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<div class="title">Reductions, visitors and broadcasting<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 <a class="el" href="namespaceEigen.html" title="Namespace containing all symbols from the Eigen library. ">Eigen</a>'s reductions, visitors and broadcasting and how they are used with <a class="el" href="classEigen_1_1MatrixBase.html">matrices </a> and <a class="el" href="classEigen_1_1ArrayBase.html">arrays </a>.</p>
<h1><a class="anchor" id="TutorialReductionsVisitorsBroadcastingReductions"></a>
Reductions</h1>
<p>In <a class="el" href="namespaceEigen.html" title="Namespace containing all symbols from the Eigen library. ">Eigen</a>, a reduction is a function taking a matrix or array, and returning a single scalar value. One of the most used reductions is <a class="el" href="classEigen_1_1DenseBase.html#a3a3b3fb530d3364ecef0bf9c9daf0983">.sum() </a>, returning the sum of all the coefficients inside a given matrix or array.</p>
<table class="example">
<tr>
<th>Example:</th><th>Output: </th></tr>
<tr>
<td><div class="fragment"><div class="line"><span class="preprocessor">#include <iostream></span></div>
<div class="line"><span class="preprocessor">#include <Eigen/Dense></span></div>
<div class="line"></div>
<div class="line"><span class="keyword">using namespace </span>std;</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::Matrix2d</a> mat;</div>
<div class="line"> mat << 1, 2,</div>
<div class="line"> 3, 4;</div>
<div class="line"> cout << <span class="stringliteral">"Here is mat.sum(): "</span> << mat.<a class="code" href="classEigen_1_1DenseBase.html#a3a3b3fb530d3364ecef0bf9c9daf0983">sum</a>() << endl;</div>
<div class="line"> cout << <span class="stringliteral">"Here is mat.prod(): "</span> << mat.<a class="code" href="classEigen_1_1DenseBase.html#a6bdcbfa7e3b07d3246ad80de7170b0f5">prod</a>() << endl;</div>
<div class="line"> cout << <span class="stringliteral">"Here is mat.mean(): "</span> << mat.<a class="code" href="classEigen_1_1DenseBase.html#a0af2b3991862a079e3efaef3e4d17d96">mean</a>() << endl;</div>
<div class="line"> cout << <span class="stringliteral">"Here is mat.minCoeff(): "</span> << mat.<a class="code" href="classEigen_1_1DenseBase.html#add6cb2d85282829eb9adc9565ce784d6">minCoeff</a>() << endl;</div>
<div class="line"> cout << <span class="stringliteral">"Here is mat.maxCoeff(): "</span> << mat.<a class="code" href="classEigen_1_1DenseBase.html#a878f0dae18b28d8158c5f1c232edced2">maxCoeff</a>() << endl;</div>
<div class="line"> cout << <span class="stringliteral">"Here is mat.trace(): "</span> << mat.<a class="code" href="classEigen_1_1MatrixBase.html#a71696dd0adbf4731561fd60e55c3a96e">trace</a>() << endl;</div>
<div class="line">}</div>
</div><!-- fragment --> </td><td><pre class="fragment">Here is mat.sum(): 10
Here is mat.prod(): 24
Here is mat.mean(): 2.5
Here is mat.minCoeff(): 1
Here is mat.maxCoeff(): 4
Here is mat.trace(): 5
</pre> </td></tr>
</table>
<p>The <em>trace</em> of a matrix, as returned by the function <code>trace()</code>, is the sum of the diagonal coefficients and can equivalently be computed <code>a.diagonal().sum()</code>.</p>
<h2><a class="anchor" id="TutorialReductionsVisitorsBroadcastingReductionsNorm"></a>
Norm computations</h2>
<p>The (Euclidean a.k.a. <img class="formulaInl" alt="$\ell^2$" src="form_190.png"/>) squared norm of a vector can be obtained <a class="el" href="classEigen_1_1MatrixBase.html#a229bd5cc6237359a1d85401743476ede">squaredNorm() </a>. It is equal to the dot product of the vector by itself, and equivalently to the sum of squared absolute values of its coefficients.</p>
