1 namespace Eigen { 2 3 /** \page TutorialLinearAlgebra Tutorial page 6 - Linear algebra and decompositions 4 \ingroup Tutorial 5 6 \li \b Previous: \ref TutorialAdvancedInitialization 7 \li \b Next: \ref TutorialReductionsVisitorsBroadcasting 8 9 This tutorial explains how to solve linear systems, compute various decompositions such as LU, 10 QR, %SVD, eigendecompositions... for more advanced topics, don't miss our special page on 11 \ref TopicLinearAlgebraDecompositions "this topic". 12 13 \b Table \b of \b contents 14 - \ref TutorialLinAlgBasicSolve 15 - \ref TutorialLinAlgSolutionExists 16 - \ref TutorialLinAlgEigensolving 17 - \ref TutorialLinAlgInverse 18 - \ref TutorialLinAlgLeastsquares 19 - \ref TutorialLinAlgSeparateComputation 20 - \ref TutorialLinAlgRankRevealing 21 22 23 \section TutorialLinAlgBasicSolve Basic linear solving 24 25 \b The \b problem: You have a system of equations, that you have written as a single matrix equation 26 \f[ Ax \: = \: b \f] 27 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. 28 29 \b The \b solution: You can choose between various decompositions, depending on what your matrix \a A looks like, 30 and depending on whether you favor speed or accuracy. However, let's start with an example that works in all cases, 31 and is a good compromise: 32 <table class="example"> 33 <tr><th>Example:</th><th>Output:</th></tr> 34 <tr> 35 <td>\include TutorialLinAlgExSolveColPivHouseholderQR.cpp </td> 36 <td>\verbinclude TutorialLinAlgExSolveColPivHouseholderQR.out </td> 37 </tr> 38 </table> 39 40 In this example, the colPivHouseholderQr() method returns an object of class ColPivHouseholderQR. Since here the 41 matrix is of type Matrix3f, this line could have been replaced by: 42 \code 43 ColPivHouseholderQR<Matrix3f> dec(A); 44 Vector3f x = dec.solve(b); 45 \endcode 46 47 Here, ColPivHouseholderQR is a QR decomposition with column pivoting. It's a good compromise for this tutorial, as it 48 works for all matrices while being quite fast. Here is a table of some other decompositions that you can choose from, 49 depending on your matrix and the trade-off you want to make: 50 51 <table class="manual"> 52 <tr> 53 <th>Decomposition</th> 54 <th>Method</th> 55 <th>Requirements on the matrix</th> 56 <th>Speed</th> 57 <th>Accuracy</th> 58 </tr> 59 <tr> 60 <td>PartialPivLU</td> 61 <td>partialPivLu()</td> 62 <td>Invertible</td> 63 <td>++</td> 64 <td>+</td> 65 </tr> 66 <tr class="alt"> 67 <td>FullPivLU</td> 68 <td>fullPivLu()</td> 69 <td>None</td> 70 <td>-</td> 71 <td>+++</td> 72 </tr> 73 <tr> 74 <td>HouseholderQR</td> 75 <td>householderQr()</td> 76 <td>None</td> 77 <td>++</td> 78 <td>+</td> 79 </tr> 80 <tr class="alt"> 81 <td>ColPivHouseholderQR</td> 82 <td>colPivHouseholderQr()</td> 83 <td>None</td> 84 <td>+</td> 85 <td>++</td> 86 </tr> 87 <tr> 88 <td>FullPivHouseholderQR</td> 89 <td>fullPivHouseholderQr()</td> 90 <td>None</td> 91 <td>-</td> 92 <td>+++</td> 93 </tr> 94 <tr class="alt"> 95 <td>LLT</td> 96 <td>llt()</td> 97 <td>Positive definite</td> 98 <td>+++</td> 99 <td>+</td> 100 </tr> 101 <tr> 102 <td>LDLT</td> 103 <td>ldlt()</td> 104 <td>Positive or negative semidefinite</td> 105 <td>+++</td> 106 <td>++</td> 107 </tr> 108 </table> 109 110 All of these decompositions offer a solve() method that works as in the above example. 111 112 For example, if your matrix is positive definite, the above table says that a very good 113 choice is then the LDLT decomposition. Here's an example, also demonstrating that using a general 114 matrix (not a vector) as right hand side is possible. 