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<a href="classEigen_1_1HouseholderSequence-members.html">List of all members</a> |
<a href="#pub-methods">Public Member Functions</a> |
<a href="#pro-methods">Protected Member Functions</a> </div>
<div class="headertitle">
<div class="title">HouseholderSequence< VectorsType, CoeffsType, Side > Class Template Reference<div class="ingroups"><a class="el" href="group__Householder__Module.html">Householder module</a></div></div> </div>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><h3>template<typename VectorsType, typename CoeffsType, int Side = OnTheLeft><br/>
class Eigen::HouseholderSequence< VectorsType, CoeffsType, Side ></h3>
<p>Sequence of Householder reflections acting on subspaces with decreasing size. </p>
<p>This is defined in the Householder module.</p>
<div class="fragment"><div class="line"><span class="preprocessor">#include <Eigen/Householder></span> </div>
</div><!-- fragment --><dl class="tparams"><dt>Template Parameters</dt><dd>
<table class="tparams">
<tr><td class="paramname">VectorsType</td><td>type of matrix containing the Householder vectors </td></tr>
<tr><td class="paramname">CoeffsType</td><td>type of vector containing the Householder coefficients </td></tr>
<tr><td class="paramname">Side</td><td>either OnTheLeft (the default) or OnTheRight</td></tr>
</table>
</dd>
</dl>
<p>This class represents a product sequence of Householder reflections where the first Householder reflection acts on the whole space, the second Householder reflection leaves the one-dimensional subspace spanned by the first unit vector invariant, the third Householder reflection leaves the two-dimensional subspace spanned by the first two unit vectors invariant, and so on up to the last reflection which leaves all but one dimensions invariant and acts only on the last dimension. Such sequences of Householder reflections are used in several algorithms to zero out certain parts of a matrix. Indeed, the methods <a class="el" href="classEigen_1_1HessenbergDecomposition.html#ad13845d7490115664924b3dc208ec369" title="Reconstructs the orthogonal matrix Q in the decomposition. ">HessenbergDecomposition::matrixQ()</a>, <a class="el" href="classEigen_1_1Tridiagonalization.html#ad13845d7490115664924b3dc208ec369" title="Returns the unitary matrix Q in the decomposition. ">Tridiagonalization::matrixQ()</a>, <a class="el" href="classEigen_1_1HouseholderQR.html#a520fc76215d7722387a76e7b7a484ee0">HouseholderQR::householderQ()</a>, and <a class="el" href="classEigen_1_1ColPivHouseholderQR.html#a97150efba7e2da32f48a7f823f047a89">ColPivHouseholderQR::householderQ()</a> all return a HouseholderSequence.</p>
<p>More precisely, the class HouseholderSequence represents an <img class="formulaInl" alt="$ n \times n $" src="form_133.png"/> matrix <img class="formulaInl" alt="$ H $" src="form_74.png"/> of the form <img class="formulaInl" alt="$ H = \prod_{i=0}^{n-1} H_i $" src="form_134.png"/> where the i-th Householder reflection is <img class="formulaInl" alt="$ H_i = I - h_i v_i v_i^* $" src="form_135.png"/>. The i-th Householder coefficient <img class="formulaInl" alt="$ h_i $" src="form_83.png"/> is a scalar and the i-th Householder vector <img class="formulaInl" alt="$ v_i $" src="form_85.png"/> is a vector of the form </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ v_i = [\underbrace{0, \ldots, 0}_{i-1\mbox{ zeros}}, 1, \underbrace{*, \ldots,*}_{n-i\mbox{ arbitrary entries}} ]. \]" src="form_136.png"/>
</p>
<p> The last <img class="formulaInl" alt="$ n-i $" src="form_137.png"/> entries of <img class="formulaInl" alt="$ v_i $" src="form_85.png"/> are called the essential part of the Householder vector.</p>
