1 namespace Eigen { 2 3 /** \page TutorialMatrixClass Tutorial page 1 - The %Matrix class 4 5 \ingroup Tutorial 6 7 \li \b Previous: \ref GettingStarted 8 \li \b Next: \ref TutorialMatrixArithmetic 9 10 We assume that you have already read the quick \link GettingStarted "getting started" \endlink tutorial. 11 This page is the first one in a much longer multi-page tutorial. 12 13 \b Table \b of \b contents 14 - \ref TutorialMatrixFirst3Params 15 - \ref TutorialMatrixVectors 16 - \ref TutorialMatrixDynamic 17 - \ref TutorialMatrixConstructors 18 - \ref TutorialMatrixCoeffAccessors 19 - \ref TutorialMatrixCommaInitializer 20 - \ref TutorialMatrixSizesResizing 21 - \ref TutorialMatrixAssignment 22 - \ref TutorialMatrixFixedVsDynamic 23 - \ref TutorialMatrixOptTemplParams 24 - \ref TutorialMatrixTypedefs 25 26 In Eigen, all matrices and vectors are objects of the Matrix template class. 27 Vectors are just a special case of matrices, with either 1 row or 1 column. 28 29 \section TutorialMatrixFirst3Params The first three template parameters of Matrix 30 31 The Matrix class takes six template parameters, but for now it's enough to 32 learn about the first three first parameters. The three remaining parameters have default 33 values, which for now we will leave untouched, and which we 34 \ref TutorialMatrixOptTemplParams "discuss below". 35 36 The three mandatory template parameters of Matrix are: 37 \code 38 Matrix<typename Scalar, int RowsAtCompileTime, int ColsAtCompileTime> 39 \endcode 40 \li \c Scalar is the scalar type, i.e. the type of the coefficients. 41 That is, if you want a matrix of floats, choose \c float here. 42 See \ref TopicScalarTypes "Scalar types" for a list of all supported 43 scalar types and for how to extend support to new types. 44 \li \c RowsAtCompileTime and \c ColsAtCompileTime are the number of rows 45 and columns of the matrix as known at compile time (see 46 \ref TutorialMatrixDynamic "below" for what to do if the number is not 47 known at compile time). 48 49 We offer a lot of convenience typedefs to cover the usual cases. For example, \c Matrix4f is 50 a 4x4 matrix of floats. Here is how it is defined by Eigen: 51 \code 52 typedef Matrix<float, 4, 4> Matrix4f; 53 \endcode 54 We discuss \ref TutorialMatrixTypedefs "below" these convenience typedefs. 55 56 \section TutorialMatrixVectors Vectors 57 58 As mentioned above, in Eigen, vectors are just a special case of 59 matrices, with either 1 row or 1 column. The case where they have 1 column is the most common; 60 such vectors are called column-vectors, often abbreviated as just vectors. In the other case 61 where they have 1 row, they are called row-vectors. 62 63 For example, the convenience typedef \c Vector3f is a (column) vector of 3 floats. It is defined as follows by Eigen: 64 \code 65 typedef Matrix<float, 3, 1> Vector3f; 66 \endcode 67 We also offer convenience typedefs for row-vectors, for example: 68 \code 69 typedef Matrix<int, 1, 2> RowVector2i; 70 \endcode 71 72 \section TutorialMatrixDynamic The special value Dynamic 73 74 Of course, Eigen is not limited to matrices whose dimensions are known at compile time. 75 The \c RowsAtCompileTime and \c ColsAtCompileTime template parameters can take the special 76 value \c Dynamic which indicates that the size is unknown at compile time, so must 77 be handled as a run-time variable. In Eigen terminology, such a size is referred to as a 78 \em dynamic \em size; while a size that is known at compile time is called a 79 \em fixed \em size. For example, the convenience typedef \c MatrixXd, meaning 80 a matrix of doubles with dynamic size, is defined as follows: 81 \code 82 typedef Matrix<double, Dynamic, Dynamic> MatrixXd; 83 \endcode 84 And similarly, we define a self-explanatory typedef \c VectorXi as follows: 85 \code 86 typedef Matrix<int, Dynamic, 1> VectorXi; 87 \endcode 88 You can perfectly have e.g. a fixed number of rows with a dynamic number of columns, as in: 89 \code 90 Matrix<float, 3, Dynamic> 91 \endcode 92 93 \section TutorialMatrixConstructors Constructors 94 95 A default constructor is always available, never performs any dynamic memory allocation, and never initializes the matrix coefficients. You can do: 96 \code 97 Matrix3f a; 98 MatrixXf b; 99 \endcode 100 Here, 101 \li \c a is a 3x3 matrix, with a static float[9] array of uninitialized coefficients, 102 \li \c b is a dynamic-size matrix whose size is currently 0x0, and whose array of 103 coefficients hasn't yet been allocated at all. 104 105 Constructors taking sizes are also available. For matrices, the number of rows is always passed first. 