresponding
    (possibly non-zero) eigenvalues are given by ``s**2``.

    If `a` has more than two dimensions, then broadcasting rules apply, as
    explained in :ref:`routines.linalg-broadcasting`. This means that SVD is
    working in "stacked" mode: it iterates over all indices of the first
    ``a.ndim - 2`` dimensions and for each combination SVD is applied to the
    last two indices. The matrix `a` can be reconstructed from the
    decomposition with either ``(u * s[..., None, :]) @ vh`` or
    ``u @ (s[..., None] * vh)``. (The ``@`` operator can be replaced by the
    function ``np.matmul`` for python versions below 3.5.)

    If `a` is a ``matrix`` object (as opposed to an ``ndarray``), then so are
    all the return values.

    Examples
    --------
    >>> a = np.random.randn(9, 6) + 1j*np.random.randn(9, 6)
    >>> b = np.random.randn(2, 7, 8, 3) + 1j*np.random.randn(2, 7, 8, 3)

    Reconstruction based on full SVD, 2D case:

    >>> u, s, vh = np.linalg.svd(a, full_matrices=True)
    >>> u.shape, s.shape, vh.shape
    ((9, 9), (6,), (6, 6))
    >>> np.allclose(a, np.dot(u[:, :6] * s, vh))
    True
    >>> smat = np.zeros((9, 6), dtype=complex)
    >>> smat[:6, :6] = np.diag(s)
    >>> np.allclose(a, np.dot(u, np.dot(smat, vh)))
    True

    Reconstruction based on reduced SVD, 2D case:

    >>> u, s, vh = np.linalg.svd(a, full_matrices=False)
    >>> u.shape, s.shape, vh.shape
    ((9, 6), (6,), (6, 6))
    >>> np.allclose(a, np.dot(u * s, vh))
    True
    >>> smat = np.diag(s)
    >>> np.allclose(a, np.dot(u, np.dot(smat, vh)))
    True

    Reconstruction based on full SVD, 4D case:

    >>> u, s, vh = np.linalg.svd(b, full_matrices=True)
    >>> u.shape, s.shape, vh.shape
    ((2, 7, 8, 8), (2, 7, 3), (2, 7, 3, 3))
    >>> np.allclose(b, np.matmul(u[..., :3] * s[..., None, :], vh))
    True
    >>> np.allclose(b, np.matmul(u[..., :3], s[..., None] * vh))
    True

    Reconstruction based on reduced SVD, 4D case:

    >>> u, s, vh = np.linalg.svd(b, full_matrices=False)
    >>> u.shape, s.shape, vh.shape
    ((2, 7, 8, 3), (2, 7, 3), (2, 7, 3, 3))
    >>> np.allclose(b, np.matmul(u * s[..., None, :], vh))
    True
    >>> np.allclose(b, np.matmul(u, s[..., None] * vh))
    True