<p>Typical usages are listed below, where H is a <a class="el" href="classEigen_1_1HouseholderSequence.html" title="Sequence of Householder reflections acting on subspaces with decreasing size. ">HouseholderSequence</a>: </p>
<div class="fragment"><div class="line">* A.applyOnTheRight(H); <span class="comment">// A = A * H</span></div>
<div class="line">* A.applyOnTheLeft(H); <span class="comment">// A = H * A</span></div>
<div class="line">* A.applyOnTheRight(H.adjoint()); <span class="comment">// A = A * H^*</span></div>
<div class="line">* A.applyOnTheLeft(H.adjoint()); <span class="comment">// A = H^* * A</span></div>
<div class="line">* MatrixXd Q = H; <span class="comment">// conversion to a dense matrix</span></div>
<div class="line">* </div>
</div><!-- fragment --><p> In addition to the adjoint, you can also apply the inverse (=adjoint), the transpose, and the conjugate operators.</p>
<p>See the documentation for <a class="el" href="classEigen_1_1HouseholderSequence.html#a3202eb0eaf16ad05ef90ca741bbf1cbc" title="Constructor. ">HouseholderSequence(const VectorsType&, const CoeffsType&)</a> for an example.</p>
<dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1MatrixBase.html#a9aaee4fba54baa7d0ca508e094f2d6f9">MatrixBase::applyOnTheLeft()</a>, <a class="el" href="classEigen_1_1MatrixBase.html#a6e5b1e71a1916ddd01e7c324831c40dc">MatrixBase::applyOnTheRight()</a> </dd></dl>
</div><div id="dynsection-0" onclick="return toggleVisibility(this)" class="dynheader closed" style="cursor:pointer;">
<img id="dynsection-0-trigger" src="closed.png" alt="+"/> Inheritance diagram for HouseholderSequence< VectorsType, CoeffsType, Side >:</div>
<div id="dynsection-0-summary" class="dynsummary" style="display:block;">
</div>
<div id="dynsection-0-content" class="dyncontent" style="display:none;">
<div class="center"><img src="classEigen_1_1HouseholderSequence__inherit__graph.png" border="0" usemap="#HouseholderSequence_3_01VectorsType_00_01CoeffsType_00_01Side_01_4_inherit__map" alt="Inheritance graph"/></div>
<map name="HouseholderSequence_3_01VectorsType_00_01CoeffsType_00_01Side_01_4_inherit__map" id="HouseholderSequence_3_01VectorsType_00_01CoeffsType_00_01Side_01_4_inherit__map">
<area shape="rect" id="node2" href="structEigen_1_1EigenBase.html" title="{EigenBase\< HouseholderSequence\l\< VectorsType, CoeffsType,\l Side \> \>\n||+ addTo()\l+ applyThisOnTheLeft()\l+ applyThisOnTheRight()\l+ cols()\l+ const_cast_derived()\l+ const_derived()\l+ derived()\l+ derived()\l+ evalTo()\l+ rows()\l+ size()\l+ subTo()\l}" alt="" coords="5,6,232,274"/></map>
</div>
<table class="memberdecls">
<tr class="heading"><td colspan="2"><h2 class="groupheader"><a name="pub-methods"></a>
Public Member Functions</h2></td></tr>
<tr class="memitem:a5dc0922b29ae5e2a867cff75cbf7ac1a"><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a5dc0922b29ae5e2a867cff75cbf7ac1a"></a>
<a class="el" href="classEigen_1_1HouseholderSequence.html">ConjugateReturnType</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1HouseholderSequence.html#a5dc0922b29ae5e2a867cff75cbf7ac1a">adjoint</a> () const </td></tr>
<tr class="memdesc:a5dc0922b29ae5e2a867cff75cbf7ac1a"><td class="mdescLeft"> </td><td class="mdescRight">Adjoint (conjugate transpose) of the Householder sequence. <br/></td></tr>
<tr class="separator:a5dc0922b29ae5e2a867cff75cbf7ac1a"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:aaca1908a5ec508a25ff0a8bca803e5f3"><td class="memItemLeft" align="right" valign="top">Index </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1HouseholderSequence.html#aaca1908a5ec508a25ff0a8bca803e5f3">cols</a> () const </td></tr>
<tr class="memdesc:aaca1908a5ec508a25ff0a8bca803e5f3"><td class="mdescLeft"> </td><td class="mdescRight">Number of columns of transformation viewed as a matrix. <a href="#aaca1908a5ec508a25ff0a8bca803e5f3">More...</a><br/></td></tr>
