Module « scipy.linalg »

Fonction svd - module scipy.linalg

Signature de la fonction svd

def svd(a, full_matrices=True, compute_uv=True, overwrite_a=False, check_finite=True, lapack_driver='gesdd') 

Description

svd.__doc__

    Singular Value Decomposition.

    Factorizes the matrix `a` into two unitary matrices ``U`` and ``Vh``, and
    a 1-D array ``s`` of singular values (real, non-negative) such that
    ``a == U @ S @ Vh``, where ``S`` is a suitably shaped matrix of zeros with
    main diagonal ``s``.

    Parameters
    ----------
    a : (M, N) array_like
        Matrix to decompose.
    full_matrices : bool, optional
        If True (default), `U` and `Vh` are of shape ``(M, M)``, ``(N, N)``.
        If False, the shapes are ``(M, K)`` and ``(K, N)``, where
        ``K = min(M, N)``.
    compute_uv : bool, optional
        Whether to compute also ``U`` and ``Vh`` in addition to ``s``.
        Default is True.
    overwrite_a : bool, optional
        Whether to overwrite `a`; may improve performance.
        Default is False.
    check_finite : bool, optional
        Whether to check that the input matrix contains only finite numbers.
        Disabling may give a performance gain, but may result in problems
        (crashes, non-termination) if the inputs do contain infinities or NaNs.
    lapack_driver : {'gesdd', 'gesvd'}, optional
        Whether to use the more efficient divide-and-conquer approach
        (``'gesdd'``) or general rectangular approach (``'gesvd'``)
        to compute the SVD. MATLAB and Octave use the ``'gesvd'`` approach.
        Default is ``'gesdd'``.

        .. versionadded:: 0.18

    Returns
    -------
    U : ndarray
        Unitary matrix having left singular vectors as columns.
        Of shape ``(M, M)`` or ``(M, K)``, depending on `full_matrices`.
    s : ndarray
        The singular values, sorted in non-increasing order.
        Of shape (K,), with ``K = min(M, N)``.
    Vh : ndarray
        Unitary matrix having right singular vectors as rows.
        Of shape ``(N, N)`` or ``(K, N)`` depending on `full_matrices`.

    For ``compute_uv=False``, only ``s`` is returned.

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

    See also
    --------
    svdvals : Compute singular values of a matrix.
    diagsvd : Construct the Sigma matrix, given the vector s.

    Examples
    --------
    >>> from scipy import linalg
    >>> from numpy.random import default_rng
    >>> rng = default_rng()
    >>> m, n = 9, 6
    >>> a = rng.standard_normal((m, n)) + 1.j*rng.standard_normal((m, n))
    >>> U, s, Vh = linalg.svd(a)
    >>> U.shape,  s.shape, Vh.shape
    ((9, 9), (6,), (6, 6))

    Reconstruct the original matrix from the decomposition:

    >>> sigma = np.zeros((m, n))
    >>> for i in range(min(m, n)):
    ...     sigma[i, i] = s[i]
    >>> a1 = np.dot(U, np.dot(sigma, Vh))
    >>> np.allclose(a, a1)
    True

    Alternatively, use ``full_matrices=False`` (notice that the shape of
    ``U`` is then ``(m, n)`` instead of ``(m, m)``):

    >>> U, s, Vh = linalg.svd(a, full_matrices=False)
    >>> U.shape, s.shape, Vh.shape
    ((9, 6), (6,), (6, 6))
    >>> S = np.diag(s)
    >>> np.allclose(a, np.dot(U, np.dot(S, Vh)))
    True

    >>> s2 = linalg.svd(a, compute_uv=False)
    >>> np.allclose(s, s2)
    True