<p><a class="el" href="namespaceEigen.html" title="Namespace containing all symbols from the Eigen library. ">Eigen</a> also provides the <a class="el" href="classEigen_1_1MatrixBase.html#a0be1b433c65ce9d92c81a4718daf54e5">norm() </a> method, which returns the square root of <a class="el" href="classEigen_1_1MatrixBase.html#a229bd5cc6237359a1d85401743476ede">squaredNorm() </a>.</p>
<p>These operations can also operate on matrices; in that case, a n-by-p matrix is seen as a vector of size (n*p), so for example the <a class="el" href="classEigen_1_1MatrixBase.html#a0be1b433c65ce9d92c81a4718daf54e5">norm() </a> method returns the "Frobenius" or "Hilbert-Schmidt" norm. We refrain from speaking of the <img class="formulaInl" alt="$\ell^2$" src="form_190.png"/> norm of a matrix because that can mean different things.</p>
<p>If you want other <img class="formulaInl" alt="$\ell^p$" src="form_191.png"/> norms, use the <a class="el" href="">lpNnorm<p>() </a> method. The template parameter <em>p</em> can take the special value <em>Infinity</em> if you want the <img class="formulaInl" alt="$\ell^\infty$" src="form_192.png"/> norm, which is the maximum of the absolute values of the coefficients.</p>
<p>The following example demonstrates these methods.</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"> VectorXf v(2);</div>
<div class="line"> <a class="code" href="classEigen_1_1Matrix.html">MatrixXf</a> m(2,2), n(2,2);</div>
<div class="line"> </div>
<div class="line"> v << -1,</div>
<div class="line"> 2;</div>
<div class="line"> </div>
<div class="line"> m << 1,-2,</div>
<div class="line"> -3,4;</div>
<div class="line"></div>
<div class="line"> cout << <span class="stringliteral">"v.squaredNorm() = "</span> << v.<a class="code" href="classEigen_1_1MatrixBase.html#a229bd5cc6237359a1d85401743476ede">squaredNorm</a>() << endl;</div>
<div class="line"> cout << <span class="stringliteral">"v.norm() = "</span> << v.norm() << endl;</div>
<div class="line"> cout << <span class="stringliteral">"v.lpNorm<1>() = "</span> << v.lpNorm<1>() << endl;</div>
<div class="line"> cout << <span class="stringliteral">"v.lpNorm<Infinity>() = "</span> << v.lpNorm<<a class="code" href="namespaceEigen.html#a6cff718ef6bce746aa5e100036226625">Infinity</a>>() << endl;</div>
<div class="line"></div>
<div class="line"> cout << endl;</div>
<div class="line"> cout << <span class="stringliteral">"m.squaredNorm() = "</span> << m.<a class="code" href="classEigen_1_1MatrixBase.html#a229bd5cc6237359a1d85401743476ede">squaredNorm</a>() << endl;</div>
<div class="line"> cout << <span class="stringliteral">"m.norm() = "</span> << m.<a class="code" href="classEigen_1_1MatrixBase.html#a0be1b433c65ce9d92c81a4718daf54e5">norm</a>() << endl;</div>
<div class="line"> cout << <span class="stringliteral">"m.lpNorm<1>() = "</span> << m.lpNorm<1>() << endl;</div>
<div class="line"> cout << <span class="stringliteral">"m.lpNorm<Infinity>() = "</span> << m.lpNorm<<a class="code" href="namespaceEigen.html#a6cff718ef6bce746aa5e100036226625">Infinity</a>>() << endl;</div>
<div class="line">}</div>
</div><!-- fragment --> </td><td><pre class="fragment">v.squaredNorm() = 5
v.norm() = 2.23607
v.lpNorm<1>() = 3
v.lpNorm<Infinity>() = 2
m.squaredNorm() = 30
m.norm() = 5.47723
m.lpNorm<1>() = 10
m.lpNorm<Infinity>() = 4
</pre> </td></tr>
</table>
<h2><a class="anchor" id="TutorialReductionsVisitorsBroadcastingReductionsBool"></a>
Boolean reductions</h2>
<p>The following reductions operate on boolean values:</p>
<ul>
