1 namespace Eigen { 2 3 /** \page QuickRefPage Quick reference guide 4 5 \b Table \b of \b contents 6 - \ref QuickRef_Headers 7 - \ref QuickRef_Types 8 - \ref QuickRef_Map 9 - \ref QuickRef_ArithmeticOperators 10 - \ref QuickRef_Coeffwise 11 - \ref QuickRef_Reductions 12 - \ref QuickRef_Blocks 13 - \ref QuickRef_Misc 14 - \ref QuickRef_DiagTriSymm 15 \n 16 17 <hr> 18 19 <a href="#" class="top">top</a> 20 \section QuickRef_Headers Modules and Header files 21 22 The Eigen library is divided in a Core module and several additional modules. Each module has a corresponding header file which has to be included in order to use the module. The \c %Dense and \c Eigen header files are provided to conveniently gain access to several modules at once. 23 24 <table class="manual"> 25 <tr><th>Module</th><th>Header file</th><th>Contents</th></tr> 26 <tr><td>\link Core_Module Core \endlink</td><td>\code#include <Eigen/Core>\endcode</td><td>Matrix and Array classes, basic linear algebra (including triangular and selfadjoint products), array manipulation</td></tr> 27 <tr class="alt"><td>\link Geometry_Module Geometry \endlink</td><td>\code#include <Eigen/Geometry>\endcode</td><td>Transform, Translation, Scaling, Rotation2D and 3D rotations (Quaternion, AngleAxis)</td></tr> 28 <tr><td>\link LU_Module LU \endlink</td><td>\code#include <Eigen/LU>\endcode</td><td>Inverse, determinant, LU decompositions with solver (FullPivLU, PartialPivLU)</td></tr> 29 <tr><td>\link Cholesky_Module Cholesky \endlink</td><td>\code#include <Eigen/Cholesky>\endcode</td><td>LLT and LDLT Cholesky factorization with solver</td></tr> 30 <tr class="alt"><td>\link Householder_Module Householder \endlink</td><td>\code#include <Eigen/Householder>\endcode</td><td>Householder transformations; this module is used by several linear algebra modules</td></tr> 31 <tr><td>\link SVD_Module SVD \endlink</td><td>\code#include <Eigen/SVD>\endcode</td><td>SVD decomposition with least-squares solver (JacobiSVD)</td></tr> 32 <tr class="alt"><td>\link QR_Module QR \endlink</td><td>\code#include <Eigen/QR>\endcode</td><td>QR decomposition with solver (HouseholderQR, ColPivHouseholderQR, FullPivHouseholderQR)</td></tr> 33 <tr><td>\link Eigenvalues_Module Eigenvalues \endlink</td><td>\code#include <Eigen/Eigenvalues>\endcode</td><td>Eigenvalue, eigenvector decompositions (EigenSolver, SelfAdjointEigenSolver, ComplexEigenSolver)</td></tr> 34 <tr class="alt"><td>\link Sparse_Module Sparse \endlink</td><td>\code#include <Eigen/Sparse>\endcode</td><td>%Sparse matrix storage and related basic linear algebra (SparseMatrix, DynamicSparseMatrix, SparseVector)</td></tr> 35 <tr><td></td><td>\code#include <Eigen/Dense>\endcode</td><td>Includes Core, Geometry, LU, Cholesky, SVD, QR, and Eigenvalues header files</td></tr> 36 <tr class="alt"><td></td><td>\code#include <Eigen/Eigen>\endcode</td><td>Includes %Dense and %Sparse header files (the whole Eigen library)</td></tr> 37 </table> 38 39 <a href="#" class="top">top</a> 40 \section QuickRef_Types Array, matrix and vector types 41 42 43 \b Recall: Eigen provides two kinds of dense objects: mathematical matrices and vectors which are both represented by the template class Matrix, and general 1D and 2D arrays represented by the template class Array: 44 \code 45 typedef Matrix<Scalar, RowsAtCompileTime, ColsAtCompileTime, Options> MyMatrixType; 46 typedef Array<Scalar, RowsAtCompileTime, ColsAtCompileTime, Options> MyArrayType; 47 \endcode 48 49 \li \c Scalar is the scalar type of the coefficients (e.g., \c float, \c double, \c bool, \c int, etc.). 