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<div class="header">
<div class="headertitle">
<div class="title">Block operations<div class="ingroups"><a class="el" href="group__DenseMatrixManipulation__chapter.html">Dense matrix and array manipulation</a></div></div> </div>
</div><!--header-->
<div class="contents">
<p>This page explains the essentials of block operations. A block is a rectangular part of a matrix or array. Blocks expressions can be used both as rvalues and as lvalues. As usual with <a class="el" href="namespaceEigen.html" title="Namespace containing all symbols from the Eigen library. ">Eigen</a> expressions, this abstraction has zero runtime cost provided that you let your compiler optimize.</p>
<h1><a class="anchor" id="TutorialBlockOperationsUsing"></a>
Using block operations</h1>
<p>The most general block operation in <a class="el" href="namespaceEigen.html" title="Namespace containing all symbols from the Eigen library. ">Eigen</a> is called <a class="el" href="classEigen_1_1DenseBase.html#a1dbaa2fc7b809720407130f48dfacf8f">.block() </a>. There are two versions, whose syntax is as follows:</p>
<table class="manual">
<tr>
<th><b>Block</b> <b>operation</b> </th><th>Version constructing a <br/>
dynamic-size block expression </th><th>Version constructing a <br/>
fixed-size block expression </th></tr>
<tr>
<td>Block of size <code>(p,q)</code>, starting at <code>(i,j)</code> </td><td><div class="fragment"><div class="line">matrix.block(i,j,p,q);</div>
</div><!-- fragment --> </td><td><div class="fragment"><div class="line">matrix.block<p,q>(i,j);</div>
</div><!-- fragment --> </td></tr>
</table>
<p>As always in <a class="el" href="namespaceEigen.html" title="Namespace containing all symbols from the Eigen library. ">Eigen</a>, indices start at 0.</p>
<p>Both versions can be used on fixed-size and dynamic-size matrices and arrays. These two expressions are semantically equivalent. The only difference is that the fixed-size version will typically give you faster code if the block size is small, but requires this size to be known at compile time.</p>
<p>The following program uses the dynamic-size and fixed-size versions to print the values of several blocks inside a matrix.</p>
<table class="example">
<tr>
<th>Example:</th><th>Output: </th></tr>
<tr>
<td><div class="fragment"><div class="line"><span class="preprocessor">#include <Eigen/Dense></span></div>
<div class="line"><span class="preprocessor">#include <iostream></span></div>
<div class="line"></div>
<div class="line"><span class="keyword">using namespace </span>std;</div>
<div class="line"></div>
<div class="line"><span class="keywordtype">int</span> main()</div>
<div class="line">{</div>
<div class="line"> <a class="code" href="classEigen_1_1Matrix.html">Eigen::MatrixXf</a> m(4,4);</div>
<div class="line"> m << 1, 2, 3, 4,</div>
<div class="line"> 5, 6, 7, 8,</div>
<div class="line"> 9,10,11,12,</div>
<div class="line"> 13,14,15,16;</div>
<div class="line"> cout << <span class="stringliteral">"Block in the middle"</span> << endl;</div>
<div class="line"> cout << m.block<2,2>(1,1) << endl << endl;</div>
<div class="line"> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 1; i <= 3; ++i)</div>
<div class="line"> {</div>
<div class="line"> cout << <span class="stringliteral">"Block of size "</span> << i << <span class="stringliteral">"x"</span> << i << endl;</div>
<div class="line"> cout << m.block(0,0,i,i) << endl << endl;</div>
<div class="line"> }</div>
<div class="line">}</div>
</div><!-- fragment --> </td><td><pre class="fragment">Block in the middle
6 7
10 11
Block of size 1x1
1
Block of size 2x2
1 2
5 6
Block of size 3x3
