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# 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', rng=None, options=None, *, random_state=None)

  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:
    A sparse matrix or LinearOperator

      Matrix to decompose.

    k int, optional

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

    ncv int, optional

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

    tol float, 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.

    v0 ndarray, 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

    maxiter int, 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.

    rng `numpy.random.Generator`, optional

      Pseudorandom number generator state. When rng is None, a new `numpy.random.Generator` is created using entropy from the operating system. Types other than `numpy.random.Generator` are passed to `numpy.random.default_rng` to instantiate a `Generator`.

    options dict, optional

      A dictionary of solver-specific options. No solver-specific options are currently supported; this parameter is reserved for future use.

  Returns:
    u ndarray, shape=(M, k)

      Unitary matrix having left singular vectors as columns.

    s ndarray, shape=(k,)

      The singular values.

    vh ndarray, 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.

      >>> import numpy as np
      >>> from scipy.stats import ortho_group
      >>> from scipy.sparse import csc_array, diags_array
      >>> from scipy.sparse.linalg import svds
      >>> rng = np.random.default_rng()
      >>> orthogonal = csc_array(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_array(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.toarray(), 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.toarray())
      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