lu
==============================================

Purpose
----------------

Computes the LU decomposition of a square matrix with partial
(row) pivoting, such that: :math:`X = LU`.

Format
----------------
.. function:: { l, u } = lu(x)

    :param x: square nonsingular matrix
    :type x: NxN matrix

    :return l: NxN "scrambled" lower triangular matrix. This is
        a lower triangular matrix that has been reordered based on the row pivoting.

    :rtype l: NxN matrix

    :return u: upper triangular matrix

    :rtype u: NxN matrix

Examples
----------------

::

    // Set seed for repeatable random numbers
    rndseed 13;

    // Print format, display 4 digits after decimal point
    format /rd 10,4;

    // Create random A matrix
    A = rndn(3, 3);

    { L, U } = lu(A);
    A2 = L*U;

::

          -0.0195     0.4054    -0.0874
    A =   -1.2948     0.1734     1.9712
           0.5408    -0.1294     0.7646

           0.0150     1.0000     0.0000
    L =    1.0000     0.0000     0.0000
          -0.4177    -0.1414     1.0000

          -1.2948     0.1734     1.9712
    U =    0.0000     0.4028    -0.1170
           0.0000     0.0000     1.5714

          -0.0195     0.4054    -0.0874
    A2 =  -1.2948     0.1734     1.9712 
           0.5408    -0.1294     0.7646

.. seealso:: Functions :func:`crout`, :func:`croutp`, :func:`chol`