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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 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 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)

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 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 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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