<tr class="separator:aaca1908a5ec508a25ff0a8bca803e5f3"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:aab6c2241712d7faf07019a0d86d3fb79"><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="aab6c2241712d7faf07019a0d86d3fb79"></a>
<a class="el" href="classEigen_1_1HouseholderSequence.html">ConjugateReturnType</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1HouseholderSequence.html#aab6c2241712d7faf07019a0d86d3fb79">conjugate</a> () const </td></tr>
<tr class="memdesc:aab6c2241712d7faf07019a0d86d3fb79"><td class="mdescLeft"> </td><td class="mdescRight">Complex conjugate of the Householder sequence. <br/></td></tr>
<tr class="separator:aab6c2241712d7faf07019a0d86d3fb79"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:aa84222add803ad7c9db07dd4dd91d5d9"><td class="memItemLeft" align="right" valign="top"><a class="el" href="classEigen_1_1HouseholderSequence.html">HouseholderSequence</a><br class="typebreak"/>
< VectorsType, CoeffsType, <br class="typebreak"/>
Side > & </td><td class="memItemRight" valign="bottom"><a class="el" href="structEigen_1_1EigenBase.html#aa84222add803ad7c9db07dd4dd91d5d9">derived</a> ()</td></tr>
<tr class="separator:aa84222add803ad7c9db07dd4dd91d5d9"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a2850de52161dfe2bb6e380f1892276b8"><td class="memItemLeft" align="right" valign="top">const <a class="el" href="classEigen_1_1HouseholderSequence.html">HouseholderSequence</a><br class="typebreak"/>
< VectorsType, CoeffsType, <br class="typebreak"/>
Side > & </td><td class="memItemRight" valign="bottom"><a class="el" href="structEigen_1_1EigenBase.html#a2850de52161dfe2bb6e380f1892276b8">derived</a> () const</td></tr>
<tr class="separator:a2850de52161dfe2bb6e380f1892276b8"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:aaeaaa79416e82ffed90e36543cc1f564"><td class="memItemLeft" align="right" valign="top">const <a class="el" href="classEigen_1_1Block.html">EssentialVectorType</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1HouseholderSequence.html#aaeaaa79416e82ffed90e36543cc1f564">essentialVector</a> (Index k) const </td></tr>
<tr class="memdesc:aaeaaa79416e82ffed90e36543cc1f564"><td class="mdescLeft"> </td><td class="mdescRight">Essential part of a Householder vector. <a href="#aaeaaa79416e82ffed90e36543cc1f564">More...</a><br/></td></tr>
<tr class="separator:aaeaaa79416e82ffed90e36543cc1f564"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a3202eb0eaf16ad05ef90ca741bbf1cbc"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1HouseholderSequence.html#a3202eb0eaf16ad05ef90ca741bbf1cbc">HouseholderSequence</a> (const VectorsType &v, const CoeffsType &h)</td></tr>
<tr class="memdesc:a3202eb0eaf16ad05ef90ca741bbf1cbc"><td class="mdescLeft"> </td><td class="mdescRight">Constructor. <a href="#a3202eb0eaf16ad05ef90ca741bbf1cbc">More...</a><br/></td></tr>
<tr class="separator:a3202eb0eaf16ad05ef90ca741bbf1cbc"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a17449d69230251d4fdcf3e43a1b3ddda"><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a17449d69230251d4fdcf3e43a1b3ddda"></a>
 </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1HouseholderSequence.html#a17449d69230251d4fdcf3e43a1b3ddda">HouseholderSequence</a> (const <a class="el" href="classEigen_1_1HouseholderSequence.html">HouseholderSequence</a> &other)</td></tr>
<tr class="memdesc:a17449d69230251d4fdcf3e43a1b3ddda"><td class="mdescLeft"> </td><td class="mdescRight">Copy constructor. <br/></td></tr>
<tr class="separator:a17449d69230251d4fdcf3e43a1b3ddda"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ad13f4e783d2a24b55b5280e0a9096df1"><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="ad13f4e783d2a24b55b5280e0a9096df1"></a>
