"""SVD decomposition functions."""
import numpy as np
from numpy import zeros, r_, diag, dot, arccos, arcsin, where, clip
# Local imports.
from ._misc import LinAlgError, _datacopied
from .lapack import get_lapack_funcs, _compute_lwork
from ._decomp import _asarray_validated
__all__ = ['svd', 'svdvals', 'diagsvd', 'orth', 'subspace_angles', 'null_space']
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 (default), `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
--------
>>> import numpy as np
>>> from scipy import linalg
>>> rng = np.random.default_rng()
>>> m, n = 9, 6
>>> a = rng.standard_normal((m, n)) + 1.j*rng.standard_normal((m, n))
>>> U, s, Vh = linalg.svd(a)
>>> U.shape, s.shape, Vh.shape
((9, 9), (6,), (6, 6))
Reconstruct the original matrix from the decomposition:
>>> sigma = np.zeros((m, n))
>>> for i in range(min(m, n)):
... sigma[i, i] = s[i]
>>> a1 = np.dot(U, np.dot(sigma, Vh))
>>> np.allclose(a, a1)
True
Alternatively, use ``full_matrices=False`` (notice that the shape of
``U`` is then ``(m, n)`` instead of ``(m, m)``):
>>> U, s, Vh = linalg.svd(a, full_matrices=False)
>>> U.shape, s.shape, Vh.shape
((9, 6), (6,), (6, 6))
>>> S = np.diag(s)
>>> 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
# accommodate empty matrix
if a1.size == 0:
u0, s0, v0 = svd(np.eye(2, dtype=a1.dtype))
s = np.empty_like(a1, shape=(0,), dtype=s0.dtype)
if full_matrices:
u = np.empty_like(a1, shape=(m, m), dtype=u0.dtype)
u[...] = np.identity(m)
v = np.empty_like(a1, shape=(n, n), dtype=v0.dtype)
v[...] = np.identity(n)
else:
u = np.empty_like(a1, shape=(m, 0), dtype=u0.dtype)
v = np.empty_like(a1, shape=(0, n), dtype=v0.dtype)
if compute_uv:
return u, s, v
else:
return s
overwrite_a = overwrite_a or (_datacopied(a1, a))
if not isinstance(lapack_driver, str):
raise TypeError('lapack_driver must be a string')
if lapack_driver not in ('gesdd', 'gesvd'):
message = f'lapack_driver must be "gesdd" or "gesvd", not "{lapack_driver}"'
raise ValueError(message)
if lapack_driver == 'gesdd' and compute_uv:
# XXX: revisit int32 when ILP64 lapack becomes a thing
max_mn, min_mn = (m, n) if m > n else (n, m)
if full_matrices:
if max_mn*max_mn > np.iinfo(np.int32).max:
raise ValueError(f"Indexing a matrix size {max_mn} x {max_mn} "
" would incur integer overflow in LAPACK.")
else:
sz = max(m * min_mn, n * min_mn)
if max(m * min_mn, n * min_mn) > np.iinfo(np.int32).max:
raise ValueError(f"Indexing a matrix of {sz} elements would "
"incur an in integer overflow in LAPACK.")
funcs = (lapack_driver, lapack_driver + '_lwork')
gesXd, gesXd_lwork = get_lapack_funcs(funcs, (a1,), ilp64='preferred')
# 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 %dth 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.
See Also
--------
svd : Compute the full singular value decomposition of a matrix.
diagsvd : Construct the Sigma matrix, given the vector s.
Examples
--------
>>> import numpy as np
>>> from scipy.linalg import svdvals
>>> m = np.array([[1.0, 0.0],
... [2.0, 3.0],
... [1.0, 1.0],
... [0.0, 2.0],
... [1.0, 0.0]])
>>> svdvals(m)
array([ 4.28091555, 1.63516424])
We can verify the maximum singular value of `m` by computing the maximum
length of `m.dot(u)` over all the unit vectors `u` in the (x,y) plane.
We approximate "all" the unit vectors with a large sample. Because
of linearity, we only need the unit vectors with angles in [0, pi].
