svds(solver=’propack’)#
- scipy.sparse.linalg.svds(A, k=6, ncv=None, tol=0, which='LM', v0=None, maxiter=None, return_singular_vectors=True, solver='arpack', random_state=None, options={})
Partial singular value decomposition of a sparse matrix using PROPACK.
Compute the largest or smallest k singular values and corresponding singular vectors of a sparse matrix A. The order in which the singular values are returned is not guaranteed.
In the descriptions below, let
M, N = A.shape
.- Parameters
- Asparse matrix or LinearOperator
Matrix to decompose. If A is a
LinearOperator
object, it must define bothmatvec
andrmatvec
methods.- kint, default: 6
Number of singular values and singular vectors to compute. Must satisfy
1 <= k <= min(M, N)
.- ncvint, optional
Ignored.
- tolfloat, optional
The desired relative accuracy for computed singular values. Zero (default) means machine precision.
- which{‘LM’, ‘SM’}
Which k singular values to find: either the largest magnitude (‘LM’) or smallest magnitude (‘SM’) singular values. Note that choosing
which='SM'
will force theirl
option to be setTrue
.- v0ndarray, optional
Starting vector for iterations: must be of length
A.shape[0]
. If not specified, PROPACK will generate a starting vector.- maxiterint, optional
Maximum number of iterations / maximal dimension of the Krylov subspace. Default is
10 * k
.- return_singular_vectors{True, False, “u”, “vh”}
Singular values are always computed and returned; this parameter controls the computation and return of singular vectors.
True
: return singular vectors.False
: do not return singular vectors."u"
: compute only the left singular vectors; returnNone
for the right singular vectors."vh"
: compute only the right singular vectors; returnNone
for the left singular vectors.
- solver{‘arpack’, ‘propack’, ‘lobpcg’}, optional
This is the solver-specific documentation for
solver='propack'
. ‘arpack’ and ‘lobpcg’ are also supported.- random_state{None, int,
numpy.random.Generator
, numpy.random.RandomState
}, optionalPseudorandom number generator state used to generate resamples.
If random_state is
None
(or np.random), thenumpy.random.RandomState
singleton is used. If random_state is an int, a newRandomState
instance is used, seeded with random_state. If random_state is already aGenerator
orRandomState
instance then that instance is used.- optionsdict, optional
A dictionary of solver-specific options. No solver-specific options are currently supported; this parameter is reserved for future use.
- Returns
- undarray, shape=(M, k)
Unitary matrix having left singular vectors as columns.
- sndarray, shape=(k,)
The singular values.
- vhndarray, shape=(k, N)
Unitary matrix having right singular vectors as rows.
Notes
This is an interface to the Fortran library PROPACK [1]. The current default is to run with IRL mode disabled unless seeking the smallest singular values/vectors (
which='SM'
).References
- 1
Larsen, Rasmus Munk. “PROPACK-Software for large and sparse SVD calculations.” Available online. URL http://sun. stanford. edu/rmunk/PROPACK (2004): 2008-2009.
Examples
Construct a matrix
A
from singular values and vectors.>>> from scipy.stats import ortho_group >>> from scipy.sparse import csc_matrix, diags >>> from scipy.sparse.linalg import svds >>> rng = np.random.default_rng() >>> orthogonal = csc_matrix(ortho_group.rvs(10, random_state=rng)) >>> s = [0.0001, 0.001, 3, 4, 5] # singular values >>> u = orthogonal[:, :5] # left singular vectors >>> vT = orthogonal[:, 5:].T # right singular vectors >>> A = u @ diags(s) @ vT
With only three singular values/vectors, the SVD approximates the original matrix.
>>> u2, s2, vT2 = svds(A, k=3, solver='propack') >>> A2 = u2 @ np.diag(s2) @ vT2 >>> np.allclose(A2, A.todense(), atol=1e-3) True
With all five singular values/vectors, we can reproduce the original matrix.
>>> u3, s3, vT3 = svds(A, k=5, solver='propack') >>> A3 = u3 @ np.diag(s3) @ vT3 >>> np.allclose(A3, A.todense()) True
The singular values match the expected singular values, and the singular vectors are as expected up to a difference in sign.
>>> (np.allclose(s3, s) and ... np.allclose(np.abs(u3), np.abs(u.toarray())) and ... np.allclose(np.abs(vT3), np.abs(vT.toarray()))) True
The singular vectors are also orthogonal. >>> (np.allclose(u3.T @ u3, np.eye(5)) and … np.allclose(vT3 @ vT3.T, np.eye(5))) True