1namespace Eigen {
2
3/** \eigenManualPage TutorialBlockOperations Block operations
4
5This page explains the essentials of block operations.
6A block is a rectangular part of a matrix or array. Blocks expressions can be used both
7as rvalues and as lvalues. As usual with Eigen expressions, this abstraction has zero runtime cost
8provided that you let your compiler optimize.
9
10\eigenAutoToc
11
12\section TutorialBlockOperationsUsing Using block operations
13
14The most general block operation in Eigen is called \link DenseBase::block() .block() \endlink.
15There are two versions, whose syntax is as follows:
16
17<table class="manual">
18<tr><th>\b %Block \b operation</td>
19<th>Version constructing a \n dynamic-size block expression</th>
20<th>Version constructing a \n fixed-size block expression</th></tr>
21<tr><td>%Block of size <tt>(p,q)</tt>, starting at <tt>(i,j)</tt></td>
22    <td>\code
23matrix.block(i,j,p,q);\endcode </td>
24    <td>\code 
25matrix.block<p,q>(i,j);\endcode </td>
26</tr>
27</table>
28
29As always in Eigen, indices start at 0.
30
31Both versions can be used on fixed-size and dynamic-size matrices and arrays.
32These two expressions are semantically equivalent.
33The only difference is that the fixed-size version will typically give you faster code if the block size is small,
34but requires this size to be known at compile time.
35
36The following program uses the dynamic-size and fixed-size versions to print the values of several blocks inside a
37matrix.
38
39<table class="example">
40<tr><th>Example:</th><th>Output:</th></tr>
41<tr><td>
42\include Tutorial_BlockOperations_print_block.cpp
43</td>
44<td>
45\verbinclude Tutorial_BlockOperations_print_block.out
46</td></tr></table>
47
48In the above example the \link DenseBase::block() .block() \endlink function was employed as a \em rvalue, i.e.
49it was only read from. However, blocks can also be used as \em lvalues, meaning that you can assign to a block.
50
51This is illustrated in the following example. This example also demonstrates blocks in arrays, which works exactly like the above-demonstrated blocks in matrices.
52
53<table class="example">
54<tr><th>Example:</th><th>Output:</th></tr>
55<tr><td>
56\include Tutorial_BlockOperations_block_assignment.cpp
57</td>
58<td>
59\verbinclude Tutorial_BlockOperations_block_assignment.out
60</td></tr></table>
61
62While the \link DenseBase::block() .block() \endlink method can be used for any block operation, there are
63other methods for special cases, providing more specialized API and/or better performance. On the topic of performance, all what
64matters is that you give Eigen as much information as possible at compile time. For example, if your block is a single whole column in a matrix,
65using the specialized \link DenseBase::col() .col() \endlink function described below lets Eigen know that, which can give it optimization opportunities.
66
67The rest of this page describes these specialized methods.
68
69\section TutorialBlockOperationsSyntaxColumnRows Columns and rows
70
71Individual columns and rows are special cases of blocks. Eigen provides methods to easily address them:
72\link DenseBase::col() .col() \endlink and \link DenseBase::row() .row()\endlink.
73
74<table class="manual">
75<tr><th>%Block operation</th>
76<th>Method</th>
77<tr><td>i<sup>th</sup> row
78                    \link DenseBase::row() * \endlink</td>
79    <td>\code
80matrix.row(i);\endcode </td>
81</tr>
82<tr><td>j<sup>th</sup> column
83                    \link DenseBase::col() * \endlink</td>
84    <td>\code
85matrix.col(j);\endcode </td>
86</tr>
87</table>
88
89The argument for \p col() and \p row() is the index of the column or row to be accessed. As always in Eigen, indices start at 0.
90
91<table class="example">
92<tr><th>Example:</th><th>Output:</th></tr>
93<tr><td>
94\include Tutorial_BlockOperations_colrow.cpp
95</td>
96<td>
97\verbinclude Tutorial_BlockOperations_colrow.out
98</td></tr></table>
99
100That example also demonstrates that block expressions (here columns) can be used in arithmetic like any other expression.
101
102
103\section TutorialBlockOperationsSyntaxCorners Corner-related operations
104
105Eigen also provides special methods for blocks that are flushed against one of the corners or sides of a
106matrix or array. For instance, \link DenseBase::topLeftCorner() .topLeftCorner() \endlink can be used to refer
107to a block in the top-left corner of a matrix.
