package scipy

Get an attribute of this module as a Py.Object.t. This is useful to pass a Python function to another function.

Return available LAPACK function objects from names.

Arrays are used to determine the optimal prefix of LAPACK routines.

Parameters ---------- names : str or sequence of str Name(s) of LAPACK functions without type prefix.

arrays : sequence of ndarrays, optional Arrays can be given to determine optimal prefix of LAPACK routines. If not given, double-precision routines will be used, otherwise the most generic type in arrays will be used.

dtype : str or dtype, optional Data-type specifier. Not used if `arrays` is non-empty.

Returns ------- funcs : list List containing the found function(s).

Notes ----- This routine automatically chooses between Fortran/C interfaces. Fortran code is used whenever possible for arrays with column major order. In all other cases, C code is preferred.

In LAPACK, the naming convention is that all functions start with a type prefix, which depends on the type of the principal matrix. These can be one of 's', 'd', 'c', 'z' for the numpy types float32, float64, complex64, complex128 respectively, and are stored in attribute ``typecode`` of the returned functions.

Examples -------- Suppose we would like to use '?lange' routine which computes the selected norm of an array. We pass our array in order to get the correct 'lange' flavor.

>>> import scipy.linalg as LA >>> a = np.random.rand(3,2) >>> x_lange = LA.get_lapack_funcs('lange', (a,)) >>> x_lange.typecode 'd' >>> x_lange = LA.get_lapack_funcs('lange',(a*1j,)) >>> x_lange.typecode 'z'

Several LAPACK routines work best when its internal WORK array has the optimal size (big enough for fast computation and small enough to avoid waste of memory). This size is determined also by a dedicated query to the function which is often wrapped as a standalone function and commonly denoted as ``###_lwork``. Below is an example for ``?sysv``

>>> import scipy.linalg as LA >>> a = np.random.rand(1000,1000) >>> b = np.random.rand(1000,1)*1j >>> # We pick up zsysv and zsysv_lwork due to b array ... xsysv, xlwork = LA.get_lapack_funcs(('sysv', 'sysv_lwork'), (a, b)) >>> opt_lwork, _ = xlwork(a.shape0) # returns a complex for 'z' prefix >>> udut, ipiv, x, info = xsysv(a, b, lwork=int(opt_lwork.real))

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) lwork : int, optional Work array size, lwork >= a.shape1. 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. 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'``. R : float or complex ndarray Of shape (M, N), or (K, N) for ``mode='economic'``. ``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 -------- >>> from scipy import linalg >>> a = np.random.randn(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(d1: <= d:-1) True >>> np.allclose(a:, p4, 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,))

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.shape0``, if mode is 'right', ``a.shape0 == c.shape1``. 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.shape0`` = ``a.shape0`` 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 -------- >>> 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

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.shape1. 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 -------- >>> from scipy import linalg >>> a = np.random.randn(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))

Call a LAPACK routine, determining lwork automatically and handling error return values