<li><a class="el" href="classEigen_1_1DenseBase.html#aea914316af61df197f21629e14e7870a">all() </a> returns <b>true</b> if all of the coefficients in a given <a class="el" href="classEigen_1_1Matrix.html" title="The matrix class, also used for vectors and row-vectors. ">Matrix</a> or <a class="el" href="classEigen_1_1Array.html" title="General-purpose arrays with easy API for coefficient-wise operations. ">Array</a> evaluate to <b>true</b> .</li>
<li><a class="el" href="classEigen_1_1DenseBase.html#a42571e028736ca9103bac8b50f269824">any() </a> returns <b>true</b> if at least one of the coefficients in a given <a class="el" href="classEigen_1_1Matrix.html" title="The matrix class, also used for vectors and row-vectors. ">Matrix</a> or <a class="el" href="classEigen_1_1Array.html" title="General-purpose arrays with easy API for coefficient-wise operations. ">Array</a> evaluates to <b>true</b> .</li>
<li><a class="el" href="classEigen_1_1DenseBase.html#aa671b5ea336ba21a7644d3fa6577ee00">count() </a> returns the number of coefficients in a given <a class="el" href="classEigen_1_1Matrix.html" title="The matrix class, also used for vectors and row-vectors. ">Matrix</a> or <a class="el" href="classEigen_1_1Array.html" title="General-purpose arrays with easy API for coefficient-wise operations. ">Array</a> that evaluate to <b>true</b>.</li>
</ul>
<p>These are typically used in conjunction with the coefficient-wise comparison and equality operators provided by <a class="el" href="classEigen_1_1Array.html" title="General-purpose arrays with easy API for coefficient-wise operations. ">Array</a>. For instance, <code>array > 0</code> is an Array of the same size as <code>array</code> , with <b>true</b> at those positions where the corresponding coefficient of <code>array</code> is positive. Thus, <code>(array > 0).all()</code> tests whether all coefficients of <code>array</code> are positive. This can be seen in the following example:</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"> ArrayXXf a(2,2);</div>
<div class="line"> </div>
<div class="line"> a << 1,2,</div>
<div class="line"> 3,4;</div>
<div class="line"></div>
<div class="line"> cout << <span class="stringliteral">"(a > 0).all() = "</span> << (a > 0).all() << endl;</div>
<div class="line"> cout << <span class="stringliteral">"(a > 0).any() = "</span> << (a > 0).any() << endl;</div>
<div class="line"> cout << <span class="stringliteral">"(a > 0).count() = "</span> << (a > 0).count() << endl;</div>
<div class="line"> cout << endl;</div>
<div class="line"> cout << <span class="stringliteral">"(a > 2).all() = "</span> << (a > 2).all() << endl;</div>
<div class="line"> cout << <span class="stringliteral">"(a > 2).any() = "</span> << (a > 2).any() << endl;</div>
<div class="line"> cout << <span class="stringliteral">"(a > 2).count() = "</span> << (a > 2).count() << endl;</div>
<div class="line">}</div>
</div><!-- fragment --> </td><td><pre class="fragment">(a > 0).all() = 1
(a > 0).any() = 1
(a > 0).count() = 4
(a > 2).all() = 0
(a > 2).any() = 1
(a > 2).count() = 2
</pre> </td></tr>
</table>
<h2><a class="anchor" id="TutorialReductionsVisitorsBroadcastingReductionsUserdefined"></a>
User defined reductions</h2>
<p>TODO</p>
<p>In the meantime you can have a look at the DenseBase::redux() function.</p>
<h1><a class="anchor" id="TutorialReductionsVisitorsBroadcastingVisitors"></a>
Visitors</h1>
