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 Since here the 29 matrix is of type Matrix3f, this line could have been replaced by: 30 \code 31 ColPivHouseholderQR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<br/>on the matrix</th> 44 <th>Speed<br/> (small-to-medium)</th> 45 <th>Speed<br/> (large)</th> 46 <th>Accuracy</th> 47 </tr> 48 <tr> 49 <td>PartialPivLU</td> 50 <td>partialPivLu()</td> 51 <td>Invertible</td> 52 <td>++</td> 53 <td>++</td> 54 <td>+</td> 55 </tr> 56 <tr class="alt"> 57 <td>FullPivLU</td> 58 <td>fullPivLu()</td> 59 <td>None</td> 60 <td>-</td> 61 <td>- -</td> 62 <td>+++</td> 63 </tr> 64 <tr> 65 <td>HouseholderQR</td> 66 <td>householderQr()</td> 67 <td>None</td> 68 <td>++</td> 69 <td>++</td> 70 <td>+</td> 71 </tr> 72 <tr class="alt"> 73 <td>ColPivHouseholderQR</td> 74 <td>colPivHouseholderQr()</td> 75 <td>None</td> 76 <td>++</td> 77 <td>-</td> 78 <td>+++</td> 79 </tr> 80 <tr> 81 <td>FullPivHouseholderQR</td> 82 <td>fullPivHouseholderQr()</td> 83 <td>None</td> 84 <td>-</td> 85 <td>- -</td> 86 <td>+++</td> 87 </tr> 88 <tr class="alt"> 89 <td>LLT</td> 90 <td>llt()</td> 91 <td>Positive definite</td> 92 <td>+++</td> 93 <td>+++</td> 94 <td>+</td> 95 </tr> 96 <tr> 97 <td>LDLT</td> 98 <td>ldlt()</td> 99 <td>Positive or negative<br/> semidefinite</td> 100 <td>+++</td> 101 <td>+</td> 102 <td>++</td> 103 </tr> 104 <tr class="alt"> 105 <td>JacobiSVD</td> 106 <td>jacobiSvd()</td> 107 <td>None</td> 108 <td>- -</td> 109 <td>- - -</td> 110 <td>+++</td> 111 </tr> 112 </table> 113 114 All of these decompositions offer a solve() method that works as in the above example. 115 116 For example, if your matrix is positive definite, the above table says that a very good 117 choice is then the LLT or LDLT decomposition. Here's an example, also demonstrating that using a general 118 matrix (not a vector) as right hand side is possible. 119 120 <table class="example"> 121 <tr><th>Example:</th><th>Output:</th></tr> 122 <tr> 123 <td>\include TutorialLinAlgExSolveLDLT.cpp </td> 124 <td>\verbinclude TutorialLinAlgExSolveLDLT.out </td> 125 </tr> 126 </table> 127 128 For a \ref TopicLinearAlgebraDecompositions "much more complete table" comparing all decompositions supported by Eigen (notice that Eigen 129 supports many other decompositions), see our special page on 130 \ref TopicLinearAlgebraDecompositions "this topic". 131 132 \section TutorialLinAlgSolutionExists Checking if a solution really exists 133 134 Only you know what error margin you want to allow for a solution to be considered valid. 135 So Eigen lets you do this computation for yourself, if you want to, as in this example: 136 137 <table class="example"> 138 <tr><th>Example:</th><th>Output:</th></tr> 139 <tr> 140 <td>\include TutorialLinAlgExComputeSolveError.cpp </td> 141 <td>\verbinclude TutorialLinAlgExComputeSolveError.out </td> 142 </tr> 143 </table> 144 145 \section TutorialLinAlgEigensolving Computing eigenvalues and eigenvectors 146 147 You need an eigendecomposition here, see available such decompositions on \ref TopicLinearAlgebraDecompositions "this page". 148 Make sure to check if your matrix is self-adjoint, as is often the case in these problems. Here's an example using 149 SelfAdjointEigenSolver, it could easily be adapted to general matrices using EigenSolver or ComplexEigenSolver. 150 151 The computation of eigenvalues and eigenvectors does not necessarily converge, but such failure to converge is 152 very rare. The call to info() is to check for this possibility. 153 154 <table class="example"> 155 <tr><th>Example:</th><th>Output:</th></tr> 156 <tr> 157 <td>\include TutorialLinAlgSelfAdjointEigenSolver.cpp </td> 158 <td>\verbinclude TutorialLinAlgSelfAdjointEigenSolver.out </td> 159 </tr> 160 </table> 161 162 \section TutorialLinAlgInverse Computing inverse and determinant 163 164 First of all, make sure that you really want this. While inverse and determinant are fundamental mathematical concepts, 165 in \em numerical linear algebra they are not as popular as in pure mathematics. Inverse computations are often 166 advantageously replaced by solve() operations, and the determinant is often \em not a good way of checking if a matrix 167 is invertible. 168 169 However, for \em very \em small matrices, the above is not true, and inverse and determinant can be very useful. 170 171 While certain decompositions, such as PartialPivLU and FullPivLU, offer inverse() and determinant() methods, you can also 172 call inverse() and determinant() directly on a matrix. If your matrix is of a very small fixed size (at most 4x4) this 173 allows Eigen to avoid performing a LU decomposition, and instead use formulas that are more efficient on such small matrices. 