def svd(a,full_matrices=1,compute_uv=1,overwrite_a=0):
"""Compute singular value decomposition (SVD) of matrix a.
Description:
Singular value decomposition of a matrix a is
a = u * sigma * v^H,
where v^H denotes conjugate(transpose(v)), u,v are unitary
matrices, sigma is zero matrix with a main diagonal containing
real non-negative singular values of the matrix a.
Inputs:
a -- An M x N matrix.
compute_uv -- If zero, then only the vector of singular values
is returned.
Outputs:
u -- An M x M unitary matrix [compute_uv=1].
s -- An min(M,N) vector of singular values in descending order,
sigma = diagsvd(s).
vh -- An N x N unitary matrix [compute_uv=1], vh = v^H.
"""
# A hack until full_matrices == 0 support is fixed here.
if full_matrices == 0:
import numpy.linalg
return numpy.linalg.svd(a, full_matrices=0, compute_uv=compute_uv)
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2:
raise ValueError, 'expected matrix'
m,n = a1.shape
overwrite_a = overwrite_a or (_datanotshared(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,
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 %-th argument of internal gesdd'%(-info)
if compute_uv:
return u,s,v
else:
return s
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
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 : ndarray
Matrix to decompose, of shape ``(M,N)``.
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.
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_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 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
"""
# A hack until full_matrices == 0 support is fixed here.
if full_matrices == 0:
import numpy.linalg
return numpy.linalg.svd(a, full_matrices=0, compute_uv=compute_uv)
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2:
raise ValueError('expected matrix')
m, n = a1.shape
overwrite_a = overwrite_a or (_datanotshared(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,
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 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
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 : ndarray
Matrix to decompose, of shape ``(M,N)``.
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.
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_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 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
"""
# A hack until full_matrices == 0 support is fixed here.
if full_matrices == 0:
import numpy.linalg
return numpy.linalg.svd(a, full_matrices=0, compute_uv=compute_uv)
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,))
gesdd_info = get_func_info(gesdd)
if gesdd_info.module_name[:7] == 'flapack':
lwork = calc_lwork.gesdd(gesdd_info.prefix, m, n, compute_uv)[1]
u,s,v,info = gesdd(a1,compute_uv = compute_uv, lwork = lwork,
overwrite_a = overwrite_a)
else: # 'clapack'
raise NotImplementedError('calling gesdd from %s' % gesdd_info.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 svd(a, full_matrices=True, compute_uv=True, overwrite_a=False,
check_finite=True, returns="U, S, V"):
"""
Singular Value Decomposition.
Factorizes the matrix a into two unitary matrices U and Vh, and
a diagonal matrix S of suitable shape with non-negative real
numbers on the diagonal.
.
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)``.
overwrite_a : bool, optional
Whether to overwrite `a`; may improve performance.
Default is False.
check_finite : boolean, 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: string, optional
Select the returned values, among ``U``, ``S``, ``s``, ``V`` and ``Vh``,.
Default is ``"U, S, Vh"``.
Returns
-------
The selection of return values is configurable by the ``returns`` parameter.
U : ndarray
Unitary matrix having left singular vectors as columns.
Of shape ``(M,M)`` or ``(M,K)``, depending on `full_matrices`.
S : ndarray
A matrix with the singular values of ``a``, sorted in non-increasing
order, in the main diagonal and zeros elsewhere.
Of shape ``(M,N)`` or ``(K,K)``, depending on `full_matrices`.
V : ndarray
Unitary matrix having right singular vectors as columns.
Of shape ``(N,N)`` or ``(N, K)`` depending on `full_matrices`.
s : ndarray, not returned by default
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 ``(N, K)`` depending on `full_matrices`.
Raises
------
LinAlgError
If SVD computation does not converge.
Examples
--------
>>> import numpy as np
>>> from wish.examples import svd
>>> a = np.random.randn(9, 6) + 1.j*np.random.randn(9, 6)
>>> U, S, V = svd(a)
>>> U.shape, S.shape, V.shape
((9, 9), (9, 6), (6, 6))
>>> U, S, Vh = svd(a, full_matrices=False, returns="U, S, Vh")
>>> U.shape, S.shape, Vh.shape
((9, 9), (9, 6), (6, 6))
>>> np.allclose(a, np.dot(U, np.dot(S, Vh)))
True
>>> s = svd(a, returns="s")
>>> np.allclose(s, np.diagonal(S))
True
"""
wishlist = wish.make(returns)
for name in wishlist:
if name not in ["U", "S", "V", "s", "Vh"]:
error = "unexpected return value {name!r}"
raise TypeError(error.format(name=name))
if check_finite:
a1 = asarray_chkfinite(a)
else:
a1 = asarray(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,))
lwork = calc_lwork.gesdd(gesdd.typecode, m, n, compute_uv)[1]
compute_uv = "U" in wishlist or "V" in wishlist or "Vh" in wishlist
U,s,Vh,info = gesdd(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 "V" in wishlist:
V = Vh.transpose().conjugate()
if "S" in wishlist:
S = _diagsvd(s, U.shape[1], Vh.shape[0])
# f**k, maybe not computed. Use shape of A.
# Interaction with full_matrices?
return wishlist.grant()