<a class="el" href="classEigen_1_1HouseholderSequence.html">ConjugateReturnType</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1HouseholderSequence.html#ad13f4e783d2a24b55b5280e0a9096df1">inverse</a> () const </td></tr>
<tr class="memdesc:ad13f4e783d2a24b55b5280e0a9096df1"><td class="mdescLeft"> </td><td class="mdescRight">Inverse of the Householder sequence (equals the adjoint). <br/></td></tr>
<tr class="separator:ad13f4e783d2a24b55b5280e0a9096df1"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:ab03433b49fb0a856a189b52c61f4872d"><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="ab03433b49fb0a856a189b52c61f4872d"></a>
Index </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1HouseholderSequence.html#ab03433b49fb0a856a189b52c61f4872d">length</a> () const </td></tr>
<tr class="memdesc:ab03433b49fb0a856a189b52c61f4872d"><td class="mdescLeft"> </td><td class="mdescRight">Returns the length of the Householder sequence. <br/></td></tr>
<tr class="separator:ab03433b49fb0a856a189b52c61f4872d"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a6abbb28d5386d39fc294ab8b5ead5375"><td class="memTemplParams" colspan="2">template<typename OtherDerived > </td></tr>
<tr class="memitem:a6abbb28d5386d39fc294ab8b5ead5375"><td class="memTemplItemLeft" align="right" valign="top">internal::matrix_type_times_scalar_type<br class="typebreak"/>
< Scalar, OtherDerived >::Type </td><td class="memTemplItemRight" valign="bottom"><a class="el" href="classEigen_1_1HouseholderSequence.html#a6abbb28d5386d39fc294ab8b5ead5375">operator*</a> (const <a class="el" href="classEigen_1_1MatrixBase.html">MatrixBase</a>< OtherDerived > &other) const </td></tr>
<tr class="memdesc:a6abbb28d5386d39fc294ab8b5ead5375"><td class="mdescLeft"> </td><td class="mdescRight">Computes the product of a Householder sequence with a matrix. <a href="#a6abbb28d5386d39fc294ab8b5ead5375">More...</a><br/></td></tr>
<tr class="separator:a6abbb28d5386d39fc294ab8b5ead5375"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a5552abd83dbd03c85cea6d61fd8875a5"><td class="memItemLeft" align="right" valign="top">Index </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1HouseholderSequence.html#a5552abd83dbd03c85cea6d61fd8875a5">rows</a> () const </td></tr>
<tr class="memdesc:a5552abd83dbd03c85cea6d61fd8875a5"><td class="mdescLeft"> </td><td class="mdescRight">Number of rows of transformation viewed as a matrix. <a href="#a5552abd83dbd03c85cea6d61fd8875a5">More...</a><br/></td></tr>
<tr class="separator:a5552abd83dbd03c85cea6d61fd8875a5"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a11ea4599a61fff6b27734e4dc31f2701"><td class="memItemLeft" align="right" valign="top"><a class="el" href="classEigen_1_1HouseholderSequence.html">HouseholderSequence</a> & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1HouseholderSequence.html#a11ea4599a61fff6b27734e4dc31f2701">setLength</a> (Index <a class="el" href="classEigen_1_1HouseholderSequence.html#ab03433b49fb0a856a189b52c61f4872d">length</a>)</td></tr>
<tr class="memdesc:a11ea4599a61fff6b27734e4dc31f2701"><td class="mdescLeft"> </td><td class="mdescRight">Sets the length of the Householder sequence. <a href="#a11ea4599a61fff6b27734e4dc31f2701">More...</a><br/></td></tr>
<tr class="separator:a11ea4599a61fff6b27734e4dc31f2701"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:af60a8214544812a06d520bd6605c539b"><td class="memItemLeft" align="right" valign="top"><a class="el" href="classEigen_1_1HouseholderSequence.html">HouseholderSequence</a> & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1HouseholderSequence.html#af60a8214544812a06d520bd6605c539b">setShift</a> (Index <a class="el" href="classEigen_1_1HouseholderSequence.html#a83a0b0137bca9a5429116067ca0a1e0a">shift</a>)</td></tr>
