venv/lib/python3.11/site-packages/scipy/linalg/_decomp_qr.py

"""QR decomposition functions."""
import numpy as np

# Local imports
from .lapack import get_lapack_funcs
from ._misc import _datacopied

__all__ = ['qr', 'qr_multiply', 'rq']


def safecall(f, name, *args, **kwargs):
    """Call a LAPACK routine, determining lwork automatically and handling
    error return values"""
    lwork = kwargs.get("lwork", None)
    if lwork in (None, -1):
        kwargs['lwork'] = -1
        ret = f(*args, **kwargs)
        kwargs['lwork'] = ret[-2][0].real.astype(np.int_)
    ret = f(*args, **kwargs)
    if ret[-1] < 0:
        raise ValueError("illegal value in %dth argument of internal %s"
                         % (-ret[-1], name))
    return ret[:-2]


def qr(a, overwrite_a=False, lwork=None, mode='full', pivoting=False,
       check_finite=True):
    """
    Compute QR decomposition of a matrix.

    Calculate the decomposition ``A = Q R`` where Q is unitary/orthogonal
    and R upper triangular.

    Parameters
    ----------
    a : (M, N) array_like
        Matrix to be decomposed
    overwrite_a : bool, optional
        Whether data in `a` is overwritten (may improve performance if
        `overwrite_a` is set to True by reusing the existing input data
        structure rather than creating a new one.)
    lwork : int, optional
        Work array size, lwork >= a.shape[1]. If None or -1, an optimal size
        is computed.
    mode : {'full', 'r', 'economic', 'raw'}, optional
        Determines what information is to be returned: either both Q and R
        ('full', default), only R ('r') or both Q and R but computed in
        economy-size ('economic', see Notes). The final option 'raw'
        (added in SciPy 0.11) makes the function return two matrices
        (Q, TAU) in the internal format used by LAPACK.
    pivoting : bool, optional
        Whether or not factorization should include pivoting for rank-revealing
        qr decomposition. If pivoting, compute the decomposition
        ``A[:, P] = Q @ R`` as above, but where P is chosen such that the
        diagonal of R is non-increasing. Equivalently, albeit less efficiently,
        an explicit P matrix may be formed explicitly by permuting the rows or columns
        (depending on the side of the equation on which it is to be used) of
        an identity matrix. See Examples.
    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.

    Returns
    -------
    Q : float or complex ndarray
        Of shape (M, M), or (M, K) for ``mode='economic'``. Not returned
        if ``mode='r'``. Replaced by tuple ``(Q, TAU)`` if ``mode='raw'``.
    R : float or complex ndarray
        Of shape (M, N), or (K, N) for ``mode in ['economic', 'raw']``.
        ``K = min(M, N)``.
    P : int ndarray
        Of shape (N,) for ``pivoting=True``. Not returned if
        ``pivoting=False``.

    Raises
    ------
    LinAlgError
        Raised if decomposition fails

    Notes
    -----
    This is an interface to the LAPACK routines dgeqrf, zgeqrf,
    dorgqr, zungqr, dgeqp3, and zgeqp3.

    If ``mode=economic``, the shapes of Q and R are (M, K) and (K, N) instead
    of (M,M) and (M,N), with ``K=min(M,N)``.

    Examples
    --------
    >>> import numpy as np
    >>> from scipy import linalg
    >>> rng = np.random.default_rng()
    >>> a = rng.standard_normal((9, 6))

    >>> q, r = linalg.qr(a)
    >>> np.allclose(a, np.dot(q, r))
    True
    >>> q.shape, r.shape
    ((9, 9), (9, 6))

    >>> r2 = linalg.qr(a, mode='r')
    >>> np.allclose(r, r2)
    True

    >>> q3, r3 = linalg.qr(a, mode='economic')
    >>> q3.shape, r3.shape
    ((9, 6), (6, 6))

    >>> q4, r4, p4 = linalg.qr(a, pivoting=True)
    >>> d = np.abs(np.diag(r4))
    >>> np.all(d[1:] <= d[:-1])
    True
    >>> np.allclose(a[:, p4], np.dot(q4, r4))
    True
    >>> P = np.eye(p4.size)[p4]
    >>> np.allclose(a, np.dot(q4, r4) @ P)
    True
    >>> np.allclose(a @ P.T, np.dot(q4, r4))
    True
    >>> q4.shape, r4.shape, p4.shape
    ((9, 9), (9, 6), (6,))

