| namespace Eigen { |
| |
| /** \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 |
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| |
| \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. |
| |
| */ |
| |
| } |