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namespace Eigen { | |

/** \eigenManualPage TutorialLinearAlgebra Linear algebra and decompositions | |

This page explains how to solve linear systems, compute various decompositions such as LU, | |

QR, %SVD, eigendecompositions... After reading this page, don't miss our | |

\link TopicLinearAlgebraDecompositions catalogue \endlink of dense matrix decompositions. | |

\eigenAutoToc | |

\section TutorialLinAlgBasicSolve Basic linear solving | |

\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="example"> | |

<tr><th>Example:</th><th>Output:</th></tr> | |

<tr> | |

<td>\include TutorialLinAlgExSolveColPivHouseholderQR.cpp </td> | |

<td>\verbinclude TutorialLinAlgExSolveColPivHouseholderQR.out </td> | |

</tr> | |

</table> | |

In this example, the colPivHouseholderQr() method returns an object of class ColPivHouseholderQR. Since here the | |

matrix is of type Matrix3f, this line could have been replaced by: | |

\code | |

ColPivHouseholderQR<Matrix3f> 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 class="manual"> | |

<tr> | |

<th>Decomposition</th> | |

<th>Method</th> | |

<th>Requirements<br/>on the matrix</th> | |

<th>Speed<br/> (small-to-medium)</th> | |

<th>Speed<br/> (large)</th> | |

<th>Accuracy</th> | |

</tr> | |

<tr> | |

<td>PartialPivLU</td> | |

<td>partialPivLu()</td> | |

<td>Invertible</td> | |

<td>++</td> | |

<td>++</td> | |

<td>+</td> | |

</tr> | |

<tr class="alt"> | |

<td>FullPivLU</td> | |

<td>fullPivLu()</td> | |

<td>None</td> | |

<td>-</td> | |

<td>- -</td> | |

<td>+++</td> | |

</tr> | |

<tr> | |

<td>HouseholderQR</td> | |

<td>householderQr()</td> | |

<td>None</td> | |

<td>++</td> | |

<td>++</td> | |

<td>+</td> | |

</tr> | |

<tr class="alt"> | |

<td>ColPivHouseholderQR</td> | |

<td>colPivHouseholderQr()</td> | |

<td>None</td> | |

<td>+</td> | |

<td>-</td> | |

<td>+++</td> | |

</tr> | |

<tr> | |

<td>FullPivHouseholderQR</td> | |

<td>fullPivHouseholderQr()</td> | |

<td>None</td> | |

<td>-</td> | |

<td>- -</td> | |

<td>+++</td> | |

</tr> | |

<tr class="alt"> | |

<td>CompleteOrthogonalDecomposition</td> | |

<td>completeOrthogonalDecomposition()</td> | |

<td>None</td> | |

<td>+</td> | |

<td>-</td> | |

<td>+++</td> | |

</tr> | |

<tr class="alt"> | |

<td>LLT</td> | |

<td>llt()</td> | |

<td>Positive definite</td> | |

<td>+++</td> | |

<td>+++</td> | |

<td>+</td> | |

</tr> | |

<tr> | |

<td>LDLT</td> | |

<td>ldlt()</td> | |

<td>Positive or negative<br/> semidefinite</td> | |

<td>+++</td> | |

<td>+</td> | |

<td>++</td> | |

</tr> | |

<tr class="alt"> | |

<td>BDCSVD</td> | |

<td>bdcSvd()</td> | |

<td>None</td> | |

<td>-</td> | |

<td>-</td> | |

<td>+++</td> | |

</tr> | |

<tr class="alt"> | |

<td>JacobiSVD</td> | |

<td>jacobiSvd()</td> | |

<td>None</td> | |

<td>-</td> | |

<td>- - -</td> | |

<td>+++</td> | |

</tr> | |

</table> | |

To get an overview of the true relative speed of the different decompositions, check this \link DenseDecompositionBenchmark benchmark \endlink. | |

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

For example, if your matrix is positive definite, the above table says that a very good | |

choice is then the LLT or LDLT decomposition. Here's an example, also demonstrating that using a general | |

matrix (not a vector) as right hand side is possible. | |

<table class="example"> | |

<tr><th>Example:</th><th>Output:</th></tr> | |

<tr> | |

<td>\include TutorialLinAlgExSolveLDLT.cpp </td> | |

<td>\verbinclude TutorialLinAlgExSolveLDLT.out </td> | |

</tr> | |

</table> | |

For a \ref TopicLinearAlgebraDecompositions "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". | |

\section TutorialLinAlgSolutionExists Checking if a solution really exists | |

Only you know what error margin you want to allow for a solution to be considered valid. | |

So Eigen lets you do this computation for yourself, if you want to, as in this example: | |

<table class="example"> | |

<tr><th>Example:</th><th>Output:</th></tr> | |

<tr> | |

<td>\include TutorialLinAlgExComputeSolveError.cpp </td> | |

<td>\verbinclude TutorialLinAlgExComputeSolveError.out </td> | |

</tr> | |

</table> | |

\section TutorialLinAlgEigensolving Computing eigenvalues and eigenvectors | |

You need an eigendecomposition here, see available such decompositions on \ref TopicLinearAlgebraDecompositions "this page". | |

Make sure to check if your matrix is self-adjoint, as is often the case in these problems. Here's an example using | |

SelfAdjointEigenSolver, it could easily be adapted to general matrices using EigenSolver or ComplexEigenSolver. | |