<p>Visitors are useful when one wants to obtain the location of a coefficient inside a <a class="el" href="classEigen_1_1Matrix.html" title="The matrix class, also used for vectors and row-vectors. ">Matrix</a> or <a class="el" href="classEigen_1_1Array.html" title="General-purpose arrays with easy API for coefficient-wise operations. ">Array</a>. The simplest examples are <a class="el" href="classEigen_1_1DenseBase.html#a878f0dae18b28d8158c5f1c232edced2">maxCoeff(&x,&y) </a> and <a class="el" href="classEigen_1_1DenseBase.html#add6cb2d85282829eb9adc9565ce784d6">minCoeff(&x,&y)</a>, which can be used to find the location of the greatest or smallest coefficient in a <a class="el" href="classEigen_1_1Matrix.html" title="The matrix class, also used for vectors and row-vectors. ">Matrix</a> or <a class="el" href="classEigen_1_1Array.html" title="General-purpose arrays with easy API for coefficient-wise operations. ">Array</a>.</p>
<p>The arguments passed to a visitor are pointers to the variables where the row and column position are to be stored. These variables should be of type <a class="el" href="classEigen_1_1DenseBase.html#a4d4873e91be950c079f067fa97fd5c40">Index </a>, as shown 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 <iostream></span></div>
<div class="line"><span class="preprocessor">#include <Eigen/Dense></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"> <a class="code" href="classEigen_1_1Matrix.html">Eigen::MatrixXf</a> m(2,2);</div>
<div class="line"> </div>
<div class="line"> m << 1, 2,</div>
<div class="line"> 3, 4;</div>
<div class="line"></div>
<div class="line"> <span class="comment">//get location of maximum</span></div>
<div class="line"> <a class="code" href="classEigen_1_1DenseBase.html#a4d4873e91be950c079f067fa97fd5c40">MatrixXf::Index</a> maxRow, maxCol;</div>
<div class="line"> <span class="keywordtype">float</span> max = m.<a class="code" href="classEigen_1_1DenseBase.html#a878f0dae18b28d8158c5f1c232edced2">maxCoeff</a>(&maxRow, &maxCol);</div>
<div class="line"></div>
<div class="line"> <span class="comment">//get location of minimum</span></div>
<div class="line"> <a class="code" href="classEigen_1_1DenseBase.html#a4d4873e91be950c079f067fa97fd5c40">MatrixXf::Index</a> minRow, minCol;</div>
<div class="line"> <span class="keywordtype">float</span> min = m.<a class="code" href="classEigen_1_1DenseBase.html#add6cb2d85282829eb9adc9565ce784d6">minCoeff</a>(&minRow, &minCol);</div>
<div class="line"></div>
<div class="line"> cout << <span class="stringliteral">"Max: "</span> << max << <span class="stringliteral">", at: "</span> <<</div>
<div class="line"> maxRow << <span class="stringliteral">","</span> << maxCol << endl;</div>
<div class="line"> cout << <span class="stringliteral">"Min: "</span> << min << <span class="stringliteral">", at: "</span> <<</div>
<div class="line"> minRow << <span class="stringliteral">","</span> << minCol << endl;</div>
<div class="line">}</div>
</div><!-- fragment --> </td><td><pre class="fragment">Max: 4, at: 1,1
Min: 1, at: 0,0
</pre> </td></tr>
</table>
<p>Note that both functions also return the value of the minimum or maximum coefficient if needed, as if it was a typical reduction operation.</p>
<h1><a class="anchor" id="TutorialReductionsVisitorsBroadcastingPartialReductions"></a>
Partial reductions</h1>
<p>Partial reductions are reductions that can operate column- or row-wise on a <a class="el" href="classEigen_1_1Matrix.html" title="The matrix class, also used for vectors and row-vectors. ">Matrix</a> or <a class="el" href="classEigen_1_1Array.html" title="General-purpose arrays with easy API for coefficient-wise operations. ">Array</a>, applying the reduction operation on each column or row and returning a column or row-vector with the corresponding values. Partial reductions are applied with <a class="el" href="classEigen_1_1DenseBase.html#abe7ae69362c464b6721adbb47c655874">colwise() </a> or <a class="el" href="classEigen_1_1DenseBase.html#a8a7fd1e8004d4bd93a7ea36957aa8e99">rowwise() </a>.</p>