<tr class="memdesc:af60a8214544812a06d520bd6605c539b"><td class="mdescLeft"> </td><td class="mdescRight">Sets the shift of the Householder sequence. <a href="#af60a8214544812a06d520bd6605c539b">More...</a><br/></td></tr>
<tr class="separator:af60a8214544812a06d520bd6605c539b"><td class="memSeparator" colspan="2"> </td></tr>
<tr class="memitem:a83a0b0137bca9a5429116067ca0a1e0a"><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a83a0b0137bca9a5429116067ca0a1e0a"></a>
Index </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1HouseholderSequence.html#a83a0b0137bca9a5429116067ca0a1e0a">shift</a> () const </td></tr>
<tr class="memdesc:a83a0b0137bca9a5429116067ca0a1e0a"><td class="mdescLeft"> </td><td class="mdescRight">Returns the shift of the Householder sequence. <br/></td></tr>
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<tr class="memitem:a76f5bc8a03ec105ab4be1f2b91b7b5d5"><td class="memItemLeft" align="right" valign="top">Index </td><td class="memItemRight" valign="bottom"><a class="el" href="structEigen_1_1EigenBase.html#a76f5bc8a03ec105ab4be1f2b91b7b5d5">size</a> () const</td></tr>
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<a class="el" href="classEigen_1_1HouseholderSequence.html">HouseholderSequence</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1HouseholderSequence.html#a675ccc58e030cd6a2f43864257737ea0">transpose</a> () const </td></tr>
<tr class="memdesc:a675ccc58e030cd6a2f43864257737ea0"><td class="mdescLeft"> </td><td class="mdescRight">Transpose of the Householder sequence. <br/></td></tr>
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<tr class="heading"><td colspan="2"><h2 class="groupheader"><a name="pro-methods"></a>
Protected Member Functions</h2></td></tr>
<tr class="memitem:a10457dacb3949bc6dd5001878641819e"><td class="memItemLeft" align="right" valign="top"><a class="el" href="classEigen_1_1HouseholderSequence.html">HouseholderSequence</a> & </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1HouseholderSequence.html#a10457dacb3949bc6dd5001878641819e">setTrans</a> (bool <a class="el" href="classEigen_1_1HouseholderSequence.html#a60112ec072cfadceb8e909715cc33a1c">trans</a>)</td></tr>
<tr class="memdesc:a10457dacb3949bc6dd5001878641819e"><td class="mdescLeft"> </td><td class="mdescRight">Sets the transpose flag. <a href="#a10457dacb3949bc6dd5001878641819e">More...</a><br/></td></tr>
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bool </td><td class="memItemRight" valign="bottom"><a class="el" href="classEigen_1_1HouseholderSequence.html#a60112ec072cfadceb8e909715cc33a1c">trans</a> () const </td></tr>
<tr class="memdesc:a60112ec072cfadceb8e909715cc33a1c"><td class="mdescLeft"> </td><td class="mdescRight">Returns the transpose flag. <br/></td></tr>
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<h2 class="groupheader">Constructor & Destructor Documentation</h2>
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<td>(</td>
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<td class="paramtype">const CoeffsType & </td>
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<p>Constructor. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[in]</td><td class="paramname">v</td><td>Matrix containing the essential parts of the Householder vectors </td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">h</td><td>Vector containing the Householder coefficients</td></tr>
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<p>Constructs the Householder sequence with coefficients given by <code>h</code> and vectors given by <code>v</code>. The i-th Householder coefficient <img class="formulaInl" alt="$ h_i $" src="form_83.png"/> is given by <code>h(i)</code> and the essential part of the i-th Householder vector <img class="formulaInl" alt="$ v_i $" src="form_85.png"/> is given by <code>v(k,i)</code> with <code>k</code> > <code>i</code> (the subdiagonal part of the i-th column). If <code>v</code> has fewer columns than rows, then the Householder sequence contains as many Householder reflections as there are columns.</p>
