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Module « scipy.linalg.interpolative »

Fonction interp_decomp - module scipy.linalg.interpolative

Signature de la fonction interp_decomp

def interp_decomp(A, eps_or_k, rand=True, rng=None) 

Description

help(scipy.linalg.interpolative.interp_decomp)

Compute ID of a matrix.

An ID of a matrix `A` is a factorization defined by a rank `k`, a column
index array `idx`, and interpolation coefficients `proj` such that::

    numpy.dot(A[:,idx[:k]], proj) = A[:,idx[k:]]

The original matrix can then be reconstructed as::

    numpy.hstack([A[:,idx[:k]],
                                numpy.dot(A[:,idx[:k]], proj)]
                            )[:,numpy.argsort(idx)]

or via the routine :func:`reconstruct_matrix_from_id`. This can
equivalently be written as::

    numpy.dot(A[:,idx[:k]],
                        numpy.hstack([numpy.eye(k), proj])
                      )[:,np.argsort(idx)]

in terms of the skeleton and interpolation matrices::

    B = A[:,idx[:k]]

and::

    P = numpy.hstack([numpy.eye(k), proj])[:,np.argsort(idx)]

respectively. See also :func:`reconstruct_interp_matrix` and
:func:`reconstruct_skel_matrix`.

The ID can be computed to any relative precision or rank (depending on the
value of `eps_or_k`). If a precision is specified (`eps_or_k < 1`), then
this function has the output signature::

    k, idx, proj = interp_decomp(A, eps_or_k)

Otherwise, if a rank is specified (`eps_or_k >= 1`), then the output
signature is::

    idx, proj = interp_decomp(A, eps_or_k)

..  This function automatically detects the form of the input parameters
    and passes them to the appropriate backend. For details, see
    :func:`_backend.iddp_id`, :func:`_backend.iddp_aid`,
    :func:`_backend.iddp_rid`, :func:`_backend.iddr_id`,
    :func:`_backend.iddr_aid`, :func:`_backend.iddr_rid`,
    :func:`_backend.idzp_id`, :func:`_backend.idzp_aid`,
    :func:`_backend.idzp_rid`, :func:`_backend.idzr_id`,
    :func:`_backend.idzr_aid`, and :func:`_backend.idzr_rid`.

Parameters
----------
A : :class:`numpy.ndarray` or :class:`scipy.sparse.linalg.LinearOperator` with `rmatvec`
    Matrix to be factored
eps_or_k : float or int
    Relative error (if ``eps_or_k < 1``) or rank (if ``eps_or_k >= 1``) of
    approximation.
rand : bool, optional
    Whether to use random sampling if `A` is of type :class:`numpy.ndarray`
    (randomized algorithms are always used if `A` is of type
    :class:`scipy.sparse.linalg.LinearOperator`).
rng : `numpy.random.Generator`, optional
    Pseudorandom number generator state. When `rng` is None, a new
    `numpy.random.Generator` is created using entropy from the
    operating system. Types other than `numpy.random.Generator` are
    passed to `numpy.random.default_rng` to instantiate a ``Generator``.
    If `rand` is ``False``, the argument is ignored.

Returns
-------
k : int
    Rank required to achieve specified relative precision if
    ``eps_or_k < 1``.
idx : :class:`numpy.ndarray`
    Column index array.
proj : :class:`numpy.ndarray`
    Interpolation coefficients.

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