50 \li \c RowsAtCompileTime and \c ColsAtCompileTime are the number of rows and columns of the matrix as known at compile-time or \c Dynamic. 51 \li \c Options can be \c ColMajor or \c RowMajor, default is \c ColMajor. (see class Matrix for more options) 52 53 All combinations are allowed: you can have a matrix with a fixed number of rows and a dynamic number of columns, etc. The following are all valid: 54 \code 55 Matrix<double, 6, Dynamic> // Dynamic number of columns (heap allocation) 56 Matrix<double, Dynamic, 2> // Dynamic number of rows (heap allocation) 57 Matrix<double, Dynamic, Dynamic, RowMajor> // Fully dynamic, row major (heap allocation) 58 Matrix<double, 13, 3> // Fully fixed (static allocation) 59 \endcode 60 61 In most cases, you can simply use one of the convenience typedefs for \ref matrixtypedefs "matrices" and \ref arraytypedefs "arrays". Some examples: 62 <table class="example"> 63 <tr><th>Matrices</th><th>Arrays</th></tr> 64 <tr><td>\code 65 Matrix<float,Dynamic,Dynamic> <=> MatrixXf 66 Matrix<double,Dynamic,1> <=> VectorXd 67 Matrix<int,1,Dynamic> <=> RowVectorXi 68 Matrix<float,3,3> <=> Matrix3f 69 Matrix<float,4,1> <=> Vector4f 70 \endcode</td><td>\code 71 Array<float,Dynamic,Dynamic> <=> ArrayXXf 72 Array<double,Dynamic,1> <=> ArrayXd 73 Array<int,1,Dynamic> <=> RowArrayXi 74 Array<float,3,3> <=> Array33f 75 Array<float,4,1> <=> Array4f 76 \endcode</td></tr> 77 </table> 78 79 Conversion between the matrix and array worlds: 80 \code 81 Array44f a1, a1; 82 Matrix4f m1, m2; 83 m1 = a1 * a2; // coeffwise product, implicit conversion from array to matrix. 84 a1 = m1 * m2; // matrix product, implicit conversion from matrix to array. 85 a2 = a1 + m1.array(); // mixing array and matrix is forbidden 86 m2 = a1.matrix() + m1; // and explicit conversion is required. 87 ArrayWrapper<Matrix4f> m1a(m1); // m1a is an alias for m1.array(), they share the same coefficients 88 MatrixWrapper<Array44f> a1m(a1); 89 \endcode 90 91 In the rest of this document we will use the following symbols to emphasize the features which are specifics to a given kind of object: 92 \li <a name="matrixonly"><a/>\matrixworld linear algebra matrix and vector only 93 \li <a name="arrayonly"><a/>\arrayworld array objects only 94 95 \subsection QuickRef_Basics Basic matrix manipulation 96 97 <table class="manual"> 98 <tr><th></th><th>1D objects</th><th>2D objects</th><th>Notes</th></tr> 99 <tr><td>Constructors</td> 100 <td>\code 101 Vector4d v4; 102 Vector2f v1(x, y); 103 Array3i v2(x, y, z); 104 Vector4d v3(x, y, z, w); 105 106 VectorXf v5; // empty object 107 ArrayXf v6(size); 108 \endcode</td><td>\code 109 Matrix4f m1; 110 111 112 113 114 MatrixXf m5; // empty object 115 MatrixXf m6(nb_rows, nb_columns); 116 \endcode</td><td class="note"> 117 By default, the coefficients \n are left uninitialized</td></tr> 118 <tr class="alt"><td>Comma initializer</td> 119 <td>\code 120 Vector3f v1; v1 << x, y, z; 121 ArrayXf v2(4); v2 << 1, 2, 3, 4; 122 123 \endcode</td><td>\code 124 Matrix3f m1; m1 << 1, 2, 3, 125 4, 5, 6, 126 7, 8, 9; 127 \endcode</td><td></td></tr> 128 129 <tr><td>Comma initializer (bis)</td> 130 <td colspan="2"> 131 \include Tutorial_commainit_02.cpp 132 </td> 133 <td> 134 output: 135 \verbinclude Tutorial_commainit_02.out 136 </td> 137 </tr> 138 139 <tr class="alt"><td>Runtime info</td> 140 <td>\code 141 vector.size(); 142 143 vector.innerStride(); 144 vector.data(); 145 \endcode</td><td>\code 146 matrix.rows(); matrix.cols(); 147 