.. _LU_decompHW:

LU Decomposition Homework
==========================

#. How can LU decomposition improve the computational efficiency of finding
   solutions to :math:`\mathbf{A}\,\bm{x} = \bm{b}` for a set of several
   :math:`\bm{b}` vectors?

#. Start with the matrix

   .. math:: \mathbf{A} = \begin{bmatrix}
                         6  & 9 & -2 \\ 
                         -3  & 0 & 1 \\ 
                         6  & 4 & 6
                      \end{bmatrix}.

   Using LU decomposition, find the matrix factors of :math:`\bf{A}`
   (:math:`\bf{L}`, :math:`\bf{U}`, and :math:`\bf{P}`). What are the
   determinants of each? Find the determinant of :math:`\bf{A}` by means
   LU decomposition, ``det(P)*prod(diag(U))``. Verify that ``det(A)``
   gives the same answer.

#. Using LU decomposition with Python, find the solution to the following
   system of equations (:math:`\mathbf{A}\,\bm{x} = \bm{b}`). Use
   :math:`\bf{L}`, :math:`\bf{U}`, and :math:`\bf{P}` in your solution.

   ::

        A = np.array([[-8, 2, 8, -1], [2, -6, 6, 4], [-3, 9, 14, 11], [-9, 8, -4, 13]]).astype(float)
        b = np.array([[11, 5, 1, 4]]).astype(float).T