    >>> q5, r5, p5 = linalg.qr(a, mode='economic', pivoting=True)
    >>> q5.shape, r5.shape, p5.shape
    ((9, 6), (6, 6), (6,))
    >>> P = np.eye(6)[:, p5]
    >>> np.allclose(a @ P, np.dot(q5, r5))
    True

    """
    # 'qr' was the old default, equivalent to 'full'. Neither 'full' nor
    # 'qr' are used below.
    # 'raw' is used internally by qr_multiply
    if mode not in ['full', 'qr', 'r', 'economic', 'raw']:
        raise ValueError("Mode argument should be one of ['full', 'r', "
                         "'economic', 'raw']")

    if check_finite:
        a1 = np.asarray_chkfinite(a)
    else:
        a1 = np.asarray(a)
    if len(a1.shape) != 2:
        raise ValueError("expected a 2-D array")

    M, N = a1.shape

    # accommodate empty arrays
    if a1.size == 0:
        K = min(M, N)

        if mode not in ['economic', 'raw']:
            Q = np.empty_like(a1, shape=(M, M))
            Q[...] = np.identity(M)
            R = np.empty_like(a1)
        else:
            Q = np.empty_like(a1, shape=(M, K))
            R = np.empty_like(a1, shape=(K, N))

        if pivoting:
            Rj = R, np.arange(N, dtype=np.int32)
        else:
            Rj = R,

        if mode == 'r':
            return Rj
        elif mode == 'raw':
            qr = np.empty_like(a1, shape=(M, N))
            tau = np.zeros_like(a1, shape=(K,))
            return ((qr, tau),) + Rj
        return (Q,) + Rj

    overwrite_a = overwrite_a or (_datacopied(a1, a))

    if pivoting:
        geqp3, = get_lapack_funcs(('geqp3',), (a1,))
        qr, jpvt, tau = safecall(geqp3, "geqp3", a1, overwrite_a=overwrite_a)
        jpvt -= 1  # geqp3 returns a 1-based index array, so subtract 1
    else:
        geqrf, = get_lapack_funcs(('geqrf',), (a1,))
        qr, tau = safecall(geqrf, "geqrf", a1, lwork=lwork,
                           overwrite_a=overwrite_a)

    if mode not in ['economic', 'raw'] or M < N:
        R = np.triu(qr)
    else:
        R = np.triu(qr[:N, :])

    if pivoting:
        Rj = R, jpvt
    else:
        Rj = R,

    if mode == 'r':
        return Rj
    elif mode == 'raw':
        return ((qr, tau),) + Rj

    gor_un_gqr, = get_lapack_funcs(('orgqr',), (qr,))

    if M < N:
        Q, = safecall(gor_un_gqr, "gorgqr/gungqr", qr[:, :M], tau,
                      lwork=lwork, overwrite_a=1)
    elif mode == 'economic':
        Q, = safecall(gor_un_gqr, "gorgqr/gungqr", qr, tau, lwork=lwork,
                      overwrite_a=1)
    else:
        t = qr.dtype.char
        qqr = np.empty((M, M), dtype=t)
        qqr[:, :N] = qr
        Q, = safecall(gor_un_gqr, "gorgqr/gungqr", qqr, tau, lwork=lwork,
                      overwrite_a=1)

    return (Q,) + Rj


def qr_multiply(a, c, mode='right', pivoting=False, conjugate=False,
                overwrite_a=False, overwrite_c=False):
    """
    Calculate the QR decomposition and multiply Q with a matrix.

    Calculate the decomposition ``A = Q R`` where Q is unitary/orthogonal
    and R upper triangular. Multiply Q with a vector or a matrix c.

