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"""SVD decomposition functions."""
from __future__ import division, print_function, absolute_import
import numpy
from numpy import zeros, r_, diag
# Local imports.
from .misc import LinAlgError, _datacopied
from .lapack import get_lapack_funcs, _compute_lwork
from .decomp import _asarray_validated
from scipy._lib.six import string_types
__all__ = ['svd', 'svdvals', 'diagsvd', 'orth']
def svd(a, full_matrices=True, compute_uv=True, overwrite_a=False,
check_finite=True, lapack_driver='gesdd'):
"""
Singular Value Decomposition.
Factorizes the matrix a into two unitary matrices U and Vh, and
a 1-D array s of singular values (real, non-negative) such that
``a == U*S*Vh``, where S is a suitably shaped matrix of zeros with
main diagonal s.
Parameters
----------
a : (M, N) array_like
Matrix to decompose.
full_matrices : bool, optional
If True, `U` and `Vh` are of shape ``(M,M)``, ``(N,N)``.
If False, the shapes are ``(M,K)`` and ``(K,N)``, where
``K = min(M,N)``.
compute_uv : bool, optional
Whether to compute also `U` and `Vh` in addition to `s`.
Default is True.
overwrite_a : bool, optional
Whether to overwrite `a`; may improve performance.
Default is False.
check_finite : bool, optional
Whether to check that the input matrix contains only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
lapack_driver : {'gesdd', 'gesvd'}, optional
Whether to use the more efficient divide-and-conquer approach
(``'gesdd'``) or general rectangular approach (``'gesvd'``)
to compute the SVD. MATLAB and Octave use the ``'gesvd'`` approach.
Default is ``'gesdd'``.
.. versionadded:: 0.18
Returns
-------
U : ndarray
Unitary matrix having left singular vectors as columns.
Of shape ``(M,M)`` or ``(M,K)``, depending on `full_matrices`.
s : ndarray
The singular values, sorted in non-increasing order.
Of shape (K,), with ``K = min(M, N)``.
Vh : ndarray
Unitary matrix having right singular vectors as rows.
Of shape ``(N,N)`` or ``(K,N)`` depending on `full_matrices`.
For ``compute_uv=False``, only `s` is returned.
Raises
------
LinAlgError
If SVD computation does not converge.
See also
--------
svdvals : Compute singular values of a matrix.
diagsvd : Construct the Sigma matrix, given the vector s.
Examples
--------
>>> from scipy import linalg
>>> a = np.random.randn(9, 6) + 1.j*np.random.randn(9, 6)
>>> U, s, Vh = linalg.svd(a)
>>> U.shape, Vh.shape, s.shape
((9, 9), (6, 6), (6,))
>>> U, s, Vh = linalg.svd(a, full_matrices=False)
>>> U.shape, Vh.shape, s.shape
((9, 6), (6, 6), (6,))
>>> S = linalg.diagsvd(s, 6, 6)
>>> np.allclose(a, np.dot(U, np.dot(S, Vh)))
True
>>> s2 = linalg.svd(a, compute_uv=False)
>>> np.allclose(s, s2)
True
"""
a1 = _asarray_validated(a, check_finite=check_finite)
if len(a1.shape) != 2:
raise ValueError('expected matrix')
m, n = a1.shape
overwrite_a = overwrite_a or (_datacopied(a1, a))
if not isinstance(lapack_driver, string_types):
raise TypeError('lapack_driver must be a string')
if lapack_driver not in ('gesdd', 'gesvd'):
raise ValueError('lapack_driver must be "gesdd" or "gesvd", not "%s"'
% (lapack_driver,))
funcs = (lapack_driver, lapack_driver + '_lwork')
gesXd, gesXd_lwork = get_lapack_funcs(funcs, (a1,))
# compute optimal lwork
lwork = _compute_lwork(gesXd_lwork, a1.shape[0], a1.shape[1],
compute_uv=compute_uv, full_matrices=full_matrices)
# perform decomposition
u, s, v, info = gesXd(a1, compute_uv=compute_uv, lwork=lwork,
full_matrices=full_matrices, overwrite_a=overwrite_a)
if info > 0:
raise LinAlgError("SVD did not converge")
if info < 0:
raise ValueError('illegal value in %d-th argument of internal gesdd'
% -info)
if compute_uv:
return u, s, v
else:
return s
def svdvals(a, overwrite_a=False, check_finite=True):
"""
Compute singular values of a matrix.
Parameters
----------
a : (M, N) array_like
Matrix to decompose.
overwrite_a : bool, optional
Whether to overwrite `a`; may improve performance.
Default is False.
check_finite : bool, optional
Whether to check that the input matrix contains only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
s : (min(M, N),) ndarray
The singular values, sorted in decreasing order.
Raises
------
LinAlgError
If SVD computation does not converge.
Notes
-----
``svdvals(a)`` only differs from ``svd(a, compute_uv=False)`` by its
handling of the edge case of empty ``a``, where it returns an
empty sequence:
>>> a = np.empty((0, 2))
>>> from scipy.linalg import svdvals
>>> svdvals(a)
array([], dtype=float64)
See also
--------
svd : Compute the full singular value decomposition of a matrix.
diagsvd : Construct the Sigma matrix, given the vector s.
"""
a = _asarray_validated(a, check_finite=check_finite)
if a.size:
return svd(a, compute_uv=0, overwrite_a=overwrite_a,
check_finite=False)
elif len(a.shape) != 2:
raise ValueError('expected matrix')
else:
return numpy.empty(0)
def diagsvd(s, M, N):
"""
Construct the sigma matrix in SVD from singular values and size M, N.
Parameters
----------
s : (M,) or (N,) array_like
Singular values
M : int
Size of the matrix whose singular values are `s`.
N : int
Size of the matrix whose singular values are `s`.
Returns
-------
S : (M, N) ndarray
The S-matrix in the singular value decomposition
"""
part = diag(s)
typ = part.dtype.char
MorN = len(s)
if MorN == M:
return r_['-1', part, zeros((M, N-M), typ)]
elif MorN == N:
return r_[part, zeros((M-N, N), typ)]
else:
raise ValueError("Length of s must be M or N.")
# Orthonormal decomposition
def orth(A):
"""
Construct an orthonormal basis for the range of A using SVD
Parameters
----------
A : (M, N) array_like
Input array
Returns
-------
Q : (M, K) ndarray
Orthonormal basis for the range of A.
K = effective rank of A, as determined by automatic cutoff
See also
--------
svd : Singular value decomposition of a matrix
"""
u, s, vh = svd(A, full_matrices=False)
M, N = A.shape
eps = numpy.finfo(float).eps
tol = max(M, N) * numpy.amax(s) * eps
num = numpy.sum(s > tol, dtype=int)
Q = u[:, :num]
return Q
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