interp_decomp — SciPy v1.15.3 Manual (original) (raw)
scipy.linalg.interpolative.
scipy.linalg.interpolative.interp_decomp(A, eps_or_k, rand=True, rng=None)[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:
and:
P = numpy.hstack([numpy.eye(k), proj])[:,np.argsort(idx)]
respectively. See also reconstruct_interp_matrix andreconstruct_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:
Anumpy.ndarray or scipy.sparse.linalg.LinearOperator with rmatvec
Matrix to be factored
eps_or_kfloat or int
Relative error (if eps_or_k < 1) or rank (if eps_or_k >= 1) of approximation.
randbool, optional
Whether to use random sampling if A is of type numpy.ndarray(randomized algorithms are always used if A is of typescipy.sparse.linalg.LinearOperator).
rngnumpy.random.Generator, optional
Pseudorandom number generator state. When rng is None, a newnumpy.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:
kint
Rank required to achieve specified relative precision ifeps_or_k < 1.
Column index array.
projnumpy.ndarray
Interpolation coefficients.