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 Since here the 29 matrix is of type Matrix3f, this line could have been replaced by: 30 \code 31 ColPivHouseholderQR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