===============================================
Triangular Solver (for Matrix Vector Equations)                         [TOC]
===============================================

As an example we consider the system of linear equations $Ax = b$ with

---- LATEX ---------------------------------------------------------------------
A =
\left(\begin{array}{ccc}
      4 & 5 & 6 \\
      1 & 2 & 3 \\
      7 & 8 & 1 \\
      \end{array}\right)
\quad\text{and}
\quad b =
\left(\begin{array}{c}
      32 \\
      14 \\
      26 \\
      \end{array}\right)
--------------------------------------------------------------------------------

An $LU$ factorization without pivoting is given by

---- LATEX ---------------------------------------------------------------------
L =
\left(\begin{array}{ccc}
      1             &   0   &   0   \\
      \frac{1}{4}   &   1   &   0   \\
      \frac{7}{4}   &   -1  &   1   \\
      \end{array}\right)
\quad\text{and}
\quad U =
\left(\begin{array}{ccc}
      4 &          5  & 6           \\
      0 & \frac{3}{4} & \frac{3}{2} \\
      0 &           0 &          -8 \\
      \end{array}\right)
\quad\text{and}
--------------------------------------------------------------------------------

Hence the problem $Ax = LUx = b$ can be solved by first computing

---- LATEX ---------------------------------------------------------------------
y = L^{-1} b
--------------------------------------------------------------------------------

and then

---- LATEX ---------------------------------------------------------------------
x = U^{-1} y
--------------------------------------------------------------------------------


Skeleton for simple Example
===========================
:import: session13/simple_dtrsv_ex.c [fold]

Exercise:
=========
- Solve the above system of linear equations with pencil and paper:

  - Solve $Ly = b$ with forward substitution.
  - Solve $Ux = y$ with backward substitution.

- Make yourself familiar with the structure of the program.  Compile and run it:

  - For testing the *col major case* compile it with

    ---- SHELL(path=session13) -------------------------------------------------
    gcc -Wall -std=c11 -o simple_dtrsv_ex simple_dtrsv_ex.c
    ----------------------------------------------------------------------------

  - For testing the *row major case* compile it with

    ---- SHELL(path=session13) -------------------------------------------------
    gcc -Wall -std=c11 -o simple_dtrsv_ex -DCOLMAJOR=0 simple_dtrsv_ex.c
    ----------------------------------------------------------------------------

- Change the program such that $A$ gets initialized as

  ---- LATEX -------------------------------------------------------------------
  A =
  \left(\begin{array}{rrrr}
  4             & 5             & 6             &   32  \\
  \frac{1}{4}   & \frac{3}{4}   & \frac{3}{2}   &   14  \\
  \frac{7}{4}   & -1            & -8            &   26  \\
  \end{array}\right)
  ------------------------------------------------------------------------------

- Implement function `dtrsv`.  You implementation should be optimized for the
  row major and the col major case:

  - For col major cases it should be based on `daxpy` function calls.
  - For row major cases it should be based on `ddot` function calls.