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# scipy.linalg.lu_factor

scipy.linalg.lu_factor(a, overwrite_a=False, check_finite=True)

  Compute pivoted LU decomposition of a matrix.

  The decomposition is:

      A = P L U

  where P is a permutation matrix, L lower triangular with unit diagonal elements, and U upper triangular.

  Parameters:
    a(M, N) array_like

      Matrix to decompose

    overwrite_a bool, optional

      Whether to overwrite data in A (may increase performance)

    check_finite bool, optional

      Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.

  Returns:
    lu(M, N) ndarray

      Matrix containing U in its upper triangle, and L in its lower triangle. The unit diagonal elements of L are not stored.

    piv(K,) ndarray

      Pivot indices representing the permutation matrix P: row i of matrix was interchanged with row piv[i]. Of shape `(K,)`, with `K = min(M, N)`.

  See also

  `lu`

    gives lu factorization in more user-friendly format

  `lu_solve`

    solve an equation system using the LU factorization of a matrix

  Notes

  This is a wrapper to the `*GETRF` routines from LAPACK. Unlike `lu`, it outputs the L and U factors into a single array and returns pivot indices instead of a permutation matrix.

  While the underlying `*GETRF` routines return 1-based pivot indices, the `piv` array returned by `lu_factor` contains 0-based indices.

  Examples

      >>> import numpy as np
      >>> from scipy.linalg import lu_factor
      >>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]])
      >>> lu, piv = lu_factor(A)
      >>> piv
      array([2, 2, 3, 3], dtype=int32)

  Convert LAPACK’s `piv` array to NumPy index and test the permutation

      >>> def pivot_to_permutation(piv):
      ...     perm = np.arange(len(piv))
      ...     for i in range(len(piv)):
      ...         perm[i], perm[piv[i]] = perm[piv[i]], perm[i]
      ...     return perm
      ...
      >>> p_inv = pivot_to_permutation(piv)
      >>> p_inv
      array([2, 0, 3, 1])
      >>> L, U = np.tril(lu, k=-1) + np.eye(4), np.triu(lu)
      >>> np.allclose(A[p_inv] - L @ U, np.zeros((4, 4)))
      True

  The P matrix in P L U is defined by the inverse permutation and can be recovered using argsort:

      >>> p = np.argsort(p_inv)
      >>> p
      array([1, 3, 0, 2])
      >>> np.allclose(A - L[p] @ U, np.zeros((4, 4)))
      True

  or alternatively:

      >>> P = np.eye(4)[p]
      >>> np.allclose(A - P @ L @ U, np.zeros((4, 4)))
      True