115 116 <table class="example"> 117 <tr><th>Example:</th><th>Output:</th></tr> 118 <tr> 119 <td>\include TutorialLinAlgExSolveLDLT.cpp </td> 120 <td>\verbinclude TutorialLinAlgExSolveLDLT.out </td> 121 </tr> 122 </table> 123 124 For a \ref TopicLinearAlgebraDecompositions "much more complete table" comparing all decompositions supported by Eigen (notice that Eigen 125 supports many other decompositions), see our special page on 126 \ref TopicLinearAlgebraDecompositions "this topic". 127 128 \section TutorialLinAlgSolutionExists Checking if a solution really exists 129 130 Only you know what error margin you want to allow for a solution to be considered valid. 131 So Eigen lets you do this computation for yourself, if you want to, as in this example: 132 133 <table class="example"> 134 <tr><th>Example:</th><th>Output:</th></tr> 135 <tr> 136 <td>\include TutorialLinAlgExComputeSolveError.cpp </td> 137 <td>\verbinclude TutorialLinAlgExComputeSolveError.out </td> 138 </tr> 139 </table> 140 141 \section TutorialLinAlgEigensolving Computing eigenvalues and eigenvectors 142 143 You need an eigendecomposition here, see available such decompositions on \ref TopicLinearAlgebraDecompositions "this page". 144 Make sure to check if your matrix is self-adjoint, as is often the case in these problems. Here's an example using 145 SelfAdjointEigenSolver, it could easily be adapted to general matrices using EigenSolver or ComplexEigenSolver. 146 147 The computation of eigenvalues and eigenvectors does not necessarily converge, but such failure to converge is 148 very rare. The call to info() is to check for this possibility. 149 150 <table class="example"> 151 <tr><th>Example:</th><th>Output:</th></tr> 152 <tr> 153 <td>\include TutorialLinAlgSelfAdjointEigenSolver.cpp </td> 154 <td>\verbinclude TutorialLinAlgSelfAdjointEigenSolver.out </td> 155 </tr> 156 </table> 157 158 \section TutorialLinAlgInverse Computing inverse and determinant 159 160 First of all, make sure that you really want this. While inverse and determinant are fundamental mathematical concepts, 161 in \em numerical linear algebra they are not as popular as in pure mathematics. Inverse computations are often 162 advantageously replaced by solve() operations, and the determinant is often \em not a good way of checking if a matrix 163 is invertible. 164 165 However, for \em very \em small matrices, the above is not true, and inverse and determinant can be very useful. 166 167 While certain decompositions, such as PartialPivLU and FullPivLU, offer inverse() and determinant() methods, you can also 168 call inverse() and determinant() directly on a matrix. If your matrix is of a very small fixed size (at most 4x4) this 169 allows Eigen to avoid performing a LU decomposition, and instead use formulas that are more efficient on such small matrices. 170 171 Here is an example: 172 <table class="example"> 173 <tr><th>Example:</th><th>Output:</th></tr> 174 <tr> 175 <td>\include TutorialLinAlgInverseDeterminant.cpp </td> 176 <td>\verbinclude TutorialLinAlgInverseDeterminant.out </td> 177 </tr> 178 </table> 179 180 \section TutorialLinAlgLeastsquares Least squares solving 181 182 The best way to do least squares solving is with a SVD decomposition. Eigen provides one as the JacobiSVD class, and its solve() 183 is doing least-squares solving. 184 185 Here is an example: 186 <table class="example"> 187 <tr><th>Example:</th><th>Output:</th></tr> 188 <tr> 189 <td>\include TutorialLinAlgSVDSolve.cpp </td> 190 <td>\verbinclude TutorialLinAlgSVDSolve.out </td> 191 </tr> 192 </table> 193 194 Another way, potentially faster but less reliable, is to use a LDLT decomposition 195 of the normal matrix. In any case, just read any reference text on least squares, and it will be very easy for you 196 to implement any linear least squares computation on top of Eigen. 