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=None)

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 both `matvec` and `rmatvec` 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 the `irl` option to be set `True`.

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; return `None` for the right singular vectors.

• `"vh"`: compute only the right singular vectors; return `None` 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`,

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 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.

```>>> import numpy as np
>>> 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
```