    Parameters
    ----------
    a : (M, N), array_like
        Input array
    c : array_like
        Input array to be multiplied by ``q``.
    mode : {'left', 'right'}, optional
        ``Q @ c`` is returned if mode is 'left', ``c @ Q`` is returned if
        mode is 'right'.
        The shape of c must be appropriate for the matrix multiplications,
        if mode is 'left', ``min(a.shape) == c.shape[0]``,
        if mode is 'right', ``a.shape[0] == c.shape[1]``.
    pivoting : bool, optional
        Whether or not factorization should include pivoting for rank-revealing
        qr decomposition, see the documentation of qr.
    conjugate : bool, optional
        Whether Q should be complex-conjugated. This might be faster
        than explicit conjugation.
    overwrite_a : bool, optional
        Whether data in a is overwritten (may improve performance)
    overwrite_c : bool, optional
        Whether data in c is overwritten (may improve performance).
        If this is used, c must be big enough to keep the result,
        i.e. ``c.shape[0]`` = ``a.shape[0]`` if mode is 'left'.

    Returns
    -------
    CQ : ndarray
        The product of ``Q`` and ``c``.
    R : (K, N), ndarray
        R array of the resulting QR factorization where ``K = min(M, N)``.
    P : (N,) ndarray
        Integer pivot array. Only returned when ``pivoting=True``.

    Raises
    ------
    LinAlgError
        Raised if QR decomposition fails.

    Notes
    -----
    This is an interface to the LAPACK routines ``?GEQRF``, ``?ORMQR``,
    ``?UNMQR``, and ``?GEQP3``.

    .. versionadded:: 0.11.0

    Examples
    --------
    >>> import numpy as np
    >>> from scipy.linalg import qr_multiply, qr
    >>> A = np.array([[1, 3, 3], [2, 3, 2], [2, 3, 3], [1, 3, 2]])
    >>> qc, r1, piv1 = qr_multiply(A, 2*np.eye(4), pivoting=1)
    >>> qc
    array([[-1.,  1., -1.],
           [-1., -1.,  1.],
           [-1., -1., -1.],
           [-1.,  1.,  1.]])
    >>> r1
    array([[-6., -3., -5.            ],
           [ 0., -1., -1.11022302e-16],
           [ 0.,  0., -1.            ]])
    >>> piv1
    array([1, 0, 2], dtype=int32)
    >>> q2, r2, piv2 = qr(A, mode='economic', pivoting=1)
    >>> np.allclose(2*q2 - qc, np.zeros((4, 3)))
    True

    """
    if mode not in ['left', 'right']:
        raise ValueError("Mode argument can only be 'left' or 'right' but "
                         f"not '{mode}'")
    c = np.asarray_chkfinite(c)
    if c.ndim < 2:
        onedim = True
        c = np.atleast_2d(c)
        if mode == "left":
            c = c.T
    else:
        onedim = False

    a = np.atleast_2d(np.asarray(a))  # chkfinite done in qr
    M, N = a.shape

    if mode == 'left':
        if c.shape[0] != min(M, N + overwrite_c*(M-N)):
            raise ValueError('Array shapes are not compatible for Q @ c'
                             f' operation: {a.shape} vs {c.shape}')
    else:
        if M != c.shape[1]:
            raise ValueError('Array shapes are not compatible for c @ Q'
                             f' operation: {c.shape} vs {a.shape}')

    raw = qr(a, overwrite_a, None, "raw", pivoting)
    Q, tau = raw[0]

    # accommodate empty arrays
    if c.size == 0:
        return (np.empty_like(c),) + raw[1:]

    gor_un_mqr, = get_lapack_funcs(('ormqr',), (Q,))
    if gor_un_mqr.typecode in ('s', 'd'):
        trans = "T"
    else:
        trans = "C"

    Q = Q[:, :min(M, N)]
    if M > N and mode == "left" and not overwrite_c:
        if conjugate:
            cc = np.zeros((c.shape[1], M), dtype=c.dtype, order="F")
            cc[:, :N] = c.T
        else:
            cc = np.zeros((M, c.shape[1]), dtype=c.dtype, order="F")
            cc[:N, :] = c
            trans = "N"
        if conjugate:
            lr = "R"
        else:
            lr = "L"
        overwrite_c = True
    elif c.flags["C_CONTIGUOUS"] and trans == "T" or conjugate:
        cc = c.T
        if mode == "left":
            lr = "R"
        else:
            lr = "L"
    else:
        trans = "N"
        cc = c
        if mode == "left":
            lr = "L"
        else:
            lr = "R"
    cQ, = safecall(gor_un_mqr, "gormqr/gunmqr", lr, trans, Q, tau, cc,
                   overwrite_c=overwrite_c)
    if trans != "N":
        cQ = cQ.T
    if mode == "right":
        cQ = cQ[:, :min(M, N)]
    if onedim:
        cQ = cQ.ravel()

    return (cQ,) + raw[1:]


def rq(a, overwrite_a=False, lwork=None, mode='full', check_finite=True):
    """
    Compute RQ decomposition of a matrix.

