==============================
Computing the QR Factorization                                    [TOC]
==============================

In this example we compute the $QR$ factorization and use it for solving a
system of linear equations.

Consider the system of linear equations $Ax=b$.  We store the coefficient matrix
$A$ and the right-hand side $b$ in a single matrix $Ab = (A,b)$.  We then
compute the $QR$ factorization of $Ab$ with __lapack::qrf__ such that
$QR = (A,b).$   Hence, we have $R = (Q^T A, Q^T b)$ where $R$ is an upper
trapezoidal matrix and $Q^T A$ upper triangular.  We finally use the triangular
solver __blas::sm__ to obtain the solution of $(Q^T A) x = (Q^T b)$.

Function __lapack::qrf__ is the FLENS port of LAPACK's __geqrf__ and
__blas::sm__ the port of the BLAS routine __trsm__.

:links:  __lapack::qrf__  -> file:flens/lapack/qr/qrf.h
         __geqrf__        -> file:flens/lapack/interface/ref_lapack/dgeqrf.f
         __blas::sm__     -> file:flens/blas/level3/sm.h
         __trsm__         -> file:flens/lapack/interface/ref_blas/dtrsm.f


Example Code
============
:import: flens/examples/lapack-geqrf.cc [stripped, downloadable]

Comments on Example Code
========================
:import: flens/examples/lapack-geqrf.cc

Compile
=======

  *--[SHELL]----------------------------------------------------------------*
  |                                                                         |
  |  cd flens/examples                                                      |
  |  clang++ -std=c++0x -Wall -I../.. -o lapack-geqrf lapack-geqrf.cc       |
  |                                                                         |
  *-------------------------------------------------------------------------*


Run
===

  *--[SHELL]----------------------------------------------------------------*
  |                                                                         |
  |  cd flens/examples                                                      |
  |  ./lapack-geqrf                                                         |
  |                                                                         |
  *-------------------------------------------------------------------------*

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