197 198 \section TutorialLinAlgSeparateComputation Separating the computation from the construction 199 200 In the above examples, the decomposition was computed at the same time that the decomposition object was constructed. 201 There are however situations where you might want to separate these two things, for example if you don't know, 202 at the time of the construction, the matrix that you will want to decompose; or if you want to reuse an existing 203 decomposition object. 204 205 What makes this possible is that: 206 \li all decompositions have a default constructor, 207 \li all decompositions have a compute(matrix) method that does the computation, and that may be called again 208 on an already-computed decomposition, reinitializing it. 209 210 For example: 211 212 <table class="example"> 213 <tr><th>Example:</th><th>Output:</th></tr> 214 <tr> 215 <td>\include TutorialLinAlgComputeTwice.cpp </td> 216 <td>\verbinclude TutorialLinAlgComputeTwice.out </td> 217 </tr> 218 </table> 219 220 Finally, you can tell the decomposition constructor to preallocate storage for decomposing matrices of a given size, 221 so that when you subsequently decompose such matrices, no dynamic memory allocation is performed (of course, if you 222 are using fixed-size matrices, no dynamic memory allocation happens at all). This is done by just 223 passing the size to the decomposition constructor, as in this example: 224 \code 225 HouseholderQR<MatrixXf> qr(50,50); 226 MatrixXf A = MatrixXf::Random(50,50); 227 qr.compute(A); // no dynamic memory allocation 228 \endcode 229 230 \section TutorialLinAlgRankRevealing Rank-revealing decompositions 231 232 Certain decompositions are rank-revealing, i.e. are able to compute the rank of a matrix. These are typically 233 also the decompositions that behave best in the face of a non-full-rank matrix (which in the square case means a 234 singular matrix). On \ref TopicLinearAlgebraDecompositions "this table" you can see for all our decompositions 235 whether they are rank-revealing or not. 236 237 Rank-revealing decompositions offer at least a rank() method. They can also offer convenience methods such as isInvertible(), 238 and some are also providing methods to compute the kernel (null-space) and image (column-space) of the matrix, as is the 239 case with FullPivLU: 240 241 <table class="example"> 242 <tr><th>Example:</th><th>Output:</th></tr> 243 <tr> 244 <td>\include TutorialLinAlgRankRevealing.cpp </td> 245 <td>\verbinclude TutorialLinAlgRankRevealing.out </td> 246 </tr> 247 </table> 248 249 Of course, any rank computation depends on the choice of an arbitrary threshold, since practically no 250 floating-point matrix is \em exactly rank-deficient. Eigen picks a sensible default threshold, which depends 251 on the decomposition but is typically the diagonal size times machine epsilon. While this is the best default we 252 could pick, only you know what is the right threshold for your application. You can set this by calling setThreshold() 253 on your decomposition object before calling rank() or any other method that needs to use such a threshold. 254 The decomposition itself, i.e. the compute() method, is independent of the threshold. You don't need to recompute the 255 decomposition after you've changed the threshold. 256 257 <table class="example"> 258 <tr><th>Example:</th><th>Output:</th></tr> 259 <tr> 260 <td>\include TutorialLinAlgSetThreshold.cpp </td> 261 <td>\verbinclude TutorialLinAlgSetThreshold.out </td> 262 </tr> 263 </table> 264 265 \li \b Next: \ref TutorialReductionsVisitorsBroadcasting 266 267 */ 268 269 } 270