matrix.innerSize(); matrix.outerSize(); 148 matrix.innerStride(); matrix.outerStride(); 149 matrix.data(); 150 \endcode</td><td class="note">Inner/Outer* are storage order dependent</td></tr> 151 <tr><td>Compile-time info</td> 152 <td colspan="2">\code 153 ObjectType::Scalar ObjectType::RowsAtCompileTime 154 ObjectType::RealScalar ObjectType::ColsAtCompileTime 155 ObjectType::Index ObjectType::SizeAtCompileTime 156 \endcode</td><td></td></tr> 157 <tr class="alt"><td>Resizing</td> 158 <td>\code 159 vector.resize(size); 160 161 162 vector.resizeLike(other_vector); 163 vector.conservativeResize(size); 164 \endcode</td><td>\code 165 matrix.resize(nb_rows, nb_cols); 166 matrix.resize(Eigen::NoChange, nb_cols); 167 matrix.resize(nb_rows, Eigen::NoChange); 168 matrix.resizeLike(other_matrix); 169 matrix.conservativeResize(nb_rows, nb_cols); 170 \endcode</td><td class="note">no-op if the new sizes match,<br/>otherwise data are lost<br/><br/>resizing with data preservation</td></tr> 171 172 <tr><td>Coeff access with \n range checking</td> 173 <td>\code 174 vector(i) vector.x() 175 vector[i] vector.y() 176 vector.z() 177 vector.w() 178 \endcode</td><td>\code 179 matrix(i,j) 180 \endcode</td><td class="note">Range checking is disabled if \n NDEBUG or EIGEN_NO_DEBUG is defined</td></tr> 181 182 <tr class="alt"><td>Coeff access without \n range checking</td> 183 <td>\code 184 vector.coeff(i) 185 vector.coeffRef(i) 186 \endcode</td><td>\code 187 matrix.coeff(i,j) 188 matrix.coeffRef(i,j) 189 \endcode</td><td></td></tr> 190 191 <tr><td>Assignment/copy</td> 192 <td colspan="2">\code 193 object = expression; 194 object_of_float = expression_of_double.cast<float>(); 195 \endcode</td><td class="note">the destination is automatically resized (if possible)</td></tr> 196 197 </table> 198 199 \subsection QuickRef_PredefMat Predefined Matrices 200 201 <table class="manual"> 202 <tr> 203 <th>Fixed-size matrix or vector</th> 204 <th>Dynamic-size matrix</th> 205 <th>Dynamic-size vector</th> 206 </tr> 207 <tr style="border-bottom-style: none;"> 208 <td> 209 \code 210 typedef {Matrix3f|Array33f} FixedXD; 211 FixedXD x; 212 213 x = FixedXD::Zero(); 214 x = FixedXD::Ones(); 215 x = FixedXD::Constant(value); 216 x = FixedXD::Random(); 217 x = FixedXD::LinSpaced(size, low, high); 218 219 x.setZero(); 220 x.setOnes(); 221 x.setConstant(value); 222 x.setRandom(); 223 x.setLinSpaced(size, low, high); 224 \endcode 225 </td> 226 <td> 227 \code 228 typedef {MatrixXf|ArrayXXf} Dynamic2D; 229 Dynamic2D x; 230 231 x = Dynamic2D::Zero(rows, cols); 232 x = Dynamic2D::Ones(rows, cols); 233 x = Dynamic2D::Constant(rows, cols, value); 234 x = Dynamic2D::Random(rows, cols); 235 N/A 236 237 x.setZero(rows, cols); 238 x.setOnes(rows, cols); 239 x.setConstant(rows, cols, value); 240 x.setRandom(rows, cols); 241 N/A 242 \endcode 243 </td> 244 <td> 245 \code 246 typedef {VectorXf|ArrayXf} Dynamic1D; 247 Dynamic1D x; 248 249 x = Dynamic1D::Zero(size); 250 x = Dynamic1D::Ones(size); 251 x = Dynamic1D::Constant(size, value); 252 x = Dynamic1D::Random(size); 253 x = Dynamic1D::LinSpaced(size, low, high); 254 255 x.setZero(size); 256 x.setOnes(size); 257 x.setConstant(size, value); 258 x.setRandom(size); 259 x.setLinSpaced(size, low, high); 260 \endcode 261 </td> 262 </tr> 263 264 <tr><td colspan="3">Identity and \link MatrixBase::Unit basis vectors \endlink \matrixworld</td></tr> 265 <tr style="border-bottom-style: none;"> 266 <td> 267 \code 268 x = FixedXD::Identity(); 269 x.setIdentity(); 270 271 Vector3f::UnitX() // 1 0 0 272 Vector3f::UnitY() // 0 