>>> t = np.linspace(0, np.pi, 2000)
>>> u = np.array([np.cos(t), np.sin(t)])
>>> np.linalg.norm(m.dot(u), axis=0).max()
4.2809152422538475
`p` is a projection matrix with rank 1. With exact arithmetic,
its singular values would be [1, 0, 0, 0].
>>> v = np.array([0.1, 0.3, 0.9, 0.3])
>>> p = np.outer(v, v)
>>> svdvals(p)
array([ 1.00000000e+00, 2.02021698e-17, 1.56692500e-17,
8.15115104e-34])
The singular values of an orthogonal matrix are all 1. Here, we
create a random orthogonal matrix by using the `rvs()` method of
`scipy.stats.ortho_group`.
>>> from scipy.stats import ortho_group
>>> orth = ortho_group.rvs(4)
>>> svdvals(orth)
array([ 1., 1., 1., 1.])
"""
return svd(a, compute_uv=0, overwrite_a=overwrite_a,
check_finite=check_finite)
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
See Also
--------
svd : Singular value decomposition of a matrix
svdvals : Compute singular values of a matrix.
Examples
--------
>>> import numpy as np
>>> from scipy.linalg import diagsvd
>>> vals = np.array([1, 2, 3]) # The array representing the computed svd
>>> diagsvd(vals, 3, 4)
array([[1, 0, 0, 0],
[0, 2, 0, 0],
[0, 0, 3, 0]])
>>> diagsvd(vals, 4, 3)
array([[1, 0, 0],
[0, 2, 0],
[0, 0, 3],
[0, 0, 0]])
"""
part = diag(s)
typ = part.dtype.char
MorN = len(s)
if MorN == M:
return np.hstack((part, zeros((M, N - M), dtype=typ)))
elif MorN == N:
return r_[part, zeros((M - N, N), dtype=typ)]
else:
raise ValueError("Length of s must be M or N.")
# Orthonormal decomposition
def orth(A, rcond=None):
"""
Construct an orthonormal basis for the range of A using SVD
Parameters
----------
A : (M, N) array_like
Input array
rcond : float, optional
Relative condition number. Singular values ``s`` smaller than
``rcond * max(s)`` are considered zero.
Default: floating point eps * max(M,N).
Returns
-------
Q : (M, K) ndarray
Orthonormal basis for the range of A.
K = effective rank of A, as determined by rcond
See Also
--------
svd : Singular value decomposition of a matrix
null_space : Matrix null space
Examples
--------
>>> import numpy as np
>>> from scipy.linalg import orth
>>> A = np.array([[2, 0, 0], [0, 5, 0]]) # rank 2 array
>>> orth(A)
array([[0., 1.],
[1., 0.]])
>>> orth(A.T)
array([[0., 1.],
[1., 0.],
[0., 0.]])
"""
u, s, vh = svd(A, full_matrices=False)
M, N = u.shape[0], vh.shape[1]
if rcond is None:
rcond = np.finfo(s.dtype).eps * max(M, N)
tol = np.amax(s, initial=0.) * rcond
num = np.sum(s > tol, dtype=int)
Q = u[:, :num]
return Q
def null_space(A, rcond=None):
"""
Construct an orthonormal basis for the null space of A using SVD
Parameters
----------
A : (M, N) array_like
Input array
rcond : float, optional
Relative condition number. Singular values ``s`` smaller than
``rcond * max(s)`` are considered zero.
Default: floating point eps * max(M,N).
Returns
-------
Z : (N, K) ndarray
Orthonormal basis for the null space of A.
