interp_decomp(A, eps_or_k, rand=True)

An ID of a matrix A is a factorization defined by a rank k, a column index array idx , and interpolation coefficients :None:None:`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 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 reconstruct_interp_matrix and reconstruct_skel_matrix .

The ID can be computed to any relative precision or rank (depending on the value of :None:None:`eps_or_k`). If a precision is specified (:None:None:`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 (:None:None:`eps_or_k >= 1`), then the output signature is:

idx, proj = interp_decomp(A, eps_or_k)

            <Comment: 
   |value: '..  This function automatically detects the form of the input parameters\n    and passes them to the appropriate backend. For details, see\n    :func:`_backend.iddp_id`, :func:`_backend.iddp_aid`,\n    :func:`_backend.iddp_rid`, :func:`_backend.iddr_id`,\n    :func:`_backend.iddr_aid`, :func:`_backend.iddr_rid`,\n    :func:`_backend.idzp_id`, :func:`_backend.idzp_aid`,\n    :func:`_backend.idzp_rid`, :func:`_backend.idzr_id`,\n    :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 :None:None:`eps_or_k < 1`) or rank (if :None:None:`eps_or_k >= 1`) of approximation.

rand : bool, optional

Whether to use random sampling if A is of type numpy.ndarray (randomized algorithms are always used if A is of type scipy.sparse.linalg.LinearOperator ).

Returns

k : int

Rank required to achieve specified relative precision if :None:None:`eps_or_k < 1`.

idx : :class:`numpy.ndarray`

Column index array.

proj : :class:`numpy.ndarray`

Interpolation coefficients.

Compute ID of a matrix.

Examples

See :

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GitHub : /scipy/linalg/interpolative.py#482
type: <class 'function'>
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