1 0 273 Vector3f::UnitZ() // 0 0 1 274 \endcode 275 </td> 276 <td> 277 \code 278 x = Dynamic2D::Identity(rows, cols); 279 x.setIdentity(rows, cols); 280 281 282 283 N/A 284 \endcode 285 </td> 286 <td>\code 287 N/A 288 289 290 VectorXf::Unit(size,i) 291 VectorXf::Unit(4,1) == Vector4f(0,1,0,0) 292 == Vector4f::UnitY() 293 \endcode 294 </td> 295 </tr> 296 </table> 297 298 299 300 \subsection QuickRef_Map Mapping external arrays 301 302 <table class="manual"> 303 <tr> 304 <td>Contiguous \n memory</td> 305 <td>\code 306 float data[] = {1,2,3,4}; 307 Map<Vector3f> v1(data); // uses v1 as a Vector3f object 308 Map<ArrayXf> v2(data,3); // uses v2 as a ArrayXf object 309 Map<Array22f> m1(data); // uses m1 as a Array22f object 310 Map<MatrixXf> m2(data,2,2); // uses m2 as a MatrixXf object 311 \endcode</td> 312 </tr> 313 <tr> 314 <td>Typical usage \n of strides</td> 315 <td>\code 316 float data[] = {1,2,3,4,5,6,7,8,9}; 317 Map<VectorXf,0,InnerStride<2> > v1(data,3); // = [1,3,5] 318 Map<VectorXf,0,InnerStride<> > v2(data,3,InnerStride<>(3)); // = [1,4,7] 319 Map<MatrixXf,0,OuterStride<3> > m2(data,2,3); // both lines |1,4,7| 320 Map<MatrixXf,0,OuterStride<> > m1(data,2,3,OuterStride<>(3)); // are equal to: |2,5,8| 321 \endcode</td> 322 </tr> 323 </table> 324 325 326 <a href="#" class="top">top</a> 327 \section QuickRef_ArithmeticOperators Arithmetic Operators 328 329 <table class="manual"> 330 <tr><td> 331 add \n subtract</td><td>\code 332 mat3 = mat1 + mat2; mat3 += mat1; 333 mat3 = mat1 - mat2; mat3 -= mat1;\endcode 334 </td></tr> 335 <tr class="alt"><td> 336 scalar product</td><td>\code 337 mat3 = mat1 * s1; mat3 *= s1; mat3 = s1 * mat1; 338 mat3 = mat1 / s1; mat3 /= s1;\endcode 339 </td></tr> 340 <tr><td> 341 matrix/vector \n products \matrixworld</td><td>\code 342 col2 = mat1 * col1; 343 row2 = row1 * mat1; row1 *= mat1; 344 mat3 = mat1 * mat2; mat3 *= mat1; \endcode 345 </td></tr> 346 <tr class="alt"><td> 347 transposition \n adjoint \matrixworld</td><td>\code 348 mat1 = mat2.transpose(); mat1.transposeInPlace(); 349 mat1 = mat2.adjoint(); mat1.adjointInPlace(); 350 \endcode 351 </td></tr> 352 <tr><td> 353 \link MatrixBase::dot() dot \endlink product \n inner product \matrixworld</td><td>\code 354 scalar = vec1.dot(vec2); 355 scalar = col1.adjoint() * col2; 356 scalar = (col1.adjoint() * col2).value();\endcode 357 </td></tr> 358 <tr class="alt"><td> 359 outer product \matrixworld</td><td>\code 360 mat = col1 * col2.transpose();\endcode 361 </td></tr> 362 363 <tr><td> 364 \link MatrixBase::norm() norm \endlink \n \link MatrixBase::normalized() normalization \endlink \matrixworld</td><td>\code 365 scalar = vec1.norm(); scalar = vec1.squaredNorm() 366 vec2 = vec1.normalized(); vec1.normalize(); // inplace \endcode 367 </td></tr> 368 369 <tr class="alt"><td> 370 \link MatrixBase::cross() cross product \endlink \matrixworld</td><td>\code 371 #include <Eigen/Geometry> 372 vec3 = vec1.cross(vec2);\endcode</td></tr> 373 </table> 374 375 <a href="#" class="top">top</a> 376 \section QuickRef_Coeffwise Coefficient-wise \& Array operators 377 Coefficient-wise operators for matrices and vectors: 378 <table class="manual"> 379 <tr><th>Matrix API \matrixworld</th><th>Via Array conversions</th></tr> 380 <tr><td>\code 381 mat1.cwiseMin(mat2) 382 mat1.cwiseMax(mat2) 383 mat1.cwiseAbs2() 384 mat1.cwiseAbs() 385 mat1.cwiseSqrt() 386 mat1.cwiseProduct(mat2) 387 mat1.cwiseQuotient(mat2)\endcode 388 </td><td>\code 389 mat1.array().min(mat2.array()) 390 mat1.array().max(mat2.array()) 391 mat1.array().abs2() 