namespace StormEigen {

/** \eigenManualPage LeastSquares Solving linear least squares systems

This page describes how to solve linear least squares systems using %Eigen. An overdetermined system
of equations, say \a Ax = \a b, has no solutions. In this case, it makes sense to search for the
vector \a x which is closest to being a solution, in the sense that the difference \a Ax - \a b is
as small as possible. This \a x is called the least square solution (if the Euclidean norm is used).

The three methods discussed on this page are the SVD decomposition, the QR decomposition and normal
equations. Of these, the SVD decomposition is generally the most accurate but the slowest, normal
equations is the fastest but least accurate, and the QR decomposition is in between.

\eigenAutoToc


\section LeastSquaresSVD Using the SVD decomposition

The \link JacobiSVD::solve() solve() \endlink method in the JacobiSVD class can be directly used to
solve linear squares systems. It is not enough to compute only the singular values (the default for
this class); you also need the singular vectors but the thin SVD decomposition suffices for
computing least squares solutions:

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr>
  <td>\include TutorialLinAlgSVDSolve.cpp </td>
  <td>\verbinclude TutorialLinAlgSVDSolve.out </td>
</tr>
</table>

This is example from the page \link TutorialLinearAlgebra Linear algebra and decompositions \endlink.


\section LeastSquaresQR Using the QR decomposition

The solve() method in QR decomposition classes also computes the least squares solution. There are
three QR decomposition classes: HouseholderQR (no pivoting, so fast but unstable),
ColPivHouseholderQR (column pivoting, thus a bit slower but more accurate) and FullPivHouseholderQR
(full pivoting, so slowest and most stable). Here is an example with column pivoting:

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr>
  <td>\include LeastSquaresQR.cpp </td>
  <td>\verbinclude LeastSquaresQR.out </td>
</tr>
</table>


\section LeastSquaresNormalEquations Using normal equations

Finding the least squares solution of \a Ax = \a b is equivalent to solving the normal equation
<i>A</i><sup>T</sup><i>Ax</i> = <i>A</i><sup>T</sup><i>b</i>. This leads to the following code

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr>
  <td>\include LeastSquaresNormalEquations.cpp </td>
  <td>\verbinclude LeastSquaresNormalEquations.out </td>
</tr>
</table>

If the matrix \a A is ill-conditioned, then this is not a good method, because the condition number
of <i>A</i><sup>T</sup><i>A</i> is the square of the condition number of \a A. This means that you
lose twice as many digits using normal equation than if you use the other methods.

*/

}