174 175 Here is an example: 176 <table class="example"> 177 <tr><th>Example:</th><th>Output:</th></tr> 178 <tr> 179 <td>\include TutorialLinAlgInverseDeterminant.cpp </td> 180 <td>\verbinclude TutorialLinAlgInverseDeterminant.out </td> 181 </tr> 182 </table> 183 184 \section TutorialLinAlgLeastsquares Least squares solving 185 186 The most accurate method to do least squares solving is with a SVD decomposition. Eigen provides one 187 as the JacobiSVD class, and its solve() is doing least-squares solving. 188 189 Here is an example: 190 <table class="example"> 191 <tr><th>Example:</th><th>Output:</th></tr> 192 <tr> 193 <td>\include TutorialLinAlgSVDSolve.cpp </td> 194 <td>\verbinclude TutorialLinAlgSVDSolve.out </td> 195 </tr> 196 </table> 197 198 Another methods, potentially faster but less reliable, are to use a Cholesky decomposition of the 199 normal matrix or a QR decomposition. Our page on \link LeastSquares least squares solving \endlink 200 has more details. 201 202 203 \section TutorialLinAlgSeparateComputation Separating the computation from the construction 204 205 In the above examples, the decomposition was computed at the same time that the decomposition object was constructed. 206 There are however situations where you might want to separate these two things, for example if you don't know, 207 at the time of the construction, the matrix that you will want to decompose; or if you want to reuse an existing 208 decomposition object. 209 210 What makes this possible is that: 211 \li all decompositions have a default constructor, 212 \li all decompositions have a compute(matrix) method that does the computation, and that may be called again 213 on an already-computed decomposition, reinitializing it. 214 215 For example: 216 217 <table class="example"> 218 <tr><th>Example:</th><th>Output:</th></tr> 219 <tr> 220 <td>\include TutorialLinAlgComputeTwice.cpp </td> 221 <td>\verbinclude TutorialLinAlgComputeTwice.out </td> 222 </tr> 223 </table> 224 225 Finally, you can tell the decomposition constructor to preallocate storage for decomposing matrices of a given size, 226 so that when you subsequently decompose such matrices, no dynamic memory allocation is performed (of course, if you 227 are using fixed-size matrices, no dynamic memory allocation happens at all). This is done by just 228 passing the size to the decomposition constructor, as in this example: 229 \code 230 HouseholderQR<MatrixXf> qr(50,50); 231 MatrixXf A = MatrixXf::Random(50,50); 232 qr.compute(A); // no dynamic memory allocation 233 \endcode 234 235 \section TutorialLinAlgRankRevealing Rank-revealing decompositions 236 237 Certain decompositions are rank-revealing, i.e. are able to compute the rank of a matrix. These are typically 238 also the decompositions that behave best in the face of a non-full-rank matrix (which in the square case means a 239 singular matrix). On \ref TopicLinearAlgebraDecompositions "this table" you can see for all our decompositions 240 whether they are rank-revealing or not. 241 242 Rank-revealing decompositions offer at least a rank() method. They can also offer convenience methods such as isInvertible(), 243 and some are also providing methods to compute the kernel (null-space) and image (column-space) of the matrix, as is the 244 case with FullPivLU: 245 246 <table class="example"> 247 <tr><th>Example:</th><th>Output:</th></tr> 248 <tr> 249 <td>\include TutorialLinAlgRankRevealing.cpp </td> 250 <td>\verbinclude TutorialLinAlgRankRevealing.out </td> 251 </tr> 252 </table> 253 254 Of course, any rank computation depends on the choice of an arbitrary threshold, since practically no 255 floating-point matrix is \em exactly rank-deficient. Eigen picks a sensible default threshold, which depends 256 on the decomposition but is typically the diagonal size times machine epsilon. While this is the best default we 257 could pick, only you know what is the right threshold for your application. You can set this by calling setThreshold() 258 on your decomposition object before calling rank() or any other method that needs to use such a threshold. 259 The decomposition itself, i.e. the compute() method, is independent of the threshold. You don't need to recompute the 260 decomposition after you've changed the threshold. 261 262 <table class="example"> 263 <tr><th>Example:</th><th>Output:</th></tr> 264 <tr> 265 <td>\include TutorialLinAlgSetThreshold.cpp </td> 266 <td>\verbinclude TutorialLinAlgSetThreshold.out </td> 267 </tr> 268 </table> 269 270 */ 271 272 } 273
