def svd(A, eps_or_k, rand=True):
"""
Compute SVD of a matrix via an ID.
An SVD of a matrix `A` is a factorization::
A = numpy.dot(U, numpy.dot(numpy.diag(S), V.conj().T))
where `U` and `V` have orthonormal columns and `S` is nonnegative.
The SVD can be computed to any relative precision or rank (depending on the
value of `eps_or_k`).
See also :func:`interp_decomp` and :func:`id_to_svd`.
.. This function automatically detects the form of the input parameters and
passes them to the appropriate backend. For details, see
:func:`backend.iddp_svd`, :func:`backend.iddp_asvd`,
:func:`backend.iddp_rsvd`, :func:`backend.iddr_svd`,
:func:`backend.iddr_asvd`, :func:`backend.iddr_rsvd`,
:func:`backend.idzp_svd`, :func:`backend.idzp_asvd`,
:func:`backend.idzp_rsvd`, :func:`backend.idzr_svd`,
:func:`backend.idzr_asvd`, and :func:`backend.idzr_rsvd`.
Parameters
----------
A : :class:`numpy.ndarray` or :class:`scipy.sparse.linalg.LinearOperator`
Matrix to be factored, given as either a :class:`numpy.ndarray` or a
:class:`scipy.sparse.linalg.LinearOperator` with the `matvec` and
`rmatvec` methods (to apply the matrix and its adjoint).
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
-------
U : :class:`numpy.ndarray`
Left singular vectors.
S : :class:`numpy.ndarray`
Singular values.
V : :class:`numpy.ndarray`
Right singular vectors.
"""
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:
U, V, S = backend.iddp_asvd(eps, A)
else:
U, V, S = backend.idzp_asvd(eps, A)
else:
if real:
U, V, S = backend.iddp_svd(eps, A)
else:
U, V, S = backend.idzp_svd(eps, A)
else:
k = int(eps_or_k)
if k > min(A.shape):
raise ValueError(
"Approximation rank %s exceeds min(A.shape) = "
" %s " % (k, min(A.shape)))
if rand:
if real:
U, V, S = backend.iddr_asvd(A, k)
else:
U, V, S = backend.idzr_asvd(A, k)
else:
if real:
U, V, S = backend.iddr_svd(A, k)
else:
U, V, S = backend.idzr_svd(A, k)
elif isinstance(A, LinearOperator):
m, n = A.shape
matvec = lambda x: A.matvec(x)
matveca = lambda x: A.rmatvec(x)
if eps_or_k < 1:
eps = eps_or_k
if real:
U, V, S = backend.iddp_rsvd(eps, m, n, matveca, matvec)
else:
U, V, S = backend.idzp_rsvd(eps, m, n, matveca, matvec)
else:
k = int(eps_or_k)
if real:
U, V, S = backend.iddr_rsvd(m, n, matveca, matvec, k)
else:
U, V, S = backend.idzr_rsvd(m, n, matveca, matvec, k)
else:
raise _TYPE_ERROR
return U, S, V
def svd(A, eps_or_k, rand=True):
"""
Compute SVD of a matrix via an ID.
An SVD of a matrix `A` is a factorization::
A = numpy.dot(U, numpy.dot(numpy.diag(S), V.conj().T))
where `U` and `V` have orthonormal columns and `S` is nonnegative.
The SVD can be computed to any relative precision or rank (depending on the
value of `eps_or_k`).
See also :func:`interp_decomp` and :func:`id_to_svd`.
.. This function automatically detects the form of the input parameters and
passes them to the appropriate backend. For details, see
:func:`backend.iddp_svd`, :func:`backend.iddp_asvd`,
:func:`backend.iddp_rsvd`, :func:`backend.iddr_svd`,
:func:`backend.iddr_asvd`, :func:`backend.iddr_rsvd`,
:func:`backend.idzp_svd`, :func:`backend.idzp_asvd`,
:func:`backend.idzp_rsvd`, :func:`backend.idzr_svd`,
:func:`backend.idzr_asvd`, and :func:`backend.idzr_rsvd`.
Parameters
----------
A : :class:`numpy.ndarray` or :class:`scipy.sparse.linalg.LinearOperator`
Matrix to be factored, given as either a :class:`numpy.ndarray` or a
:class:`scipy.sparse.linalg.LinearOperator` with the `matvec` and
`rmatvec` methods (to apply the matrix and its adjoint).
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
-------
U : :class:`numpy.ndarray`
Left singular vectors.
S : :class:`numpy.ndarray`
Singular values.
V : :class:`numpy.ndarray`
Right singular vectors.
"""
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:
U, V, S = backend.iddp_asvd(eps, A)
else:
U, V, S = backend.idzp_asvd(eps, A)
else:
if real:
U, V, S = backend.iddp_svd(eps, A)
else:
U, V, S = backend.idzp_svd(eps, A)
else:
k = int(eps_or_k)
if rand:
if real:
U, V, S = backend.iddr_asvd(A, k)
else:
U, V, S = backend.idzr_asvd(A, k)
else:
if real:
U, V, S = backend.iddr_svd(A, k)
else:
U, V, S = backend.idzr_svd(A, k)
elif isinstance(A, LinearOperator):
m, n = A.shape
matvec = lambda x: A.matvec(x)
matveca = lambda x: A.rmatvec(x)
if eps_or_k < 1:
eps = eps_or_k
if real:
U, V, S = backend.iddp_rsvd(eps, m, n, matveca, matvec)
else:
U, V, S = backend.idzp_rsvd(eps, m, n, matveca, matvec)
else:
k = int(eps_or_k)
if real:
U, V, S = backend.iddr_rsvd(m, n, matveca, matvec, k)
else:
U, V, S = backend.idzr_rsvd(m, n, matveca, matvec, k)
else:
raise _TYPE_ERROR
return U, S, V
