namespace Eigen {

/** \page TutorialAdvancedLinearAlgebra Tutorial 3/3 - Advanced linear algebra
    \ingroup Tutorial

<div class="eimainmenu">\ref index "Overview"
  | \ref TutorialCore "Core features"
  | \ref TutorialGeometry "Geometry"
  | \b Advanced \b linear \b algebra
</div>

\b Table \b of \b contents
  - \ref TutorialAdvLinearSolvers
  - \ref TutorialAdvLU
  - \ref TutorialAdvCholesky
  - \ref TutorialAdvQR
  - \ref TutorialAdvEigenProblems

\section TutorialAdvLinearSolvers Solving linear problems

This part of the tutorial focuses on solving linear problem of the form \f$ A \mathbf{x} = b \f$,
where both \f$ A \f$ and \f$ b \f$ are known, and \f$ x \f$ is the unknown. Moreover, \f$ A \f$
assumed to be a squared matrix. Of course, the methods described here can be used whenever an expression
involve the product of an inverse matrix with a vector or another matrix: \f$ A^{-1} B \f$.
Eigen offers various algorithms to this problem, and its choice mainly depends on the nature of
the matrix \f$ A \f$, such as its shape, size and numerical properties.

\subsection TutorialAdv_Triangular Triangular solver
If the matrix \f$ A \f$ is triangular (upper or lower) and invertible (the coefficients of the diagonal
are all not zero), then the problem can be solved directly using MatrixBase::solveTriangular(), or better,
MatrixBase::solveTriangularInPlace(). Here is an example:
<table class="tutorial_code"><tr><td>
\include MatrixBase_marked.cpp
</td>
<td>
output:
\include MatrixBase_marked.out
</td></tr></table>

See MatrixBase::solveTriangular() for more details.


\subsection TutorialAdv_Inverse Direct inversion (for small matrices)
If the matrix \f$ A \f$ is small (\f$ \leq 4 \f$) and invertible, then the problem can be solved
by directly computing the inverse of the matrix \f$ A \f$: \f$ \mathbf{x} = A^{-1} b \f$. With Eigen,
this can be implemented like this:

\code
#include <Eigen/LU>
Matrix4f A = Matrix4f::Random();
Vector4f b = Vector4f::Random();
Vector4f x = A.inverse() * b;
\endcode

Note that the function inverse() is defined in the LU module.
See MatrixBase::inverse() for more details.


\subsection TutorialAdv_Symmetric Cholesky (for symmetric matrices)
If the matrix \f$ A \f$ is \b symmetric, or more generally selfadjoint, and \b positive \b definite (SPD), then
the best method is to use a Cholesky decomposition.
Such SPD matrices often arise when solving overdetermined problems in a least square sense (see below).
Eigen offers two different Cholesky decompositions: a \f$ LL^T \f$ decomposition where L is a lower triangular matrix,
and a \f$ LDL^T \f$ decomposition where L is lower triangular with unit diagonal and D is a diagonal matrix.
The latter avoids square roots and is therefore slightly more stable than the former one.
\code
#include <Eigen/Cholesky>
MatrixXf D = MatrixXf::Random(8,4);
MatrixXf A = D.transpose() * D;
VectorXf b = D.transpose() * VectorXf::Random(4);
VectorXf x;
A.llt().solve(b,&x);   // using a LLT factorization
A.ldlt().solve(b,&x);  // using a LDLT factorization
\endcode
when the value of the right hand side \f$ b \f$ is not needed anymore, then it is faster to use
the \em in \em place API, e.g.:
\code
A.llt().solveInPlace(b);
\endcode
In that case the value of \f$ b \f$ is replaced by the result \f$ x \f$.

If the linear problem has to solved for various right hand side \f$ b_i \f$, but same matrix \f$ A \f$,
then it is highly recommended to perform the decomposition of \$ A \$ only once, e.g.:
\code
// ...
LLT<MatrixXf> lltOfA(A);
lltOfA.solveInPlace(b0);
lltOfA.solveInPlace(b1);
// ...
\endcode

\sa Cholesky_Module, LLT::solve(), LLT::solveInPlace(), LDLT::solve(), LDLT::solveInPlace(), class LLT, class LDLT.


\subsection TutorialAdv_LU LU decomposition (for most cases)
If the matrix \f$ A \f$ does not fit in one of the previous category, or if you are unsure about the numerical
stability of your problem, then you can use the LU solver based on a decomposition of the same name.
Actually, Eigen's LU module does not implement a standard LU decomposition, but rather a so called LU decomposition
with full pivoting and rank update which has the advantages to be numerically much more stable.
The API of the LU solver is the same than the Cholesky one, except that there is no \em in \em place variant:
\code
Matrix4f A = Matrix4f::Random();
Vector4f b = Vector4f::Random();
Vector4f x;
A.lu().solve(b, &x);
\endcode

Again, the LU decomposition can be stored to be reused or to perform other kernel operations:
\code
// ...
LU<MatrixXf> luOfA(A);
luOfA.solve(b, &x);
// ...
\endcode

\sa class LU, LU::solve(), LU_Module


\subsection TutorialAdv_LU SVD solver (for singular matrices and special cases)
Finally, Eigen also offer a solver based on a singular value decomposition (SVD). Again, the API is the
same than with Cholesky or LU:
\code
Matrix4f A = Matrix4f::Random();
Vector4f b = Vector4f::Random();
Vector4f x;
A.svd().solve(b, &x);
SVD<MatrixXf> luOfA(A);
svdOfA.solve(b, &x);
\endcode

\sa class SVD, SVD::solve(), SVD_Module




<a href="#" class="top">top</a>\section TutorialAdvLU LU
todo

\sa LU_Module, LU::solve(), class LU

<a href="#" class="top">top</a>\section TutorialAdvCholesky Cholesky
todo

\sa Cholesky_Module, LLT::solve(), LLT::solveInPlace(), LDLT::solve(), LDLT::solveInPlace(), class LLT, class LDLT

<a href="#" class="top">top</a>\section TutorialAdvQR QR
todo

\sa QR_Module, class QR

<a href="#" class="top">top</a>\section TutorialAdvEigenProblems Eigen value problems
todo

\sa class SelfAdjointEigenSolver, class EigenSolver

*/

}