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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:
Example:Output:
\include TutorialLinAlgExSolveColPivHouseholderQR.cpp \verbinclude TutorialLinAlgExSolveColPivHouseholderQR.out
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 Since here the matrix is of type Matrix3f, this line could have been replaced by: \code ColPivHouseholderQR dec(A); Vector3f x = dec.solve(b); \endcode Here, ColPivHouseholderQR is a QR decomposition with column pivoting.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:
Decomposition Method Requirements
on the matrix
Speed
(small-to-medium)
Speed
(large)
Accuracy
PartialPivLU partialPivLu() Invertible ++ ++ +
FullPivLU fullPivLu() None - - - +++
HouseholderQR householderQr() None ++ ++ +
ColPivHouseholderQR colPivHouseholderQr() None + - +++
FullPivHouseholderQR fullPivHouseholderQr() None - - - +++
CompleteOrthogonalDecomposition completeOrthogonalDecomposition() None + - +++
LLT llt() Positive definite +++ +++ +
LDLT ldlt() Positive or negative
semidefinite
+++ + ++
BDCSVD bdcSvd() None - - +++
JacobiSVD jacobiSvd() None - - - - +++
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.
Example:Output:
\include TutorialLinAlgExSolveLDLT.cpp \verbinclude TutorialLinAlgExSolveLDLT.out
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:
Example:Output:
\include TutorialLinAlgExComputeSolveError.cpp \verbinclude TutorialLinAlgExComputeSolveError.out
\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.
Example:Output:
\include TutorialLinAlgSelfAdjointEigenSolver.cpp \verbinclude TutorialLinAlgSelfAdjointEigenSolver.out
\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:
Example:Output:
\include TutorialLinAlgInverseDeterminant.cpp \verbinclude TutorialLinAlgInverseDeterminant.out
\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:
Example:Output:
\include TutorialLinAlgSVDSolve.cpp \verbinclude TutorialLinAlgSVDSolve.out
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:
Example:Output:
\include TutorialLinAlgComputeTwice.cpp \verbinclude TutorialLinAlgComputeTwice.out
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 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:
Example:Output:
\include TutorialLinAlgRankRevealing.cpp \verbinclude TutorialLinAlgRankRevealing.out
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.
Example:Output:
\include TutorialLinAlgSetThreshold.cpp \verbinclude TutorialLinAlgSetThreshold.out
*/ }