def interp_decomp(A, eps_or_k, rand=True):
"""
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`).
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.
"""
from scipy.sparse.linalg import LinearOperator
real = _is_real(A)
if isinstance(A, np.ndarray):
if eps_or_k < 1:
eps = eps_or_k
if rand:
if real:
k, idx, proj = backend.iddp_aid(eps, A)
else:
k, idx, proj = backend.idzp_aid(eps, A)
else:
if real:
k, idx, proj = backend.iddp_id(eps, A)
else:
k, idx, proj = backend.idzp_id(eps, A)
return k, idx - 1, proj
else:
k = int(eps_or_k)
if rand:
if real:
idx, proj = backend.iddr_aid(A, k)
else:
idx, proj = backend.idzr_aid(A, k)
else:
if real:
idx, proj = backend.iddr_id(A, k)
else:
idx, proj = backend.idzr_id(A, k)
return idx - 1, proj
elif isinstance(A, LinearOperator):
m, n = A.shape
matveca = A.rmatvec
if eps_or_k < 1:
eps = eps_or_k
if real:
k, idx, proj = backend.iddp_rid(eps, m, n, matveca)
else:
k, idx, proj = backend.idzp_rid(eps, m, n, matveca)
return k, idx - 1, proj
else:
k = int(eps_or_k)
if real:
idx, proj = backend.iddr_rid(m, n, matveca, k)
else:
idx, proj = backend.idzr_rid(m, n, matveca, k)
return idx - 1, proj
else:
raise _TYPE_ERROR
def interp_decomp(A, eps_or_k, rand=True):
"""
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`).
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.
"""
from scipy.sparse.linalg import LinearOperator
real = _is_real(A)
if isinstance(A, np.ndarray):
if eps_or_k < 1:
eps = eps_or_k
if rand:
if real:
k, idx, proj = backend.iddp_aid(eps, A)
else:
k, idx, proj = backend.idzp_aid(eps, A)
else:
if real:
k, idx, proj = backend.iddp_id(eps, A)
else:
k, idx, proj = backend.idzp_id(eps, A)
return k, idx - 1, proj
else:
k = int(eps_or_k)
if rand:
if real:
idx, proj = backend.iddr_aid(A, k)
else:
idx, proj = backend.idzr_aid(A, k)
else:
if real:
idx, proj = backend.iddr_id(A, k)
else:
idx, proj = backend.idzr_id(A, k)
return idx - 1, proj
elif isinstance(A, LinearOperator):
m, n = A.shape
matveca = A.rmatvec
if eps_or_k < 1:
eps = eps_or_k
if real:
k, idx, proj = backend.iddp_rid(eps, m, n, matveca)
else:
k, idx, proj = backend.idzp_rid(eps, m, n, matveca)
return k, idx - 1, proj
else:
k = int(eps_or_k)
if real:
idx, proj = backend.iddr_rid(m, n, matveca, k)
else:
idx, proj = backend.idzr_rid(m, n, matveca, k)
return idx - 1, proj
else:
raise _TYPE_ERROR