392 mat1.array().abs() 393 mat1.array().sqrt() 394 mat1.array() * mat2.array() 395 mat1.array() / mat2.array() 396 \endcode</td></tr> 397 </table> 398 399 It is also very simple to apply any user defined function \c foo using DenseBase::unaryExpr together with std::ptr_fun: 400 \code mat1.unaryExpr(std::ptr_fun(foo))\endcode 401 402 Array operators:\arrayworld 403 404 <table class="manual"> 405 <tr><td>Arithmetic operators</td><td>\code 406 array1 * array2 array1 / array2 array1 *= array2 array1 /= array2 407 array1 + scalar array1 - scalar array1 += scalar array1 -= scalar 408 \endcode</td></tr> 409 <tr><td>Comparisons</td><td>\code 410 array1 < array2 array1 > array2 array1 < scalar array1 > scalar 411 array1 <= array2 array1 >= array2 array1 <= scalar array1 >= scalar 412 array1 == array2 array1 != array2 array1 == scalar array1 != scalar 413 \endcode</td></tr> 414 <tr><td>Trigo, power, and \n misc functions \n and the STL variants</td><td>\code 415 array1.min(array2) 416 array1.max(array2) 417 array1.abs2() 418 array1.abs() std::abs(array1) 419 array1.sqrt() std::sqrt(array1) 420 array1.log() std::log(array1) 421 array1.exp() std::exp(array1) 422 array1.pow(exponent) std::pow(array1,exponent) 423 array1.square() 424 array1.cube() 425 array1.inverse() 426 array1.sin() std::sin(array1) 427 array1.cos() std::cos(array1) 428 array1.tan() std::tan(array1) 429 array1.asin() std::asin(array1) 430 array1.acos() std::acos(array1) 431 \endcode 432 </td></tr> 433 </table> 434 435 <a href="#" class="top">top</a> 436 \section QuickRef_Reductions Reductions 437 438 Eigen provides several reduction methods such as: 439 \link DenseBase::minCoeff() minCoeff() \endlink, \link DenseBase::maxCoeff() maxCoeff() \endlink, 440 \link DenseBase::sum() sum() \endlink, \link DenseBase::prod() prod() \endlink, 441 \link MatrixBase::trace() trace() \endlink \matrixworld, 442 \link MatrixBase::norm() norm() \endlink \matrixworld, \link MatrixBase::squaredNorm() squaredNorm() \endlink \matrixworld, 443 \link DenseBase::all() all() \endlink, and \link DenseBase::any() any() \endlink. 444 All reduction operations can be done matrix-wise, 445 \link DenseBase::colwise() column-wise \endlink or 446 \link DenseBase::rowwise() row-wise \endlink. Usage example: 447 <table class="manual"> 448 <tr><td rowspan="3" style="border-right-style:dashed;vertical-align:middle">\code 449 5 3 1 450 mat = 2 7 8 451 9 4 6 \endcode 452 </td> <td>\code mat.minCoeff(); \endcode</td><td>\code 1 \endcode</td></tr> 453 <tr class="alt"><td>\code mat.colwise().minCoeff(); \endcode</td><td>\code 2 3 1 \endcode</td></tr> 454 <tr style="vertical-align:middle"><td>\code mat.rowwise().minCoeff(); \endcode</td><td>\code 455 1 456 2 457 4 458 \endcode</td></tr> 459 </table> 460 461 Special versions of \link DenseBase::minCoeff(Index*,Index*) minCoeff \endlink and \link DenseBase::maxCoeff(Index*,Index*) maxCoeff \endlink: 462 \code 463 int i, j; 464 s = vector.minCoeff(&i); // s == vector[i] 465 s = matrix.maxCoeff(&i, &j); // s == matrix(i,j) 466 \endcode 467 Typical use cases of all() and any(): 468 \code 469 if((array1 > 0).all()) ... // if all coefficients of array1 are greater than 0 ... 470 if((array1 < array2).any()) ... // if there exist a pair i,j such that array1(i,j) < array2(i,j) ... 