===================================
Solving Systems of Linear Equations                               [TOC]
===================================

In this example we solve a system of linear equations $Ax = b$ were the
coefficient matrix is *symmetric positiv definite and banded*.   We first
compute the Cholesky factorization $S = L L^T$ with __lapack::pbtrf__ which is
the FLENS port of LAPACK's __dpbtrf__.  We then solve $Lu=b$ and then
$L^T x = u$ using __lapack::pbtrs__ which is the port of __dpbtrs__.

Note that we might rename __lapack::pbtrf__ to `lapack::potrf` and
__lapack::pbtrs__ to `lapack::potrs`.

:links:  __lapack::pbtrf__  -> file:flens/lapack/pb/pbtrf.h
         __lapack::pbtrs__  -> file:flens/lapack/pb/pbtrs.h
         __dpbtrf__         -> file:cxxlapack/netlib/lapack/dpbtrf.f
         __dpbtrs__         -> file:cxxlapack/netlib/lapack/dpbtrs.f


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


Comments on Example Code
========================
:import: flens/examples/lapack-pbtrs.cc [brief]


Compile
=======
Note that we need to link against an external LAPACK implementation:

  *--[SHELL]----------------------------------------------------------------*
  |                                                                         |
  |  cd flens/examples                                                      |
  |  g++ -std=c++11 -Wall -I../.. -DUSE_CXXLAPACK -framework vecLib      +++|
  |      -o lapack-pbtrs lapack-pbtrs.cc                                    |
  |                                                                         |
  *-------------------------------------------------------------------------*


Run
===

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