svds(solver=’arpack’)

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 ARPACK.

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.

kint, optional

Number of singular values and singular vectors to compute. Must satisfy 1 <= k <= min(M, N) - 1. Default is 6.

ncvint, optional

The number of Lanczos vectors generated. The default is min(n, max(2*k + 1, 20)). If specified, must satistify k + 1 < ncv < min(M, N); ncv > 2*k is recommended.

tolfloat, optional

Tolerance for 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.

v0ndarray, optional

The starting vector for iteration: an (approximate) left singular vector if N > M and a right singular vector otherwise. Must be of length min(M, N). Default: random

maxiterint, optional

Maximum number of Arnoldi update iterations allowed; default is min(M, N) * 10.

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": if M <= N, compute only the left singular vectors and return None for the right singular vectors. Otherwise, compute all singular vectors.

  • "vh": if M > N, compute only the right singular vectors and return None for the left singular vectors. Otherwise, compute all singular vectors.

solver{‘arpack’, ‘propack’, ‘lobpcg’}, optional

This is the solver-specific documentation for solver='arpack'. ‘lobpcg’ and ‘propack’ are also supported.

random_state{None, int, numpy.random.Generator,

Pseudorandom number generator state used to generate resamples.

If random_state is None (or np.random), the numpy.random.RandomState singleton is used. If random_state is an int, a new RandomState instance is used, seeded with random_state. If random_state is already a Generator or RandomState 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 a naive implementation using ARPACK as an eigensolver on A.conj().T @ A or A @ A.conj().T, depending on which one is more efficient.

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='arpack')
>>> 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='arpack')
>>> 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.todense())) and
...  np.allclose(np.abs(vT3), np.abs(vT.todense())))
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