
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 TutorialAdvSolvers
  - \ref TutorialAdvLU
  - \ref TutorialAdvCholesky
  - \ref TutorialAdvQR
  - \ref TutorialAdvEigenProblems

\section TutorialAdvSolvers 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 TutorialAdvSolvers_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 TutorialAdvSolvers_Inverse Direct inversion (for small matrices)
If the matrix \f$ A \f$ is small (\f$ \leq 4 \f$) and invertible, then a good approach is to directly compute
the inverse of the matrix \f$ A \f$, and then obtain the solution \f$ x \f$ by \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 TutorialAdvSolvers_Symmetric Cholesky (for positive definite matrices)
If the matrix \f$ A \f$ is \b positive \b definite, then
the best method is to use a Cholesky decomposition.
Such positive definite 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 TutorialAdvSolvers_LU LU decomposition (for most cases)
If the matrix \f$ A \f$ does not fit in any of the previous categories, 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 : see the section \ref TutorialAdvLU below. 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 much better numerical stability.
The API of the LU solver is the same than the Cholesky one, except that there is no \em in \em place variant:
\code
#include <Eigen/LU>
MatrixXf A = MatrixXf::Random(20,20);
VectorXf b = VectorXf::Random(20);
VectorXf 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

See the section \ref TutorialAdvLU below.

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


\subsection TutorialAdvSolvers_SVD 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
#include <Eigen/SVD>
MatrixXf A = MatrixXf::Random(20,20);
VectorXf b = VectorXf::Random(20);
VectorXf x;
A.svd().solve(b, &x);
SVD<MatrixXf> svdOfA(A);
svdOfA.solve(b, &x);
\endcode

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


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

Eigen provides a rank-revealing LU decomposition with full pivoting, which has very good numerical stability.

You can obtain the LU decomposition of a matrix by calling \link MatrixBase::lu() lu() \endlink, which is the easiest way if you're going to use the LU decomposition only once, as in
\code
#include <Eigen/LU>
MatrixXf A = MatrixXf::Random(20,20);
VectorXf b = VectorXf::Random(20);
VectorXf x;
A.lu().solve(b, &x);
\endcode

Alternatively, you can construct a named LU decomposition, which allows you to reuse it for more than one operation:
\code
#include <Eigen/LU>
MatrixXf A = MatrixXf::Random(20,20);
Eigen::LUDecomposition<MatrixXf> lu(A);
cout << "The rank of A is" << lu.rank() << endl;
if(lu.isInvertible()) {
  cout << "A is invertible, its inverse is:" << endl << lu.inverse() << endl;
}
else {
  cout << "Here's a matrix whose columns form a basis of the kernel a.k.a. nullspace of A:"
       << endl << lu.kernel() << endl;
}
\endcode

\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

*/

}