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scipy.sparse.linalg.

# spilu

scipy.sparse.linalg.spilu(A, drop_tol=None, fill_factor=None, drop_rule=None, permc_spec=None, diag_pivot_thresh=None, relax=None, panel_size=None, options=None)

  Compute an incomplete LU decomposition for a sparse, square matrix.

  The resulting object is an approximation to the inverse of A.

  Parameters:
    A(N, N) array_like

      Sparse array to factorize. Most efficient when provided in CSC format. Other formats will be converted to CSC before factorization.

    drop_tol float, optional

      Drop tolerance (`0 <= tol <= 1`) for an incomplete LU decomposition. (default: 1e-4)

      Note that drop_tol primarily affects entries generated as fill-in during the ILU factorization; for matrices that produce little or no fill-in, changing this parameter may have no visible effect on the sparsity pattern of the factors.

    fill_factor float, optional

      Specifies the fill ratio upper bound (`>= 1.0`) for ILU. (default: 10)

    drop_rule str, optional

      Comma-separated string of drop rules to use. Available rules: `basic`, `prows`, `column`, `area`, `secondary`, `dynamic`, `interp`. (Default: `basic,area`)

      See SuperLU documentation for details.

    permc_spec str, optional

      How to permute the columns of the matrix for sparsity preservation. (default: ‘COLAMD’)

      - `NATURAL`: natural ordering.

      - `MMD_ATA`: minimum degree ordering on the structure of A^T A.

      - `MMD_AT_PLUS_A`: minimum degree ordering on the structure of A^T+A.

      - `COLAMD`: approximate minimum degree column ordering

    diag_pivot_thresh float, optional

      Threshold used for a diagonal entry to be an acceptable pivot. See SuperLU user’s guide for details [1]

    relax int, optional

      Expert option for customizing the degree of relaxing supernodes. See SuperLU user’s guide for details [1]

    panel_size int, optional

      Expert option for customizing the panel size. See SuperLU user’s guide for details [1]

    options dict, optional

      Dictionary containing additional expert options to SuperLU. See SuperLU user guide [1] (section 2.4 on the ‘Options’ argument) for more details. For example, you can specify `options=dict(Equil=False, IterRefine='SINGLE'))` to turn equilibration off and perform a single iterative refinement.

  Returns:
    invA_approx scipy.sparse.linalg.SuperLU

      Object, which has a `solve` method.

  See also

  `splu`

    complete LU decomposition

  Notes

  When a real array is factorized and the returned SuperLU object’s `solve()` method is used with complex arguments an error is generated. Instead, cast the initial array to complex and then factorize.

  To improve the better approximation to the inverse, you may need to increase fill_factor AND decrease drop_tol.

  The effect of drop_tol is matrix-dependent. In particular, drop_tol does not guarantee that existing off-diagonal entries will be removed; it controls dropping of candidate entries during factorization (often fill-in). For some sparsity patterns, the ILU factors can be identical to the full LU factors even for large drop_tol.

  This function uses the SuperLU library.

  References

  [1] (1,2,3,4)

  SuperLU https://portal.nersc.gov/project/sparse/superlu/

  Examples

      >>> import numpy as np
      >>> from scipy.sparse import csc_array
      >>> from scipy.sparse.linalg import spilu
      >>> A = csc_array([[1., 0., 0.], [5., 0., 2.], [0., -1., 0.]], dtype=float)
      >>> B = spilu(A)
      >>> x = np.array([1., 2., 3.], dtype=float)
      >>> B.solve(x)
      array([ 1. , -3. , -1.5])
      >>> A.dot(B.solve(x))
      array([ 1.,  2.,  3.])
      >>> B.solve(A.dot(x))
      array([ 1.,  2.,  3.])