108
109The different possibilities are summarized in the following table:
110
111<table class="manual">
112<tr><th>%Block \b operation</td>
113<th>Version constructing a \n dynamic-size block expression</th>
114<th>Version constructing a \n fixed-size block expression</th></tr>
115<tr><td>Top-left p by q block \link DenseBase::topLeftCorner() * \endlink</td>
116    <td>\code
117matrix.topLeftCorner(p,q);\endcode </td>
118    <td>\code 
119matrix.topLeftCorner<p,q>();\endcode </td>
120</tr>
121<tr><td>Bottom-left p by q block
122              \link DenseBase::bottomLeftCorner() * \endlink</td>
123    <td>\code
124matrix.bottomLeftCorner(p,q);\endcode </td>
125    <td>\code 
126matrix.bottomLeftCorner<p,q>();\endcode </td>
127</tr>
128<tr><td>Top-right p by q block
129              \link DenseBase::topRightCorner() * \endlink</td>
130    <td>\code
131matrix.topRightCorner(p,q);\endcode </td>
132    <td>\code 
133matrix.topRightCorner<p,q>();\endcode </td>
134</tr>
135<tr><td>Bottom-right p by q block
136               \link DenseBase::bottomRightCorner() * \endlink</td>
137    <td>\code
138matrix.bottomRightCorner(p,q);\endcode </td>
139    <td>\code 
140matrix.bottomRightCorner<p,q>();\endcode </td>
141</tr>
142<tr><td>%Block containing the first q rows
143                   \link DenseBase::topRows() * \endlink</td>
144    <td>\code
145matrix.topRows(q);\endcode </td>
146    <td>\code 
147matrix.topRows<q>();\endcode </td>
148</tr>
149<tr><td>%Block containing the last q rows
150                    \link DenseBase::bottomRows() * \endlink</td>
151    <td>\code
152matrix.bottomRows(q);\endcode </td>
153    <td>\code 
154matrix.bottomRows<q>();\endcode </td>
155</tr>
156<tr><td>%Block containing the first p columns
157                    \link DenseBase::leftCols() * \endlink</td>
158    <td>\code
159matrix.leftCols(p);\endcode </td>
160    <td>\code 
161matrix.leftCols<p>();\endcode </td>
162</tr>
163<tr><td>%Block containing the last q columns
164                    \link DenseBase::rightCols() * \endlink</td>
165    <td>\code
166matrix.rightCols(q);\endcode </td>
167    <td>\code 
168matrix.rightCols<q>();\endcode </td>
169</tr>
170</table>
171
172Here is a simple example illustrating the use of the operations presented above:
173
174<table class="example">
175<tr><th>Example:</th><th>Output:</th></tr>
176<tr><td>
177\include Tutorial_BlockOperations_corner.cpp
178</td>
179<td>
180\verbinclude Tutorial_BlockOperations_corner.out
181</td></tr></table>
182
183
184\section TutorialBlockOperationsSyntaxVectors Block operations for vectors
185
186Eigen also provides a set of block operations designed specifically for the special case of vectors and one-dimensional arrays:
187
188<table class="manual">
189<tr><th> %Block operation</th>
190<th>Version constructing a \n dynamic-size block expression</th>
191<th>Version constructing a \n fixed-size block expression</th></tr>
192<tr><td>%Block containing the first \p n elements 
193                    \link DenseBase::head() * \endlink</td>
194    <td>\code
195vector.head(n);\endcode </td>
196    <td>\code 
197vector.head<n>();\endcode </td>
198</tr>
199<tr><td>%Block containing the last \p n elements
200                    \link DenseBase::tail() * \endlink</td>
201    <td>\code
202vector.tail(n);\endcode </td>
203    <td>\code 
204vector.tail<n>();\endcode </td>
205</tr>
206<tr><td>%Block containing \p n elements, starting at position \p i
207                    \link DenseBase::segment() * \endlink</td>
208    <td>\code
209vector.segment(i,n);\endcode </td>
210    <td>\code 
211vector.segment<n>(i);\endcode </td>
212</tr>
213</table>
214
215
216An example is presented below:
217<table class="example">
218<tr><th>Example:</th><th>Output:</th></tr>
219<tr><td>
220\include Tutorial_BlockOperations_vector.cpp
221</td>
222<td>
223\verbinclude Tutorial_BlockOperations_vector.out
224</td></tr></table>
225
226*/
227
228}
229