The computation of eigenvalues and eigenvectors does not necessarily converge, but such failure to converge is | |

very rare. The call to info() is to check for this possibility. | |

<table class="example"> | |

<tr><th>Example:</th><th>Output:</th></tr> | |

<tr> | |

<td>\include TutorialLinAlgSelfAdjointEigenSolver.cpp </td> | |

<td>\verbinclude TutorialLinAlgSelfAdjointEigenSolver.out </td> | |

</tr> | |

</table> | |

\section TutorialLinAlgInverse Computing inverse and determinant | |

First of all, make sure that you really want this. While inverse and determinant are fundamental mathematical concepts, | |

in \em numerical linear algebra they are not as popular as in pure mathematics. Inverse computations are often | |

advantageously replaced by solve() operations, and the determinant is often \em not a good way of checking if a matrix | |

is invertible. | |

However, for \em very \em small matrices, the above is not true, and inverse and determinant can be very useful. | |

While certain decompositions, such as PartialPivLU and FullPivLU, offer inverse() and determinant() methods, you can also | |

call inverse() and determinant() directly on a matrix. If your matrix is of a very small fixed size (at most 4x4) this | |

allows Eigen to avoid performing a LU decomposition, and instead use formulas that are more efficient on such small matrices. | |

Here is an example: | |

<table class="example"> | |

<tr><th>Example:</th><th>Output:</th></tr> | |

<tr> | |

<td>\include TutorialLinAlgInverseDeterminant.cpp </td> | |

<td>\verbinclude TutorialLinAlgInverseDeterminant.out </td> | |

</tr> | |

</table> | |

\section TutorialLinAlgLeastsquares Least squares solving | |

The most accurate method to do least squares solving is with a SVD decomposition. | |

Eigen provides two implementations. | |

The recommended one is the BDCSVD class, which scale well for large problems | |

and automatically fall-back to the JacobiSVD class for smaller problems. | |

For both classes, their solve() method is doing least-squares solving. | |

Here is an example: | |

<table class="example"> | |

<tr><th>Example:</th><th>Output:</th></tr> | |

<tr> | |

<td>\include TutorialLinAlgSVDSolve.cpp </td> | |

<td>\verbinclude TutorialLinAlgSVDSolve.out </td> | |

</tr> | |

</table> | |

Another methods, potentially faster but less reliable, are to use a Cholesky decomposition of the | |

normal matrix or a QR decomposition. Our page on \link LeastSquares least squares solving \endlink | |

has more details. | |

\section TutorialLinAlgSeparateComputation Separating the computation from the construction | |

In the above examples, the decomposition was computed at the same time that the decomposition object was constructed. | |

There are however situations where you might want to separate these two things, for example if you don't know, | |

at the time of the construction, the matrix that you will want to decompose; or if you want to reuse an existing | |

decomposition object. | |

What makes this possible is that: | |

\li all decompositions have a default constructor, | |

\li all decompositions have a compute(matrix) method that does the computation, and that may be called again | |

on an already-computed decomposition, reinitializing it. | |

For example: | |

<table class="example"> | |

<tr><th>Example:</th><th>Output:</th></tr> | |

<tr> | |

<td>\include TutorialLinAlgComputeTwice.cpp </td> | |

<td>\verbinclude TutorialLinAlgComputeTwice.out </td> | |

</tr> | |

</table> | |

Finally, you can tell the decomposition constructor to preallocate storage for decomposing matrices of a given size, | |

so that when you subsequently decompose such matrices, no dynamic memory allocation is performed (of course, if you | |

are using fixed-size matrices, no dynamic memory allocation happens at all). This is done by just | |

passing the size to the decomposition constructor, as in this example: | |

\code | |

HouseholderQR<MatrixXf> qr(50,50); | |

MatrixXf A = MatrixXf::Random(50,50); | |

qr.compute(A); // no dynamic memory allocation | |

\endcode | |

\section TutorialLinAlgRankRevealing Rank-revealing decompositions | |

Certain decompositions are rank-revealing, i.e. are able to compute the rank of a matrix. These are typically | |

also the decompositions that behave best in the face of a non-full-rank matrix (which in the square case means a | |

singular matrix). On \ref TopicLinearAlgebraDecompositions "this table" you can see for all our decompositions | |

whether they are rank-revealing or not. | |

Rank-revealing decompositions offer at least a rank() method. They can also offer convenience methods such as isInvertible(), | |

and some are also providing methods to compute the kernel (null-space) and image (column-space) of the matrix, as is the | |

case with FullPivLU: | |

<table class="example"> | |

<tr><th>Example:</th><th>Output:</th></tr> | |

<tr> | |

<td>\include TutorialLinAlgRankRevealing.cpp </td> | |

<td>\verbinclude TutorialLinAlgRankRevealing.out </td> | |

</tr> | |

</table> | |

Of course, any rank computation depends on the choice of an arbitrary threshold, since practically no | |

floating-point matrix is \em exactly rank-deficient. Eigen picks a sensible default threshold, which depends | |

on the decomposition but is typically the diagonal size times machine epsilon. While this is the best default we | |

could pick, only you know what is the right threshold for your application. You can set this by calling setThreshold() | |

on your decomposition object before calling rank() or any other method that needs to use such a threshold. | |

The decomposition itself, i.e. the compute() method, is independent of the threshold. You don't need to recompute the | |

decomposition after you've changed the threshold. | |

<table class="example"> | |

<tr><th>Example:</th><th>Output:</th></tr> | |

<tr> | |

<td>\include TutorialLinAlgSetThreshold.cpp </td> | |

<td>\verbinclude TutorialLinAlgSetThreshold.out </td> | |

</tr> | |

</table> | |

*/ | |

} |