<p>A simple example is obtaining the maximum of the elements in each column in a given matrix, storing the result in a row-vector:</p>
<table class="example">
<tr>
<th>Example:</th><th>Output: </th></tr>
<tr>
<td><div class="fragment"><div class="line"><span class="preprocessor">#include <iostream></span></div>
<div class="line"><span class="preprocessor">#include <Eigen/Dense></span></div>
<div class="line"></div>
<div class="line"><span class="keyword">using namespace </span>std;</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> mat(2,4);</div>
<div class="line"> mat << 1, 2, 6, 9,</div>
<div class="line"> 3, 1, 7, 2;</div>
<div class="line"> </div>
<div class="line"> std::cout << <span class="stringliteral">"Column's maximum: "</span> << std::endl</div>
<div class="line"> << mat.<a class="code" href="classEigen_1_1DenseBase.html#a49a617f24129ca31a27fe8a67ec20370">colwise</a>().<a class="code" href="classEigen_1_1VectorwiseOp.html#ae45bc1f13fd426ca7bea74ae4b8f99d6">maxCoeff</a>() << std::endl;</div>
<div class="line">}</div>
</div><!-- fragment --> </td><td><pre class="fragment">Column's maximum:
3 2 7 9
</pre> </td></tr>
</table>
<p>The same operation can be performed row-wise:</p>
<table class="example">
<tr>
<th>Example:</th><th>Output: </th></tr>
<tr>
<td><div class="fragment"><div class="line"><span class="preprocessor">#include <iostream></span></div>
<div class="line"><span class="preprocessor">#include <Eigen/Dense></span></div>
<div class="line"></div>
<div class="line"><span class="keyword">using namespace </span>std;</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> mat(2,4);</div>
<div class="line"> mat << 1, 2, 6, 9,</div>
<div class="line"> 3, 1, 7, 2;</div>
<div class="line"> </div>
<div class="line"> std::cout << <span class="stringliteral">"Row's maximum: "</span> << std::endl</div>
<div class="line"> << mat.<a class="code" href="classEigen_1_1DenseBase.html#a3af2f03b1d2affcec24e0748edf892cd">rowwise</a>().<a class="code" href="classEigen_1_1VectorwiseOp.html#ae45bc1f13fd426ca7bea74ae4b8f99d6">maxCoeff</a>() << std::endl;</div>
<div class="line">}</div>
</div><!-- fragment --> </td><td><pre class="fragment">Row's maximum:
9
7
</pre> </td></tr>
</table>
<p><b>Note that column-wise operations return a 'row-vector' while row-wise operations return a 'column-vector'</b></p>
<h2><a class="anchor" id="TutorialReductionsVisitorsBroadcastingPartialReductionsCombined"></a>
Combining partial reductions with other operations</h2>
<p>It is also possible to use the result of a partial reduction to do further processing. Here is another example that finds the column whose sum of elements is the maximum within a matrix. With column-wise partial reductions this can be coded as:</p>
<table class="example">
<tr>
<th>Example:</th><th>Output: </th></tr>
<tr>
<td><div class="fragment"><div class="line"><span class="preprocessor">#include <iostream></span></div>
<div class="line"><span class="preprocessor">#include <Eigen/Dense></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"><span class="keywordtype">int</span> main()</div>
<div class="line">{</div>
<div class="line"> <a class="code" href="classEigen_1_1Matrix.html">MatrixXf</a> mat(2,4);</div>
<div class="line"> mat << 1, 2, 6, 9,</div>
<div class="line"> 3, 1, 7, 2;</div>
<div class="line"> </div>
<div class="line"> <a class="code" href="classEigen_1_1DenseBase.html#a4d4873e91be950c079f067fa97fd5c40">MatrixXf::Index</a> maxIndex;</div>
<div class="line"> <span class="keywordtype">float</span> maxNorm = mat.<a class="code" href="classEigen_1_1DenseBase.html#a49a617f24129ca31a27fe8a67ec20370">colwise</a>().<a class="code" href="classEigen_1_1VectorwiseOp.html#af27cb8c9043ed6408e64f94645439362">sum</a>().maxCoeff(&maxIndex);</div>