471 \endcode 472 473 474 <a href="#" class="top">top</a>\section QuickRef_Blocks Sub-matrices 475 476 Read-write access to a \link DenseBase::col(Index) column \endlink 477 or a \link DenseBase::row(Index) row \endlink of a matrix (or array): 478 \code 479 mat1.row(i) = mat2.col(j); 480 mat1.col(j1).swap(mat1.col(j2)); 481 \endcode 482 483 Read-write access to sub-vectors: 484 <table class="manual"> 485 <tr> 486 <th>Default versions</th> 487 <th>Optimized versions when the size \n is known at compile time</th></tr> 488 <th></th> 489 490 <tr><td>\code vec1.head(n)\endcode</td><td>\code vec1.head<n>()\endcode</td><td>the first \c n coeffs </td></tr> 491 <tr><td>\code vec1.tail(n)\endcode</td><td>\code vec1.tail<n>()\endcode</td><td>the last \c n coeffs </td></tr> 492 <tr><td>\code vec1.segment(pos,n)\endcode</td><td>\code vec1.segment<n>(pos)\endcode</td> 493 <td>the \c n coeffs in \n the range [\c pos : \c pos + \c n [</td></tr> 494 <tr class="alt"><td colspan="3"> 495 496 Read-write access to sub-matrices:</td></tr> 497 <tr> 498 <td>\code mat1.block(i,j,rows,cols)\endcode 499 \link DenseBase::block(Index,Index,Index,Index) (more) \endlink</td> 500 <td>\code mat1.block<rows,cols>(i,j)\endcode 501 \link DenseBase::block(Index,Index) (more) \endlink</td> 502 <td>the \c rows x \c cols sub-matrix \n starting from position (\c i,\c j)</td></tr> 503 <tr><td>\code 504 mat1.topLeftCorner(rows,cols) 505 mat1.topRightCorner(rows,cols) 506 mat1.bottomLeftCorner(rows,cols) 507 mat1.bottomRightCorner(rows,cols)\endcode 508 <td>\code 509 mat1.topLeftCorner<rows,cols>() 510 mat1.topRightCorner<rows,cols>() 511 mat1.bottomLeftCorner<rows,cols>() 512 mat1.bottomRightCorner<rows,cols>()\endcode 513 <td>the \c rows x \c cols sub-matrix \n taken in one of the four corners</td></tr> 514 <tr><td>\code 515 mat1.topRows(rows) 516 mat1.bottomRows(rows) 517 mat1.leftCols(cols) 518 mat1.rightCols(cols)\endcode 519 <td>\code 520 mat1.topRows<rows>() 521 mat1.bottomRows<rows>() 522 mat1.leftCols<cols>() 523 mat1.rightCols<cols>()\endcode 524 <td>specialized versions of block() \n when the block fit two corners</td></tr> 525 </table> 526 527 528 529 <a href="#" class="top">top</a>\section QuickRef_Misc Miscellaneous operations 530 531 \subsection QuickRef_Reverse Reverse 532 Vectors, rows, and/or columns of a matrix can be reversed (see DenseBase::reverse(), DenseBase::reverseInPlace(), VectorwiseOp::reverse()). 533 \code 534 vec.reverse() mat.colwise().reverse() mat.rowwise().reverse() 535 vec.reverseInPlace() 536 \endcode 537 538 \subsection QuickRef_Replicate Replicate 539 Vectors, matrices, rows, and/or columns can be replicated in any direction (see DenseBase::replicate(), VectorwiseOp::replicate()) 540 \code 541 vec.replicate(times) vec.replicate<Times> 542 mat.replicate(vertical_times, horizontal_times) mat.replicate<VerticalTimes, HorizontalTimes>() 543 mat.colwise().replicate(vertical_times, horizontal_times) mat.colwise().replicate<VerticalTimes, HorizontalTimes>() 544 mat.rowwise().replicate(vertical_times, horizontal_times) mat.rowwise().replicate<VerticalTimes, HorizontalTimes>() 545 \endcode 546 547 548 <a href="#" class="top">top</a>\section QuickRef_DiagTriSymm Diagonal, Triangular, and Self-adjoint matrices 549 (matrix world \matrixworld) 550 551 \subsection QuickRef_Diagonal Diagonal matrices 552 553 <table class="example"> 554 <tr><th>Operation</th><th>Code</th></tr> 555 <tr><td> 556 view a vector \link MatrixBase::asDiagonal() as a diagonal matrix \endlink \n </td><td>\code 557 mat1 = vec1.asDiagonal();\endcode 558 </td></tr> 559 <tr><td> 560 Declare a diagonal matrix</td><td>\code 561 DiagonalMatrix<Scalar,SizeAtCompileTime> diag1(size); 562 diag1.diagonal() = vector;\endcode 563 </td></tr> 564 <tr><td>Access the \link