doc/C06_TutorialLinearAlgebra.dox

namespace Eigen {

/** \page TutorialLinearAlgebra Tutorial page 6 - Linear algebra and decompositions
    \ingroup Tutorial

\li \b Previous: TODO
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This tutorial explains how to solve linear systems, compute various decompositions such as LU,
QR, SVD, eigendecompositions... for more advanced topics, don't miss our special page on
\ref TopicLinearAlgebraDecompositions "this topic".

\section TutorialLinAlgBasicSolve How do I solve a system of linear equations?

\b The \b problem: You have a system of equations, that you have written as a single matrix equation
    \f[ Ax \: = \: b \f]
Where \a A and \a b are matrices (\a b could be a vector, as a special case). You want to find a solution \a x.

\b The \b solution: You can choose between various decompositions, depending on what your matrix \a A looks like,
and depending on whether you favor speed or accuracy. However, let's start with an example that works in all cases,
and is a good compromise:
<table class="tutorial_code">
<tr>
  <td>\include TutorialLinAlgExSolveColPivHouseholderQR.cpp </td>
  <td>output: \verbinclude TutorialLinAlgExSolveColPivHouseholderQR.out </td>
</tr>
</table>

In this example, the colPivHouseholderQr() method returns an object of class ColPivHouseholderQR. This line could
have been replaced by:
\code
ColPivHouseholderQR dec(A);
Vector3f x = dec.solve(b);
\endcode

Here, ColPivHouseholderQR is a QR decomposition with column pivoting. It's a good compromise for this tutorial, as it
works for all matrices while being quite fast. Here is a table of some other decompositions that you can choose from,
depending on your matrix and the trade-off you want to make:

<table border="1">

    <tr>
        <td>Decomposition</td>
        <td>Method</td>
        <td>Requirements on the matrix</td>
        <td>Speed</td>
        <td>Accuracy</td>
    </tr>

    <tr>
        <td>PartialPivLU</td>
        <td>partialPivLu()</td>
        <td>Invertible</td>
        <td>++</td>
        <td>+</td>
    </tr>

    <tr>
        <td>FullPivLU</td>
        <td>fullPivLu()</td>
        <td>None</td>
        <td>-</td>
        <td>+++</td>
    </tr>

    <tr>
        <td>HouseholderQR</td>
        <td>householderQr()</td>
        <td>None</td>
        <td>++</td>
        <td>+</td>
    </tr>

    <tr>
        <td>ColPivHouseholderQR</td>
        <td>colPivHouseholderQr()</td>
        <td>None</td>
        <td>+</td>
        <td>++</td>
    </tr>

    <tr>
        <td>FullPivHouseholderQR</td>
        <td>fullPivHouseholderQr()</td>
        <td>None</td>
        <td>-</td>
        <td>+++</td>
    </tr>

    <tr>
        <td>LLT</td>
        <td>llt()</td>
        <td>Positive definite</td>
        <td>+++</td>
        <td>+</td>
    </tr>

    <tr>
        <td>LDLT</td>
        <td>ldlt()</td>
        <td>Positive or negative semidefinite</td>
        <td>+++</td>
        <td>++</td>
    </tr>

</table>

All of these decompositions offer a solve() method that works as in the above example.

For a much more complete table comparing all decompositions supported by Eigen (notice that Eigen
supports many other decompositions), see our special page on
\ref TopicLinearAlgebraDecompositions "this topic".



*/

}