1 2 3
5 6 7
9 10 11
</pre> </td></tr>
</table>
<p>In the above example the <a class="el" href="classEigen_1_1DenseBase.html#a1dbaa2fc7b809720407130f48dfacf8f">.block() </a> function was employed as a <em>rvalue</em>, i.e. it was only read from. However, blocks can also be used as <em>lvalues</em>, meaning that you can assign to a block.</p>
<p>This is illustrated in the following example. This example also demonstrates blocks in arrays, which works exactly like the above-demonstrated blocks in matrices.</p>
<table class="example">
<tr>
<th>Example:</th><th>Output: </th></tr>
<tr>
<td><div class="fragment"><div class="line"><span class="preprocessor">#include <Eigen/Dense></span></div>
<div class="line"><span class="preprocessor">#include <iostream></span></div>
<div class="line"></div>
<div class="line"><span class="keyword">using namespace </span>std;</div>
<div class="line"><span class="keyword">using namespace </span>Eigen;</div>
<div class="line"></div>
<div class="line"><span class="keywordtype">int</span> main()</div>
<div class="line">{</div>
<div class="line"> Array22f m;</div>
<div class="line"> m << 1,2,</div>
<div class="line"> 3,4;</div>
<div class="line"> Array44f a = Array44f::Constant(0.6);</div>
<div class="line"> cout << <span class="stringliteral">"Here is the array a:"</span> << endl << a << endl << endl;</div>
<div class="line"> a.block<2,2>(1,1) = m;</div>
<div class="line"> cout << <span class="stringliteral">"Here is now a with m copied into its central 2x2 block:"</span> << endl << a << endl << endl;</div>
<div class="line"> a.block(0,0,2,3) = a.block(2,1,2,3);</div>
<div class="line"> cout << <span class="stringliteral">"Here is now a with bottom-right 2x3 block copied into top-left 2x2 block:"</span> << endl << a << endl << endl;</div>
<div class="line">}</div>
</div><!-- fragment --> </td><td><pre class="fragment">Here is the array a:
0.6 0.6 0.6 0.6
0.6 0.6 0.6 0.6
0.6 0.6 0.6 0.6
0.6 0.6 0.6 0.6
Here is now a with m copied into its central 2x2 block:
0.6 0.6 0.6 0.6
0.6 1 2 0.6
0.6 3 4 0.6
0.6 0.6 0.6 0.6
Here is now a with bottom-right 2x3 block copied into top-left 2x2 block:
3 4 0.6 0.6
0.6 0.6 0.6 0.6
0.6 3 4 0.6
0.6 0.6 0.6 0.6
</pre> </td></tr>
</table>
<p>While the <a class="el" href="classEigen_1_1DenseBase.html#a1dbaa2fc7b809720407130f48dfacf8f">.block() </a> method can be used for any block operation, there are other methods for special cases, providing more specialized API and/or better performance. On the topic of performance, all what matters is that you give <a class="el" href="namespaceEigen.html" title="Namespace containing all symbols from the Eigen library. ">Eigen</a> as much information as possible at compile time. For example, if your block is a single whole column in a matrix, using the specialized <a class="el" href="classEigen_1_1DenseBase.html#a58c77695de3b33405f01f2fdf3dc389d">.col() </a> function described below lets <a class="el" href="namespaceEigen.html" title="Namespace containing all symbols from the Eigen library. ">Eigen</a> know that, which can give it optimization opportunities.</p>
<p>The rest of this page describes these specialized methods.</p>
<h1><a class="anchor" id="TutorialBlockOperationsSyntaxColumnRows"></a>
Columns and rows</h1>
<p>Individual columns and rows are special cases of blocks. <a class="el" href="namespaceEigen.html" title="Namespace containing all symbols from the Eigen library. ">Eigen</a> provides methods to easily address them: <a class="el" href="classEigen_1_1DenseBase.html#a58c77695de3b33405f01f2fdf3dc389d">.col() </a> and <a class="el" href="classEigen_1_1DenseBase.html#aa8716d44f51321072ee5c88665c28813">.row()</a>.</p>
