1namespace Eigen {
2
3/** \eigenManualPage LeastSquares Solving linear least squares systems
4
5This page describes how to solve linear least squares systems using %Eigen. An overdetermined system
6of equations, say \a Ax = \a b, has no solutions. In this case, it makes sense to search for the
7vector \a x which is closest to being a solution, in the sense that the difference \a Ax - \a b is
8as small as possible. This \a x is called the least square solution (if the Euclidean norm is used).
9
10The three methods discussed on this page are the SVD decomposition, the QR decomposition and normal
11equations. Of these, the SVD decomposition is generally the most accurate but the slowest, normal
12equations is the fastest but least accurate, and the QR decomposition is in between.
13
14\eigenAutoToc
15
16
17\section LeastSquaresSVD Using the SVD decomposition
18
19The \link JacobiSVD::solve() solve() \endlink method in the JacobiSVD class can be directly used to
20solve linear squares systems. It is not enough to compute only the singular values (the default for
21this class); you also need the singular vectors but the thin SVD decomposition suffices for
22computing least squares solutions:
23
24<table class="example">
25<tr><th>Example:</th><th>Output:</th></tr>
26<tr>
27  <td>\include TutorialLinAlgSVDSolve.cpp </td>
28  <td>\verbinclude TutorialLinAlgSVDSolve.out </td>
29</tr>
30</table>
31
32This is example from the page \link TutorialLinearAlgebra Linear algebra and decompositions \endlink.
33
34
35\section LeastSquaresQR Using the QR decomposition
36
37The solve() method in QR decomposition classes also computes the least squares solution. There are
38three QR decomposition classes: HouseholderQR (no pivoting, so fast but unstable),
39ColPivHouseholderQR (column pivoting, thus a bit slower but more accurate) and FullPivHouseholderQR
40(full pivoting, so slowest and most stable). Here is an example with column pivoting:
41
42<table class="example">
43<tr><th>Example:</th><th>Output:</th></tr>
44<tr>
45  <td>\include LeastSquaresQR.cpp </td>
46  <td>\verbinclude LeastSquaresQR.out </td>
47</tr>
48</table>
49
50
51\section LeastSquaresNormalEquations Using normal equations
52
53Finding the least squares solution of \a Ax = \a b is equivalent to solving the normal equation
54<i>A</i><sup>T</sup><i>Ax</i> = <i>A</i><sup>T</sup><i>b</i>. This leads to the following code
55
56<table class="example">
57<tr><th>Example:</th><th>Output:</th></tr>
58<tr>
59  <td>\include LeastSquaresNormalEquations.cpp </td>
60  <td>\verbinclude LeastSquaresNormalEquations.out </td>
61</tr>
62</table>
63
64If the matrix \a A is ill-conditioned, then this is not a good method, because the condition number
65of <i>A</i><sup>T</sup><i>A</i> is the square of the condition number of \a A. This means that you
66lose twice as many digits using normal equation than if you use the other methods.
67
68*/
69
70}