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scipy.sparse.linalg.

# SuperLU

class scipy.sparse.linalg.SuperLU

  LU factorization of a sparse matrix.

  Factorization is represented as:

      Pr @ A @ Pc = L @ U

  To construct these `SuperLU` objects, call the `splu` and `spilu` functions.

  Notes

  Added in version 0.14.0.

  Examples

  The LU decomposition can be used to solve matrix equations. Consider:

      >>> import numpy as np
      >>> from scipy.sparse import csc_array
      >>> from scipy.sparse.linalg import splu
      >>> A = csc_array([[1,2,0,4], [1,0,0,1], [1,0,2,1], [2,2,1,0.]])

  This can be solved for a given right-hand side:

      >>> lu = splu(A)
      >>> b = np.array([1, 2, 3, 4])
      >>> x = lu.solve(b)
      >>> A.dot(x)
      array([ 1.,  2.,  3.,  4.])

  The `lu` object also contains an explicit representation of the decomposition. The permutations are represented as mappings of indices:

      >>> lu.perm_r
      array([2, 1, 3, 0], dtype=int32)  # may vary
      >>> lu.perm_c
      array([0, 1, 3, 2], dtype=int32)  # may vary

  The L and U factors are sparse matrices in CSC format:

      >>> lu.L.toarray()
      array([[ 1. ,  0. ,  0. ,  0. ],  # may vary
             [ 0.5,  1. ,  0. ,  0. ],
             [ 0.5, -1. ,  1. ,  0. ],
             [ 0.5,  1. ,  0. ,  1. ]])
      >>> lu.U.toarray()
      array([[ 2. ,  2. ,  0. ,  1. ],  # may vary
             [ 0. , -1. ,  1. , -0.5],
             [ 0. ,  0. ,  5. , -1. ],
             [ 0. ,  0. ,  0. ,  2. ]])

  The permutation matrices can be constructed:

      >>> Pr = csc_array((np.ones(4), (lu.perm_r, np.arange(4))))
      >>> Pc = csc_array((np.ones(4), (np.arange(4), lu.perm_c)))

  We can reassemble the original matrix:

      >>> (Pr.T @ (lu.L @ lu.U) @ Pc.T).toarray()
      array([[ 1.,  2.,  0.,  4.],
             [ 1.,  0.,  0.,  1.],
             [ 1.,  0.,  2.,  1.],
             [ 2.,  2.,  1.,  0.]])

  Attributes:
    `shape`

      Shape of the original matrix as a tuple of ints.

    `nnz`

      Number of nonzero elements in the matrix.

    `perm_c`

      Permutation Pc represented as an array of indices.

    `perm_r`

      Permutation Pr represented as an array of indices.

    `L`

      Lower triangular factor with unit diagonal as a `scipy.sparse.csc_array`.

    `U`

      Upper triangular factor as a `scipy.sparse.csc_array`.

  Methods

  `solve`(rhs[, trans]) | Solves linear system of equations with one or several right-hand sides.