106 For vectors, just pass the vector size. They allocate the array of coefficients 107 with the given size, but don't initialize the coefficients themselves: 108 \code 109 MatrixXf a(10,15); 110 VectorXf b(30); 111 \endcode 112 Here, 113 \li \c a is a 10x15 dynamic-size matrix, with allocated but currently uninitialized coefficients. 114 \li \c b is a dynamic-size vector of size 30, with allocated but currently uninitialized coefficients. 115 116 In order to offer a uniform API across fixed-size and dynamic-size matrices, it is legal to use these 117 constructors on fixed-size matrices, even if passing the sizes is useless in this case. So this is legal: 118 \code 119 Matrix3f a(3,3); 120 \endcode 121 and is a no-operation. 122 123 Finally, we also offer some constructors to initialize the coefficients of small fixed-size vectors up to size 4: 124 \code 125 Vector2d a(5.0, 6.0); 126 Vector3d b(5.0, 6.0, 7.0); 127 Vector4d c(5.0, 6.0, 7.0, 8.0); 128 \endcode 129 130 \section TutorialMatrixCoeffAccessors Coefficient accessors 131 132 The primary coefficient accessors and mutators in Eigen are the overloaded parenthesis operators. 133 For matrices, the row index is always passed first. For vectors, just pass one index. 134 The numbering starts at 0. This example is self-explanatory: 135 136 <table class="example"> 137 <tr><th>Example:</th><th>Output:</th></tr> 138 <tr><td> 139 \include tut_matrix_coefficient_accessors.cpp 140 </td> 141 <td> 142 \verbinclude tut_matrix_coefficient_accessors.out 143 </td></tr></table> 144 145 Note that the syntax <tt> m(index) </tt> 146 is not restricted to vectors, it is also available for general matrices, meaning index-based access 147 in the array of coefficients. This however depends on the matrix's storage order. All Eigen matrices default to 148 column-major storage order, but this can be changed to row-major, see \ref TopicStorageOrders "Storage orders". 149 150 The operator[] is also overloaded for index-based access in vectors, but keep in mind that C++ doesn't allow operator[] to 151 take more than one argument. We restrict operator[] to vectors, because an awkwardness in the C++ language 152 would make matrix[i,j] compile to the same thing as matrix[j] ! 153 154 \section TutorialMatrixCommaInitializer Comma-initialization 155 156 %Matrix and vector coefficients can be conveniently set using the so-called \em comma-initializer syntax. 157 For now, it is enough to know this example: 158 159 <table class="example"> 160 <tr><th>Example:</th><th>Output:</th></tr> 161 <tr> 162 <td>\include Tutorial_commainit_01.cpp </td> 163 <td>\verbinclude Tutorial_commainit_01.out </td> 164 </tr></table> 165 166 167 The right-hand side can also contain matrix expressions as discussed in \ref TutorialAdvancedInitialization "this page". 168 169 \section TutorialMatrixSizesResizing Resizing 170 171 The current size of a matrix can be retrieved by \link EigenBase::rows() rows()\endlink, \link EigenBase::cols() cols() \endlink and \link EigenBase::size() size()\endlink. These methods return the number of rows, the number of columns and the number of coefficients, respectively. Resizing a dynamic-size matrix is done by the \link PlainObjectBase::resize(Index,Index) resize() \endlink method. 172 173 <table class="example"> 174 <tr><th>Example:</th><th>Output:</th></tr> 175 <tr> 176 <td>\include tut_matrix_resize.cpp </td> 177 <td>\verbinclude tut_matrix_resize.out </td> 178 </tr></table> 179 180 The resize() method is a no-operation if the actual matrix size doesn't change; otherwise it is destructive: the values of the coefficients may change. 181 If you want a conservative variant of resize() which does not change the coefficients, use \link PlainObjectBase::conservativeResize() conservativeResize()\endlink, see \ref TopicResizing "this page" for more details. 182 183 All these methods are still available on fixed-size matrices, for the sake of API uniformity. Of course, you can't actually 184 resize a fixed-size matrix. Trying to change a fixed size to an actually different value will trigger an assertion failure; 185 but the following code is legal: 186 187 <table class="example"> 188 <tr><th>Example:</th><th>Output:</th></tr> 189 <tr> 190 <td>\include tut_matrix_resize_fixed_size.cpp </td> 191 <td>\verbinclude tut_matrix_resize_fixed_size.out </td> 192 </tr></table> 193 194 195 \section TutorialMatrixAssignment Assignment and resizing 196 197 Assignment is the action of copying a matrix into another, using \c operator=. Eigen resizes the matrix on the left-hand side automatically so that it matches the size of the matrix on the right-hand size. For example: 198 199 <table class="example"> 200 <tr><th>Example:</th><th>Output:</th></tr> 201 <tr> 202 <td>\include tut_matrix_assignment_resizing.cpp </td> 203 <td>\verbinclude tut_matrix_assignment_resizing.out </td> 204 </tr></table> 205 206 Of course, if the left-hand side is of fixed size, resizing it is not allowed. 