MatrixBase::diagonal() diagonal \endlink and \link MatrixBase::diagonal(Index) super/sub diagonals \endlink of a matrix as a vector (read/write)</td> 565 <td>\code 566 vec1 = mat1.diagonal(); mat1.diagonal() = vec1; // main diagonal 567 vec1 = mat1.diagonal(+n); mat1.diagonal(+n) = vec1; // n-th super diagonal 568 vec1 = mat1.diagonal(-n); mat1.diagonal(-n) = vec1; // n-th sub diagonal 569 vec1 = mat1.diagonal<1>(); mat1.diagonal<1>() = vec1; // first super diagonal 570 vec1 = mat1.diagonal<-2>(); mat1.diagonal<-2>() = vec1; // second sub diagonal 571 \endcode</td> 572 </tr> 573 574 <tr><td>Optimized products and inverse</td> 575 <td>\code 576 mat3 = scalar * diag1 * mat1; 577 mat3 += scalar * mat1 * vec1.asDiagonal(); 578 mat3 = vec1.asDiagonal().inverse() * mat1 579 mat3 = mat1 * diag1.inverse() 580 \endcode</td> 581 </tr> 582 583 </table> 584 585 \subsection QuickRef_TriangularView Triangular views 586 587 TriangularView gives a view on a triangular part of a dense matrix and allows to perform optimized operations on it. The opposite triangular part is never referenced and can be used to store other information. 588 589 \note The .triangularView() template member function requires the \c template keyword if it is used on an 590 object of a type that depends on a template parameter; see \ref TopicTemplateKeyword for details. 591 592 <table class="example"> 593 <tr><th>Operation</th><th>Code</th></tr> 594 <tr><td> 595 Reference to a triangular with optional \n 596 unit or null diagonal (read/write): 597 </td><td>\code 598 m.triangularView<Xxx>() 599 \endcode \n 600 \c Xxx = ::Upper, ::Lower, ::StrictlyUpper, ::StrictlyLower, ::UnitUpper, ::UnitLower 601 </td></tr> 602 <tr><td> 603 Writing to a specific triangular part:\n (only the referenced triangular part is evaluated) 604 </td><td>\code 605 m1.triangularView<Eigen::Lower>() = m2 + m3 \endcode 606 </td></tr> 607 <tr><td> 608 Conversion to a dense matrix setting the opposite triangular part to zero: 609 </td><td>\code 610 m2 = m1.triangularView<Eigen::UnitUpper>()\endcode 611 </td></tr> 612 <tr><td> 613 Products: 614 </td><td>\code 615 m3 += s1 * m1.adjoint().triangularView<Eigen::UnitUpper>() * m2 616 m3 -= s1 * m2.conjugate() * m1.adjoint().triangularView<Eigen::Lower>() \endcode 617 </td></tr> 618 <tr><td> 619 Solving linear equations:\n 620 \f$ M_2 := L_1^{-1} M_2 \f$ \n 621 \f$ M_3 := {L_1^*}^{-1} M_3 \f$ \n 622 \f$ M_4 := M_4 U_1^{-1} \f$ 623 </td><td>\n \code 624 L1.triangularView<Eigen::UnitLower>().solveInPlace(M2) 625 L1.triangularView<Eigen::Lower>().adjoint().solveInPlace(M3) 626 U1.triangularView<Eigen::Upper>().solveInPlace<OnTheRight>(M4)\endcode 627 </td></tr> 628 </table> 629 630 \subsection QuickRef_SelfadjointMatrix Symmetric/selfadjoint views 631 632 Just as for triangular matrix, you can reference any triangular part of a square matrix to see it as a selfadjoint 633 matrix and perform special and optimized operations. Again the opposite triangular part is never referenced and can be 634 used to store other information. 635 636 \note The .selfadjointView() template member function requires the \c template keyword if it is used on an 637 object of a type that depends on a template parameter; see \ref TopicTemplateKeyword for details. 