<div class="line"> </div>
<div class="line"> std::cout << <span class="stringliteral">"Maximum sum at position "</span> << maxIndex << std::endl;</div>
<div class="line"></div>
<div class="line"> std::cout << <span class="stringliteral">"The corresponding vector is: "</span> << std::endl;</div>
<div class="line"> std::cout << mat.<a class="code" href="classEigen_1_1DenseBase.html#a58c77695de3b33405f01f2fdf3dc389d">col</a>( maxIndex ) << std::endl;</div>
<div class="line"> std::cout << <span class="stringliteral">"And its sum is is: "</span> << maxNorm << std::endl;</div>
<div class="line">}</div>
</div><!-- fragment --> </td><td><pre class="fragment">Maximum sum at position 2
The corresponding vector is:
6
7
And its sum is is: 13
</pre> </td></tr>
</table>
<p>The previous example applies the <a class="el" href="classEigen_1_1DenseBase.html#a3a3b3fb530d3364ecef0bf9c9daf0983">sum() </a> reduction on each column though the <a class="el" href="classEigen_1_1DenseBase.html#abe7ae69362c464b6721adbb47c655874">colwise() </a> visitor, obtaining a new matrix whose size is 1x4.</p>
<p>Therefore, if </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ \mbox{m} = \begin{bmatrix} 1 & 2 & 6 & 9 \\ 3 & 1 & 7 & 2 \end{bmatrix} \]" src="form_193.png"/>
</p>
<p>then</p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ \mbox{m.colwise().sum()} = \begin{bmatrix} 4 & 3 & 13 & 11 \end{bmatrix} \]" src="form_194.png"/>
</p>
<p>The <a class="el" href="classEigen_1_1DenseBase.html#a878f0dae18b28d8158c5f1c232edced2">maxCoeff() </a> reduction is finally applied to obtain the column index where the maximum sum is found, which is the column index 2 (third column) in this case.</p>
<h1><a class="anchor" id="TutorialReductionsVisitorsBroadcastingBroadcasting"></a>
Broadcasting</h1>
<p>The concept behind broadcasting is similar to partial reductions, with the difference that broadcasting constructs an expression where a vector (column or row) is interpreted as a matrix by replicating it in one direction.</p>
<p>A simple example is to add a certain column-vector to each column in a matrix. This can be accomplished with:</p>
<table class="example">
<tr>
<th>Example:</th><th>Output: </th></tr>
<tr>
<td><div class="fragment"><div class="line"><span class="preprocessor">#include <iostream></span></div>
<div class="line"><span class="preprocessor">#include <Eigen/Dense></span></div>
<div class="line"></div>
<div class="line"><span class="keyword">using namespace </span>std;</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> mat(2,4);</div>
<div class="line"> <a class="code" href="classEigen_1_1Matrix.html">Eigen::VectorXf</a> v(2);</div>
<div class="line"> </div>
<div class="line"> mat << 1, 2, 6, 9,</div>
<div class="line"> 3, 1, 7, 2;</div>
<div class="line"> </div>
<div class="line"> v << 0,</div>
<div class="line"> 1;</div>
<div class="line"> </div>
<div class="line"> <span class="comment">//add v to each column of m</span></div>
<div class="line"> mat.<a class="code" href="classEigen_1_1DenseBase.html#a49a617f24129ca31a27fe8a67ec20370">colwise</a>() += v;</div>
<div class="line"> </div>
<div class="line"> std::cout << <span class="stringliteral">"Broadcasting result: "</span> << std::endl;</div>
<div class="line"> std::cout << mat << std::endl;</div>
<div class="line">}</div>
</div><!-- fragment --> </td><td><pre class="fragment">Broadcasting result:
1 2 6 9
4 2 8 3
</pre> </td></tr>
</table>
