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- 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
- \section LeastSquaresSVD Using the SVD decomposition
- The \link BDCSVD::solve() solve() \endlink method in the BDCSVD 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.
- */
- }
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