    Calculate the decomposition ``A = R Q`` where Q is unitary/orthogonal
    and R upper triangular.

    Parameters
    ----------
    a : (M, N) array_like
        Matrix to be decomposed
    overwrite_a : bool, optional
        Whether data in a is overwritten (may improve performance)
    lwork : int, optional
        Work array size, lwork >= a.shape[1]. If None or -1, an optimal size
        is computed.
    mode : {'full', 'r', 'economic'}, optional
        Determines what information is to be returned: either both Q and R
        ('full', default), only R ('r') or both Q and R but computed in
        economy-size ('economic', see Notes).
    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.

    Returns
    -------
    R : float or complex ndarray
        Of shape (M, N) or (M, K) for ``mode='economic'``. ``K = min(M, N)``.
    Q : float or complex ndarray
        Of shape (N, N) or (K, N) for ``mode='economic'``. Not returned
        if ``mode='r'``.

    Raises
    ------
    LinAlgError
        If decomposition fails.

    Notes
    -----
    This is an interface to the LAPACK routines sgerqf, dgerqf, cgerqf, zgerqf,
    sorgrq, dorgrq, cungrq and zungrq.

    If ``mode=economic``, the shapes of Q and R are (K, N) and (M, K) instead
    of (N,N) and (M,N), with ``K=min(M,N)``.

    Examples
    --------
    >>> import numpy as np
    >>> from scipy import linalg
    >>> rng = np.random.default_rng()
    >>> a = rng.standard_normal((6, 9))
    >>> r, q = linalg.rq(a)
    >>> np.allclose(a, r @ q)
    True
    >>> r.shape, q.shape
    ((6, 9), (9, 9))
    >>> r2 = linalg.rq(a, mode='r')
    >>> np.allclose(r, r2)
    True
    >>> r3, q3 = linalg.rq(a, mode='economic')
    >>> r3.shape, q3.shape
    ((6, 6), (6, 9))

    """
    if mode not in ['full', 'r', 'economic']:
        raise ValueError(
                 "Mode argument should be one of ['full', 'r', 'economic']")

    if check_finite:
        a1 = np.asarray_chkfinite(a)
    else:
        a1 = np.asarray(a)
    if len(a1.shape) != 2:
        raise ValueError('expected matrix')

    M, N = a1.shape

    # accommodate empty arrays
    if a1.size == 0:
        K = min(M, N)

        if not mode == 'economic':
            R = np.empty_like(a1)
            Q = np.empty_like(a1, shape=(N, N))
            Q[...] = np.identity(N)
        else:
            R = np.empty_like(a1, shape=(M, K))
            Q = np.empty_like(a1, shape=(K, N))

        if mode == 'r':
            return R
        return R, Q

    overwrite_a = overwrite_a or (_datacopied(a1, a))

    gerqf, = get_lapack_funcs(('gerqf',), (a1,))
    rq, tau = safecall(gerqf, 'gerqf', a1, lwork=lwork,
                       overwrite_a=overwrite_a)
    if not mode == 'economic' or N < M:
        R = np.triu(rq, N-M)
    else:
        R = np.triu(rq[-M:, -M:])

    if mode == 'r':
        return R

    gor_un_grq, = get_lapack_funcs(('orgrq',), (rq,))

    if N < M:
        Q, = safecall(gor_un_grq, "gorgrq/gungrq", rq[-N:], tau, lwork=lwork,
                      overwrite_a=1)
    elif mode == 'economic':
        Q, = safecall(gor_un_grq, "gorgrq/gungrq", rq, tau, lwork=lwork,
                      overwrite_a=1)
    else:
        rq1 = np.empty((N, N), dtype=rq.dtype)
        rq1[-M:] = rq
        Q, = safecall(gor_un_grq, "gorgrq/gungrq", rq1, tau, lwork=lwork,
                      overwrite_a=1)

    return R, Q