638 639 <table class="example"> 640 <tr><th>Operation</th><th>Code</th></tr> 641 <tr><td> 642 Conversion to a dense matrix: 643 </td><td>\code 644 m2 = m.selfadjointView<Eigen::Lower>();\endcode 645 </td></tr> 646 <tr><td> 647 Product with another general matrix or vector: 648 </td><td>\code 649 m3 = s1 * m1.conjugate().selfadjointView<Eigen::Upper>() * m3; 650 m3 -= s1 * m3.adjoint() * m1.selfadjointView<Eigen::Lower>();\endcode 651 </td></tr> 652 <tr><td> 653 Rank 1 and rank K update: \n 654 \f$ upper(M_1) \mathrel{{+}{=}} s_1 M_2 M_2^* \f$ \n 655 \f$ lower(M_1) \mathbin{{-}{=}} M_2^* M_2 \f$ 656 </td><td>\n \code 657 M1.selfadjointView<Eigen::Upper>().rankUpdate(M2,s1); 658 M1.selfadjointView<Eigen::Lower>().rankUpdate(M2.adjoint(),-1); \endcode 659 </td></tr> 660 <tr><td> 661 Rank 2 update: (\f$ M \mathrel{{+}{=}} s u v^* + s v u^* \f$) 662 </td><td>\code 663 M.selfadjointView<Eigen::Upper>().rankUpdate(u,v,s); 664 \endcode 665 </td></tr> 666 <tr><td> 667 Solving linear equations:\n(\f$ M_2 := M_1^{-1} M_2 \f$) 668 </td><td>\code 669 // via a standard Cholesky factorization 670 m2 = m1.selfadjointView<Eigen::Upper>().llt().solve(m2); 671 // via a Cholesky factorization with pivoting 672 m2 = m1.selfadjointView<Eigen::Lower>().ldlt().solve(m2); 673 \endcode 674 </td></tr> 675 </table> 676 677 */ 678 679 /* 680 <table class="tutorial_code"> 681 <tr><td> 682 \link MatrixBase::asDiagonal() make a diagonal matrix \endlink \n from a vector </td><td>\code 683 mat1 = vec1.asDiagonal();\endcode 684 </td></tr> 685 <tr><td> 686 Declare a diagonal matrix</td><td>\code 687 DiagonalMatrix<Scalar,SizeAtCompileTime> diag1(size); 688 diag1.diagonal() = vector;\endcode 689 </td></tr> 690 <tr><td>Access \link MatrixBase::diagonal() the diagonal and super/sub diagonals of a matrix \endlink as a vector (read/write)</td> 691 <td>\code 692 vec1 = mat1.diagonal(); mat1.diagonal() = vec1; // main diagonal 693 vec1 = mat1.diagonal(+n); mat1.diagonal(+n) = vec1; // n-th super diagonal 694 vec1 = mat1.diagonal(-n); mat1.diagonal(-n) = vec1; // n-th sub diagonal 695 vec1 = mat1.diagonal<1>(); mat1.diagonal<1>() = vec1; // first super diagonal 696 vec1 = mat1.diagonal<-2>(); mat1.diagonal<-2>() = vec1; // second sub diagonal 697 \endcode</td> 698 </tr> 699 700 <tr><td>View on a triangular part of a matrix (read/write)</td> 701 <td>\code 702 mat2 = mat1.triangularView<Xxx>(); 703 // Xxx = Upper, Lower, StrictlyUpper, StrictlyLower, UnitUpper, UnitLower 704 mat1.triangularView<Upper>() = mat2 + mat3; // only the upper part is evaluated and referenced 705 \endcode</td></tr> 706 707 <tr><td>View a triangular part as a symmetric/self-adjoint matrix (read/write)</td> 708 <td>\code 709 mat2 = mat1.selfadjointView<Xxx>(); // Xxx = Upper or Lower 710 mat1.selfadjointView<Upper>() = mat2 + mat2.adjoint(); // evaluated and write to the upper triangular part only 711 \endcode</td></tr> 712 713 </table> 714 715 Optimized products: 716 \code 717 mat3 += scalar * vec1.asDiagonal() * mat1 718 mat3 += scalar * mat1 * vec1.asDiagonal() 719 mat3.noalias() += scalar * mat1.triangularView<Xxx>() * mat2 720 mat3.noalias() += scalar * mat2 * mat1.triangularView<Xxx>() 721 mat3.noalias() += scalar * mat1.selfadjointView<Upper or Lower>() * mat2 722 mat3.noalias() += scalar * mat2 * mat1.selfadjointView<Upper or Lower>() 723 mat1.selfadjointView<Upper or Lower>().rankUpdate(mat2); 724 mat1.selfadjointView<Upper or Lower>().rankUpdate(mat2.adjoint(), scalar); 725 \endcode 726 727 Inverse products: (all are optimized) 728 \code 729 mat3 = vec1.asDiagonal().inverse() * mat1 730 mat3 = mat1 * diag1.inverse() 731 mat1.triangularView<Xxx>().solveInPlace(mat2) 732 mat1.triangularView<Xxx>().solveInPlace<OnTheRight>(mat2) 733 mat2 = mat1.selfadjointView<Upper or Lower>().llt().solve(mat2) 734 \endcode 735 736 */ 737 } 738