<table class="manual">
<tr>
<th>Block operation </th><th>Method </th></tr>
<tr>
<td>i<sup>th</sup> row <a class="el" href="classEigen_1_1DenseBase.html#aa8716d44f51321072ee5c88665c28813">* </a> </td><td><div class="fragment"><div class="line">matrix.row(i);</div>
</div><!-- fragment --> </td></tr>
<tr>
<td>j<sup>th</sup> column <a class="el" href="classEigen_1_1DenseBase.html#a58c77695de3b33405f01f2fdf3dc389d">* </a> </td><td><div class="fragment"><div class="line">matrix.col(j);</div>
</div><!-- fragment --> </td></tr>
</table>
<p>The argument for <code>col()</code> and <code>row()</code> is the index of the column or row to be accessed. As always in <a class="el" href="namespaceEigen.html" title="Namespace containing all symbols from the Eigen library. ">Eigen</a>, indices start at 0.</p>
<table class="example">
<tr>
<th>Example:</th><th>Output: </th></tr>
<tr>
<td><div class="fragment"><div class="line"><span class="preprocessor">#include <Eigen/Dense></span></div>
<div class="line"><span class="preprocessor">#include <iostream></span></div>
<div class="line"></div>
<div class="line"><span class="keyword">using namespace </span>std;</div>
<div class="line"></div>
<div class="line"><span class="keywordtype">int</span> main()</div>
<div class="line">{</div>
<div class="line"> <a class="code" href="classEigen_1_1Matrix.html">Eigen::MatrixXf</a> m(3,3);</div>
<div class="line"> m << 1,2,3,</div>
<div class="line"> 4,5,6,</div>
<div class="line"> 7,8,9;</div>
<div class="line"> cout << <span class="stringliteral">"Here is the matrix m:"</span> << endl << m << endl;</div>
<div class="line"> cout << <span class="stringliteral">"2nd Row: "</span> << m.row(1) << endl;</div>
<div class="line"> m.col(2) += 3 * m.col(0);</div>
<div class="line"> cout << <span class="stringliteral">"After adding 3 times the first column into the third column, the matrix m is:\n"</span>;</div>
<div class="line"> cout << m << endl;</div>
<div class="line">}</div>
</div><!-- fragment --> </td><td><pre class="fragment">Here is the matrix m:
1 2 3
4 5 6
7 8 9
2nd Row: 4 5 6
After adding 3 times the first column into the third column, the matrix m is:
1 2 6
4 5 18
7 8 30
</pre> </td></tr>
</table>
<p>That example also demonstrates that block expressions (here columns) can be used in arithmetic like any other expression.</p>
<h1><a class="anchor" id="TutorialBlockOperationsSyntaxCorners"></a>
Corner-related operations</h1>
<p><a class="el" href="namespaceEigen.html" title="Namespace containing all symbols from the Eigen library. ">Eigen</a> also provides special methods for blocks that are flushed against one of the corners or sides of a matrix or array. For instance, <a class="el" href="classEigen_1_1DenseBase.html#ae06d0a5c4008014fb717866aec8d30c7">.topLeftCorner() </a> can be used to refer to a block in the top-left corner of a matrix.</p>
<p>The different possibilities are summarized in the following table:</p>
<table class="manual">
<tr>
<th>Block <b>operation</b> </th><th>Version constructing a <br/>
dynamic-size block expression </th><th>Version constructing a <br/>
fixed-size block expression </th></tr>
<tr>
<td>Top-left p by q block <a class="el" href="classEigen_1_1DenseBase.html#ae06d0a5c4008014fb717866aec8d30c7">* </a> </td><td><div class="fragment"><div class="line">matrix.topLeftCorner(p,q);</div>
</div><!-- fragment --> </td><td><div class="fragment"><div class="line">matrix.topLeftCorner<p,q>();</div>
</div><!-- fragment --> </td></tr>
<tr>
<td>Bottom-left p by q block <a class="el" href="classEigen_1_1DenseBase.html#a243a44832e36d2bf5d0d8d17d1505900">* </a> </td><td><div class="fragment"><div class="line">matrix.bottomLeftCorner(p,q);</div>