K = dimension of effective null space, as determined by rcond
See Also
--------
svd : Singular value decomposition of a matrix
orth : Matrix range
Examples
--------
1-D null space:
>>> import numpy as np
>>> from scipy.linalg import null_space
>>> A = np.array([[1, 1], [1, 1]])
>>> ns = null_space(A)
>>> ns * np.copysign(1, ns[0,0]) # Remove the sign ambiguity of the vector
array([[ 0.70710678],
[-0.70710678]])
2-D null space:
>>> from numpy.random import default_rng
>>> rng = default_rng()
>>> B = rng.random((3, 5))
>>> Z = null_space(B)
>>> Z.shape
(5, 2)
>>> np.allclose(B.dot(Z), 0)
True
The basis vectors are orthonormal (up to rounding error):
>>> Z.T.dot(Z)
array([[ 1.00000000e+00, 6.92087741e-17],
[ 6.92087741e-17, 1.00000000e+00]])
"""
u, s, vh = svd(A, full_matrices=True)
M, N = u.shape[0], vh.shape[1]
if rcond is None:
rcond = np.finfo(s.dtype).eps * max(M, N)
tol = np.amax(s, initial=0.) * rcond
num = np.sum(s > tol, dtype=int)
Q = vh[num:,:].T.conj()
return Q
def subspace_angles(A, B):
r"""
Compute the subspace angles between two matrices.
Parameters
----------
A : (M, N) array_like
The first input array.
B : (M, K) array_like
The second input array.
Returns
-------
angles : ndarray, shape (min(N, K),)
The subspace angles between the column spaces of `A` and `B` in
descending order.
See Also
--------
orth
svd
Notes
-----
This computes the subspace angles according to the formula
provided in [1]_. For equivalence with MATLAB and Octave behavior,
use ``angles[0]``.
.. versionadded:: 1.0
References
----------
.. [1] Knyazev A, Argentati M (2002) Principal Angles between Subspaces
in an A-Based Scalar Product: Algorithms and Perturbation
Estimates. SIAM J. Sci. Comput. 23:2008-2040.
Examples
--------
An Hadamard matrix, which has orthogonal columns, so we expect that
the suspace angle to be :math:`\frac{\pi}{2}`:
>>> import numpy as np
>>> from scipy.linalg import hadamard, subspace_angles
>>> rng = np.random.default_rng()
>>> H = hadamard(4)
>>> print(H)
[[ 1 1 1 1]
[ 1 -1 1 -1]
[ 1 1 -1 -1]
[ 1 -1 -1 1]]
>>> np.rad2deg(subspace_angles(H[:, :2], H[:, 2:]))
array([ 90., 90.])
And the subspace angle of a matrix to itself should be zero:
>>> subspace_angles(H[:, :2], H[:, :2]) <= 2 * np.finfo(float).eps
array([ True, True], dtype=bool)
The angles between non-orthogonal subspaces are in between these extremes:
>>> x = rng.standard_normal((4, 3))
>>> np.rad2deg(subspace_angles(x[:, :2], x[:, [2]]))
array([ 55.832]) # random
"""
# Steps here omit the U and V calculation steps from the paper
# 1. Compute orthonormal bases of column-spaces
A = _asarray_validated(A, check_finite=True)
if len(A.shape) != 2:
raise ValueError(f'expected 2D array, got shape {A.shape}')
QA = orth(A)
del A
B = _asarray_validated(B, check_finite=True)
if len(B.shape) != 2:
raise ValueError(f'expected 2D array, got shape {B.shape}')
if len(B) != len(QA):
raise ValueError('A and B must have the same number of rows, got '
f'{QA.shape[0]} and {B.shape[0]}')
QB = orth(B)
del B
# 2. Compute SVD for cosine
QA_H_QB = dot(QA.T.conj(), QB)
sigma = svdvals(QA_H_QB)
# 3. Compute matrix B
if QA.shape[1] >= QB.shape[1]:
B = QB - dot(QA, QA_H_QB)
else:
B = QA - dot(QB, QA_H_QB.T.conj())
del QA, QB, QA_H_QB
# 4. Compute SVD for sine
mask = sigma ** 2 >= 0.5
if mask.any():
mu_arcsin = arcsin(clip(svdvals(B, overwrite_a=True), -1., 1.))
else:
mu_arcsin = 0.
# 5. Compute the principal angles
# with reverse ordering of sigma because smallest sigma belongs to largest
# angle theta
theta = where(mask, mu_arcsin, arccos(clip(sigma[::-1], -1., 1.)))
return theta