1 namespace Eigen { 2 3 /** \eigenManualPage TopicStorageOrders Storage orders 4 5 There are two different storage orders for matrices and two-dimensional arrays: column-major and row-major. 6 This page explains these storage orders and how to specify which one should be used. 7 8 \eigenAutoToc 9 10 11 \section TopicStorageOrdersIntro Column-major and row-major storage 12 13 The entries of a matrix form a two-dimensional grid. However, when the matrix is stored in memory, the entries 14 have to somehow be laid out linearly. There are two main ways to do this, by row and by column. 15 16 We say that a matrix is stored in \b row-major order if it is stored row by row. The entire first row is 17 stored first, followed by the entire second row, and so on. Consider for example the matrix 18 19 \f[ 20 A = \begin{bmatrix} 21 8 & 2 & 2 & 9 \\ 22 9 & 1 & 4 & 4 \\ 23 3 & 5 & 4 & 5 24 \end{bmatrix}. 25 \f] 26 27 If this matrix is stored in row-major order, then the entries are laid out in memory as follows: 28 29 \code 8 2 2 9 9 1 4 4 3 5 4 5 \endcode 30 31 On the other hand, a matrix is stored in \b column-major order if it is stored column by column, starting with 32 the entire first column, followed by the entire second column, and so on. If the above matrix is stored in 33 column-major order, it is laid out as follows: 34 35 \code 8 9 3 2 1 5 2 4 4 9 4 5 \endcode 36 37 This example is illustrated by the following Eigen code. It uses the PlainObjectBase::data() function, which 38 returns a pointer to the memory location of the first entry of the matrix. 39 40 <table class="example"> 41 <tr><th>Example</th><th>Output</th></tr> 42 <tr><td> 43 \include TopicStorageOrders_example.cpp 44 </td> 45 <td> 46 \verbinclude TopicStorageOrders_example.out 47 </td></tr></table> 48 49 50 \section TopicStorageOrdersInEigen Storage orders in Eigen 51 52 The storage order of a matrix or a two-dimensional array can be set by specifying the \c Options template 53 parameter for Matrix or Array. As \ref TutorialMatrixClass explains, the %Matrix class template has six 54 template parameters, of which three are compulsory (\c Scalar, \c RowsAtCompileTime and \c ColsAtCompileTime) 55 and three are optional (\c Options, \c MaxRowsAtCompileTime and \c MaxColsAtCompileTime). If the \c Options 56 parameter is set to \c RowMajor, then the matrix or array is stored in row-major order; if it is set to 57 \c ColMajor, then it is stored in column-major order. This mechanism is used in the above Eigen program to 58 specify the storage order. 59 60 If the storage order is not specified, then Eigen defaults to storing the entry in column-major. This is also 61 the case if one of the convenience typedefs (\c Matrix3f, \c ArrayXXd, etc.) is used. 62 63 Matrices and arrays using one storage order can be assigned to matrices and arrays using the other storage 64 order, as happens in the above program when \c Arowmajor is initialized using \c Acolmajor. Eigen will reorder 65 the entries automatically. More generally, row-major and column-major matrices can be mixed in an expression 66 as we want. 67 68 69 \section TopicStorageOrdersWhich Which storage order to choose? 70 71 So, which storage order should you use in your program? There is no simple answer to this question; it depends 72 on your application. Here are some points to keep in mind: 73 74 - Your users may expect you to use a specific storage order. Alternatively, you may use other libraries than 75 Eigen, and these other libraries may expect a certain storage order. In these cases it may be easiest and 76 fastest to use this storage order in your whole program. 77 - Algorithms that traverse a matrix row by row will go faster when the matrix is stored in row-major order 78 because of better data locality. Similarly, column-by-column traversal is faster for column-major 79 matrices. It may be worthwhile to experiment a bit to find out what is faster for your particular 80 application. 81 - The default in Eigen is column-major. Naturally, most of the development and testing of the Eigen library 82 is thus done with column-major matrices. This means that, even though we aim to support column-major and 83 row-major storage orders transparently, the Eigen library may well work best with column-major matrices. 84 85 */ 86 } 87
