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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.

  The documentation is written assuming array arguments are of specified “core” shapes. However, array argument(s) of this function may have additional “batch” dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see Batched Linear Operations for details. Note that calls with zero-size batches are unsupported and will raise a `ValueError`.

  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