<p>We can interpret the instruction <code>mat.colwise() += v</code> in two equivalent ways. It adds the vector <code>v</code> to every column of the matrix. Alternatively, it can be interpreted as repeating the vector <code>v</code> four times to form a four-by-two matrix which is then added to <code>mat:</code> </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ \begin{bmatrix} 1 & 2 & 6 & 9 \\ 3 & 1 & 7 & 2 \end{bmatrix} + \begin{bmatrix} 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 2 & 6 & 9 \\ 4 & 2 & 8 & 3 \end{bmatrix}. \]" src="form_195.png"/>
</p>
<p> The operators <code>-=</code>, <code>+</code> and <code>-</code> can also be used column-wise and row-wise. On arrays, we can also use the operators <code>*=</code>, <code>/=</code>, <code>*</code> and <code>/</code> to perform coefficient-wise multiplication and division column-wise or row-wise. These operators are not available on matrices because it is not clear what they would do. If you want multiply column 0 of a matrix <code>mat</code> with <code>v(0)</code>, column 1 with <code>v(1)</code>, and so on, then use <code>mat = mat * v.asDiagonal()</code>.</p>
<p>It is important to point out that the vector to be added column-wise or row-wise must be of type Vector, and cannot be a <a class="el" href="classEigen_1_1Matrix.html" title="The matrix class, also used for vectors and row-vectors. ">Matrix</a>. If this is not met then you will get compile-time error. This also means that broadcasting operations can only be applied with an object of type Vector, when operating with <a class="el" href="classEigen_1_1Matrix.html" title="The matrix class, also used for vectors and row-vectors. ">Matrix</a>. The same applies for the <a class="el" href="classEigen_1_1Array.html" title="General-purpose arrays with easy API for coefficient-wise operations. ">Array</a> class, where the equivalent for VectorXf is ArrayXf. As always, you should not mix arrays and matrices in the same expression.</p>
<p>To perform the same operation row-wise we can do:</p>
<table class="example">
<tr>
<th>Example:</th><th>Output: </th></tr>
<tr>
<td><div class="fragment"><div class="line"><span class="preprocessor">#include <iostream></span></div>
<div class="line"><span class="preprocessor">#include <Eigen/Dense></span></div>
<div class="line"></div>
<div class="line"><span class="keyword">using namespace </span>std;</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> mat(2,4);</div>
<div class="line"> <a class="code" href="classEigen_1_1Matrix.html">Eigen::VectorXf</a> v(4);</div>
<div class="line"> </div>
<div class="line"> mat << 1, 2, 6, 9,</div>
<div class="line"> 3, 1, 7, 2;</div>
<div class="line"> </div>
<div class="line"> v << 0,1,2,3;</div>
<div class="line"> </div>
<div class="line"> <span class="comment">//add v to each row of m</span></div>
<div class="line"> mat.<a class="code" href="classEigen_1_1DenseBase.html#a3af2f03b1d2affcec24e0748edf892cd">rowwise</a>() += v.transpose();</div>
<div class="line"> </div>
<div class="line"> std::cout << <span class="stringliteral">"Broadcasting result: "</span> << std::endl;</div>
<div class="line"> std::cout << mat << std::endl;</div>
<div class="line">}</div>
</div><!-- fragment --> </td><td><pre class="fragment">Broadcasting result:
1 3 8 12
3 2 9 5
</pre> </td></tr>
</table>
<h2><a class="anchor" id="TutorialReductionsVisitorsBroadcastingBroadcastingCombined"></a>
Combining broadcasting with other operations</h2>
<p>Broadcasting can also be combined with other operations, such as <a class="el" href="classEigen_1_1Matrix.html" title="The matrix class, also used for vectors and row-vectors. ">Matrix</a> or <a class="el" href="classEigen_1_1Array.html" title="General-purpose arrays with easy API for coefficient-wise operations. ">Array</a> operations, reductions and partial reductions.</p>