207 208 If you do not want this automatic resizing to happen (for example for debugging purposes), you can disable it, see 209 \ref TopicResizing "this page". 210 211 212 \section TutorialMatrixFixedVsDynamic Fixed vs. Dynamic size 213 214 When should one use fixed sizes (e.g. \c Matrix4f), and when should one prefer dynamic sizes (e.g. \c MatrixXf)? 215 The simple answer is: use fixed 216 sizes for very small sizes where you can, and use dynamic sizes for larger sizes or where you have to. For small sizes, 217 especially for sizes smaller than (roughly) 16, using fixed sizes is hugely beneficial 218 to performance, as it allows Eigen to avoid dynamic memory allocation and to unroll 219 loops. Internally, a fixed-size Eigen matrix is just a plain static array, i.e. doing 220 \code Matrix4f mymatrix; \endcode 221 really amounts to just doing 222 \code float mymatrix[16]; \endcode 223 so this really has zero runtime cost. By contrast, the array of a dynamic-size matrix 224 is always allocated on the heap, so doing 225 \code MatrixXf mymatrix(rows,columns); \endcode 226 amounts to doing 227 \code float *mymatrix = new float[rows*columns]; \endcode 228 and in addition to that, the MatrixXf object stores its number of rows and columns as 229 member variables. 230 231 The limitation of using fixed sizes, of course, is that this is only possible 232 when you know the sizes at compile time. Also, for large enough sizes, say for sizes 233 greater than (roughly) 32, the performance benefit of using fixed sizes becomes negligible. 234 Worse, trying to create a very large matrix using fixed sizes could result in a stack overflow, 235 since Eigen will try to allocate the array as a static array, which by default goes on the stack. 236 Finally, depending on circumstances, Eigen can also be more aggressive trying to vectorize 237 (use SIMD instructions) when dynamic sizes are used, see \ref TopicVectorization "Vectorization". 238 239 \section TutorialMatrixOptTemplParams Optional template parameters 240 241 We mentioned at the beginning of this page that the Matrix class takes six template parameters, 242 but so far we only discussed the first three. The remaining three parameters are optional. Here is 243 the complete list of template parameters: 244 \code 245 Matrix<typename Scalar, 246 int RowsAtCompileTime, 247 int ColsAtCompileTime, 248 int Options = 0, 249 int MaxRowsAtCompileTime = RowsAtCompileTime, 250 int MaxColsAtCompileTime = ColsAtCompileTime> 251 \endcode 252 \li \c Options is a bit field. Here, we discuss only one bit: \c RowMajor. It specifies that the matrices 253 of this type use row-major storage order; by default, the storage order is column-major. See the page on 254 \ref TopicStorageOrders "storage orders". For example, this type means row-major 3x3 matrices: 255 \code 256 Matrix<float, 3, 3, RowMajor> 257 \endcode 258 \li \c MaxRowsAtCompileTime and \c MaxColsAtCompileTime are useful when you want to specify that, even though 259 the exact sizes of your matrices are not known at compile time, a fixed upper bound is known at 260 compile time. The biggest reason why you might want to do that is to avoid dynamic memory allocation. 261 For example the following matrix type uses a static array of 12 floats, without dynamic memory allocation: 262 \code 263 Matrix<float, Dynamic, Dynamic, 0, 3, 4> 264 \endcode 265 266 \section TutorialMatrixTypedefs Convenience typedefs 267 268 Eigen defines the following Matrix typedefs: 269 \li MatrixNt for Matrix<type, N, N>. For example, MatrixXi for Matrix<int, Dynamic, Dynamic>. 270 \li VectorNt for Matrix<type, N, 1>. For example, Vector2f for Matrix<float, 2, 1>. 271 \li RowVectorNt for Matrix<type, 1, N>. For example, RowVector3d for Matrix<double, 1, 3>. 272 273 Where: 274 \li N can be any one of \c 2, \c 3, \c 4, or \c X (meaning \c Dynamic). 275 \li t can be any one of \c i (meaning int), \c f (meaning float), \c d (meaning double), 276 \c cf (meaning complex<float>), or \c cd (meaning complex<double>). The fact that typedefs are only 277 defined for these five types doesn't mean that they are the only supported scalar types. For example, 278 all standard integer types are supported, see \ref TopicScalarTypes "Scalar types". 279 280 \li \b Next: \ref TutorialMatrixArithmetic 281 282 */ 283 284 } 285