</div><!-- fragment --> </td><td><div class="fragment"><div class="line">matrix.bottomLeftCorner<p,q>();</div>
</div><!-- fragment --> </td></tr>
<tr>
<td>Top-right p by q block <a class="el" href="classEigen_1_1DenseBase.html#ad0088b2c1ec4dbd677dd19f589777945">* </a> </td><td><div class="fragment"><div class="line">matrix.topRightCorner(p,q);</div>
</div><!-- fragment --> </td><td><div class="fragment"><div class="line">matrix.topRightCorner<p,q>();</div>
</div><!-- fragment --> </td></tr>
<tr>
<td>Bottom-right p by q block <a class="el" href="classEigen_1_1DenseBase.html#a6563b16b601d9fd66340ab04d101f9d1">* </a> </td><td><div class="fragment"><div class="line">matrix.bottomRightCorner(p,q);</div>
</div><!-- fragment --> </td><td><div class="fragment"><div class="line">matrix.bottomRightCorner<p,q>();</div>
</div><!-- fragment --> </td></tr>
<tr>
<td>Block containing the first q rows <a class="el" href="classEigen_1_1DenseBase.html#a77933058fb3cd3319a86dd0ac4c88d38">* </a> </td><td><div class="fragment"><div class="line">matrix.topRows(q);</div>
</div><!-- fragment --> </td><td><div class="fragment"><div class="line">matrix.topRows<q>();</div>
</div><!-- fragment --> </td></tr>
<tr>
<td>Block containing the last q rows <a class="el" href="classEigen_1_1DenseBase.html#a4809c823f0179782a50409c5444b55a1">* </a> </td><td><div class="fragment"><div class="line">matrix.bottomRows(q);</div>
</div><!-- fragment --> </td><td><div class="fragment"><div class="line">matrix.bottomRows<q>();</div>
</div><!-- fragment --> </td></tr>
<tr>
<td>Block containing the first p columns <a class="el" href="classEigen_1_1DenseBase.html#a01053284fe198875df0ff63aae33eea8">* </a> </td><td><div class="fragment"><div class="line">matrix.leftCols(p);</div>
</div><!-- fragment --> </td><td><div class="fragment"><div class="line">matrix.leftCols<p>();</div>
</div><!-- fragment --> </td></tr>
<tr>
<td>Block containing the last q columns <a class="el" href="classEigen_1_1DenseBase.html#af9afaaa2c2c27716dccdc58c50449103">* </a> </td><td><div class="fragment"><div class="line">matrix.rightCols(q);</div>
</div><!-- fragment --> </td><td><div class="fragment"><div class="line">matrix.rightCols<q>();</div>
</div><!-- fragment --> </td></tr>
</table>
<p>Here is a simple example illustrating the use of the operations presented above:</p>
<table class="example">
<tr>
<th>Example:</th><th>Output: </th></tr>
<tr>
<td><div class="fragment"><div class="line"><span class="preprocessor">#include <Eigen/Dense></span></div>
<div class="line"><span class="preprocessor">#include <iostream></span></div>
<div class="line"></div>
<div class="line"><span class="keyword">using namespace </span>std;</div>
<div class="line"></div>
<div class="line"><span class="keywordtype">int</span> main()</div>
<div class="line">{</div>
<div class="line"> <a class="code" href="classEigen_1_1Matrix.html">Eigen::Matrix4f</a> m;</div>
<div class="line"> m << 1, 2, 3, 4,</div>
<div class="line"> 5, 6, 7, 8,</div>
<div class="line"> 9, 10,11,12,</div>
<div class="line"> 13,14,15,16;</div>
<div class="line"> cout << <span class="stringliteral">"m.leftCols(2) ="</span> << endl << m.<a class="code" href="classEigen_1_1DenseBase.html#ace4288d22b0312944ec93c8c46474ef6">leftCols</a>(2) << endl << endl;</div>
<div class="line"> cout << <span class="stringliteral">"m.bottomRows<2>() ="</span> << endl << m.<a class="code" href="classEigen_1_1DenseBase.html#afe61d23353a75d7034f7b44ff02aaea1">bottomRows</a><2>() << endl << endl;</div>
