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"""SVD decomposition functions."""
import numpy
from numpy import asarray_chkfinite, zeros, r_, diag
from scipy.linalg import calc_lwork
# Local imports.
from misc import LinAlgError, _datacopied
from lapack import get_lapack_funcs
__all__ = ['svd', 'svdvals', 'diagsvd', 'orth']
def svd(a, full_matrices=True, compute_uv=True, overwrite_a=False):
"""Singular Value Decomposition.
Factorizes the matrix a into two unitary matrices U and Vh and
an 1d-array s of singular values (real, non-negative) such that
a == U S Vh if S is an suitably shaped matrix of zeros whose
main diagonal is s.
Parameters
----------
a : array, shape (M, N)
Matrix to decompose
full_matrices : boolean
If true, U, Vh are shaped (M,M), (N,N)
If false, the shapes are (M,K), (K,N) where K = min(M,N)
compute_uv : boolean
Whether to compute also U, Vh in addition to s (Default: true)
overwrite_a : boolean
Whether data in a is overwritten (may improve performance)
Returns
-------
U: array, shape (M,M) or (M,K) depending on full_matrices
s: array, shape (K,)
The singular values, sorted so that s[i] >= s[i+1]. K = min(M, N)
Vh: array, 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
Examples
--------
>>> from scipy import random, linalg, allclose, dot
>>> a = random.randn(9, 6) + 1j*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)
>>> allclose(a, dot(U, dot(S, Vh)))
True
>>> s2 = linalg.svd(a, compute_uv=False)
>>> allclose(s, s2)
True
See also
--------
svdvals : return singular values of a matrix
diagsvd : return the Sigma matrix, given the vector s
"""
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2:
raise ValueError('expected matrix')
m,n = a1.shape
overwrite_a = overwrite_a or (_datacopied(a1, a))
gesdd, = get_lapack_funcs(('gesdd',), (a1,))
if gesdd.module_name[:7] == 'flapack':
lwork = calc_lwork.gesdd(gesdd.prefix, m, n, compute_uv)[1]
u,s,v,info = gesdd(a1,compute_uv = compute_uv, lwork = lwork,
full_matrices=full_matrices, overwrite_a = overwrite_a)
else: # 'clapack'
raise NotImplementedError('calling gesdd from %s' % gesdd.module_name)
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):
"""Compute singular values of a matrix.
Parameters
----------
a : array, shape (M, N)
Matrix to decompose
overwrite_a : boolean
Whether data in a is overwritten (may improve performance)
Returns
-------
s: array, shape (K,)
The singular values, sorted so that s[i] >= s[i+1]. K = min(M, N)
Raises LinAlgError if SVD computation does not converge
See also
--------
svd : return the full singular value decomposition of a matrix
diagsvd : return the Sigma matrix, given the vector s
"""
return svd(a, compute_uv=0, overwrite_a=overwrite_a)
def diagsvd(s, M, N):
"""Construct the sigma matrix in SVD from singular values and size M,N.
Parameters
----------
s : array, shape (M,) or (N,)
Singular values
M : integer
N : integer
Size of the matrix whose singular values are s
Returns
-------
S : array, shape (M, N)
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 : array, shape (M, N)
Returns
-------
Q : array, shape (M, K)
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)
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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