<p>Now that broadcasting, reductions and partial reductions have been introduced, we can dive into a more advanced example that finds the nearest neighbour of a vector <code>v</code> within the columns of matrix <code>m</code>. The Euclidean distance will be used in this example, computing the squared Euclidean distance with the partial reduction named <a class="el" href="classEigen_1_1MatrixBase.html#a229bd5cc6237359a1d85401743476ede">squaredNorm() </a>:</p>
<table class="example">
<tr>
<th>Example:</th><th>Output: </th></tr>
<tr>
<td><div class="fragment"><div class="line"><span class="preprocessor">#include <iostream></span></div>
<div class="line"><span class="preprocessor">#include <Eigen/Dense></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"> <a class="code" href="classEigen_1_1Matrix.html">Eigen::MatrixXf</a> m(2,4);</div>
<div class="line"> <a class="code" href="classEigen_1_1Matrix.html">Eigen::VectorXf</a> v(2);</div>
<div class="line"> </div>
<div class="line"> m << 1, 23, 6, 9,</div>
<div class="line"> 3, 11, 7, 2;</div>
<div class="line"> </div>
<div class="line"> v << 2,</div>
<div class="line"> 3;</div>
<div class="line"></div>
<div class="line"> <a class="code" href="classEigen_1_1DenseBase.html#a4d4873e91be950c079f067fa97fd5c40">MatrixXf::Index</a> index;</div>
<div class="line"> <span class="comment">// find nearest neighbour</span></div>
<div class="line"> (m.<a class="code" href="classEigen_1_1DenseBase.html#a49a617f24129ca31a27fe8a67ec20370">colwise</a>() - v).colwise().<a class="code" href="classEigen_1_1MatrixBase.html#a229bd5cc6237359a1d85401743476ede">squaredNorm</a>().minCoeff(&index);</div>
<div class="line"></div>
<div class="line"> cout << <span class="stringliteral">"Nearest neighbour is column "</span> << index << <span class="stringliteral">":"</span> << endl;</div>
<div class="line"> cout << m.<a class="code" href="classEigen_1_1DenseBase.html#a58c77695de3b33405f01f2fdf3dc389d">col</a>(index) << endl;</div>
<div class="line">}</div>
</div><!-- fragment --> </td><td><pre class="fragment">Nearest neighbour is column 0:
1
3
</pre> </td></tr>
</table>
<p>The line that does the job is </p>
<div class="fragment"><div class="line">(m.<a class="code" href="classEigen_1_1DenseBase.html#a49a617f24129ca31a27fe8a67ec20370">colwise</a>() - v).colwise().squaredNorm().minCoeff(&index);</div>
</div><!-- fragment --><p>We will go step by step to understand what is happening:</p>
<ul>
<li><code>m.colwise() - v</code> is a broadcasting operation, subtracting <code>v</code> from each column in <code>m</code>. The result of this operation is a new matrix whose size is the same as matrix <code>m</code>: <p class="formulaDsp">
<img class="formulaDsp" alt="\[ \mbox{m.colwise() - v} = \begin{bmatrix} -1 & 21 & 4 & 7 \\ 0 & 8 & 4 & -1 \end{bmatrix} \]" src="form_196.png"/>
</p>
</li>
<li><code>(m.colwise() - v).colwise().squaredNorm()</code> is a partial reduction, computing the squared norm column-wise. The result of this operation is a row-vector where each coefficient is the squared Euclidean distance between each column in <code>m</code> and <code>v</code>: <p class="formulaDsp">
<img class="formulaDsp" alt="\[ \mbox{(m.colwise() - v).colwise().squaredNorm()} = \begin{bmatrix} 1 & 505 & 32 & 50 \end{bmatrix} \]" src="form_197.png"/>
</p>
</li>
<li>Finally, <code>minCoeff(&index)</code> is used to obtain the index of the column in <code>m</code> that is closest to <code>v</code> in terms of Euclidean distance. </li>
</ul>
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