# scipy.linalg.interpolative.interp_decomp¶

scipy.linalg.interpolative.interp_decomp(A, eps_or_k, rand=True)[source]

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