<div class="line"> m.<a class="code" href="classEigen_1_1DenseBase.html#a6f5fc5fe9d3fb70e62d4a9b1795704a8">topLeftCorner</a>(1,3) = m.<a class="code" href="classEigen_1_1DenseBase.html#a2b9618f3c9eb4d4c9813ae8f6a8e70c5">bottomRightCorner</a>(3,1).transpose();</div>
<div class="line"> cout << <span class="stringliteral">"After assignment, m = "</span> << endl << m << endl;</div>
<div class="line">}</div>
</div><!-- fragment --> </td><td><pre class="fragment">m.leftCols(2) =
1 2
5 6
9 10
13 14
m.bottomRows<2>() =
9 10 11 12
13 14 15 16
After assignment, m =
8 12 16 4
5 6 7 8
9 10 11 12
13 14 15 16
</pre> </td></tr>
</table>
<h1><a class="anchor" id="TutorialBlockOperationsSyntaxVectors"></a>
Block operations for vectors</h1>
<p><a class="el" href="namespaceEigen.html" title="Namespace containing all symbols from the Eigen library. ">Eigen</a> also provides a set of block operations designed specifically for the special case of vectors and one-dimensional arrays:</p>
<table class="manual">
<tr>
<th>Block operation </th><th>Version constructing a <br/>
dynamic-size block expression </th><th>Version constructing a <br/>
fixed-size block expression </th></tr>
<tr>
<td>Block containing the first <code>n</code> elements <a class="el" href="classEigen_1_1DenseBase.html#a361e155375a3c7983c294641b1e91a15">* </a> </td><td><div class="fragment"><div class="line">vector.head(n);</div>
</div><!-- fragment --> </td><td><div class="fragment"><div class="line">vector.head<n>();</div>
</div><!-- fragment --> </td></tr>
<tr>
<td>Block containing the last <code>n</code> elements <a class="el" href="classEigen_1_1DenseBase.html#a9da81041f38048642ca460fc6236da79">* </a> </td><td><div class="fragment"><div class="line">vector.tail(n);</div>
</div><!-- fragment --> </td><td><div class="fragment"><div class="line">vector.tail<n>();</div>
</div><!-- fragment --> </td></tr>
<tr>
<td>Block containing <code>n</code> elements, starting at position <code>i</code> <a class="el" href="classEigen_1_1DenseBase.html#a3a182e8a52e9c6761ded3813f751ab03">* </a> </td><td><div class="fragment"><div class="line">vector.segment(i,n);</div>
</div><!-- fragment --> </td><td><div class="fragment"><div class="line">vector.segment<n>(i);</div>
</div><!-- fragment --> </td></tr>
</table>
<p>An example is presented below: </p>
<table class="example">
<tr>
<th>Example:</th><th>Output: </th></tr>
<tr>
<td><div class="fragment"><div class="line"><span class="preprocessor">#include <Eigen/Dense></span></div>
<div class="line"><span class="preprocessor">#include <iostream></span></div>
<div class="line"></div>
<div class="line"><span class="keyword">using namespace </span>std;</div>
<div class="line"></div>
<div class="line"><span class="keywordtype">int</span> main()</div>
<div class="line">{</div>
<div class="line"> Eigen::ArrayXf v(6);</div>
<div class="line"> v << 1, 2, 3, 4, 5, 6;</div>
<div class="line"> cout << <span class="stringliteral">"v.head(3) ="</span> << endl << v.<a class="code" href="classEigen_1_1DenseBase.html#a72cddecfa6737b4388ef26c3a559a716">head</a>(3) << endl << endl;</div>
<div class="line"> cout << <span class="stringliteral">"v.tail<3>() = "</span> << endl << v.<a class="code" href="classEigen_1_1DenseBase.html#a806cbb7e3c7357c2c5fbb1a7db4af16d">tail</a><3>() << endl << endl;</div>
<div class="line"> v.<a class="code" href="classEigen_1_1DenseBase.html#a3a182e8a52e9c6761ded3813f751ab03">segment</a>(1,4) *= 2;</div>
<div class="line"> cout << <span class="stringliteral">"after 'v.segment(1,4) *= 2', v ="</span> << endl << v << endl;</div>
<div class="line">}</div>
</div><!-- fragment --> </td><td><pre class="fragment">v.head(3) =
1
2
3
v.tail<3>() =
4
5
6
after 'v.segment(1,4) *= 2', v =
1
4
6
8
10
6
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