<dl class="section note"><dt>Note</dt><dd>The HouseholderSequence object stores <code>v</code> and <code>h</code> by reference.</dd></dl>
<p>Example: </p>
<div class="fragment"><div class="line"><a class="code" href="group__matrixtypedefs.html#ga45a14b423c687c3e2e8325f148e27bf3">Matrix3d</a> v = <a class="code" href="classEigen_1_1DenseBase.html#a8e759dafdd9ecc446d397b7f5435f60a">Matrix3d::Random</a>();</div>
<div class="line">cout << <span class="stringliteral">"The matrix v is:"</span> << endl;</div>
<div class="line">cout << v << endl;</div>
<div class="line"></div>
<div class="line"><a class="code" href="group__matrixtypedefs.html#ga2006332f6989f501762673e21f5128f5">Vector3d</a> v0(1, v(1,0), v(2,0));</div>
<div class="line">cout << <span class="stringliteral">"The first Householder vector is: v_0 = "</span> << v0.transpose() << endl;</div>
<div class="line"><a class="code" href="group__matrixtypedefs.html#ga2006332f6989f501762673e21f5128f5">Vector3d</a> v1(0, 1, v(2,1));</div>
<div class="line">cout << <span class="stringliteral">"The second Householder vector is: v_1 = "</span> << v1.transpose() << endl;</div>
<div class="line"><a class="code" href="group__matrixtypedefs.html#ga2006332f6989f501762673e21f5128f5">Vector3d</a> v2(0, 0, 1);</div>
<div class="line">cout << <span class="stringliteral">"The third Householder vector is: v_2 = "</span> << v2.transpose() << endl;</div>
<div class="line"></div>
<div class="line"><a class="code" href="group__matrixtypedefs.html#ga2006332f6989f501762673e21f5128f5">Vector3d</a> h = <a class="code" href="classEigen_1_1DenseBase.html#a8e759dafdd9ecc446d397b7f5435f60a">Vector3d::Random</a>();</div>
<div class="line">cout << <span class="stringliteral">"The Householder coefficients are: h = "</span> << h.transpose() << endl;</div>
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<div class="line"><a class="code" href="group__matrixtypedefs.html#ga45a14b423c687c3e2e8325f148e27bf3">Matrix3d</a> H0 = <a class="code" href="classEigen_1_1MatrixBase.html#a0650b65c6ae6c3d19a138b72a6d68568">Matrix3d::Identity</a>() - h(0) * v0 * v0.adjoint();</div>
<div class="line">cout << <span class="stringliteral">"The first Householder reflection is represented by H_0 = "</span> << endl;</div>
<div class="line">cout << H0 << endl;</div>
<div class="line"><a class="code" href="group__matrixtypedefs.html#ga45a14b423c687c3e2e8325f148e27bf3">Matrix3d</a> H1 = <a class="code" href="classEigen_1_1MatrixBase.html#a0650b65c6ae6c3d19a138b72a6d68568">Matrix3d::Identity</a>() - h(1) * v1 * v1.adjoint();</div>
<div class="line">cout << <span class="stringliteral">"The second Householder reflection is represented by H_1 = "</span> << endl;</div>
<div class="line">cout << H1 << endl;</div>
<div class="line"><a class="code" href="group__matrixtypedefs.html#ga45a14b423c687c3e2e8325f148e27bf3">Matrix3d</a> H2 = <a class="code" href="classEigen_1_1MatrixBase.html#a0650b65c6ae6c3d19a138b72a6d68568">Matrix3d::Identity</a>() - h(2) * v2 * v2.adjoint();</div>
<div class="line">cout << <span class="stringliteral">"The third Householder reflection is represented by H_2 = "</span> << endl;</div>
<div class="line">cout << H2 << endl;</div>
<div class="line">cout << <span class="stringliteral">"Their product is H_0 H_1 H_2 = "</span> << endl;</div>
<div class="line">cout << H0 * H1 * H2 << endl;</div>
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<div class="line">HouseholderSequence<Matrix3d, Vector3d> hhSeq(v, h);</div>
<div class="line"><a class="code" href="group__matrixtypedefs.html#ga45a14b423c687c3e2e8325f148e27bf3">Matrix3d</a> hhSeqAsMatrix(hhSeq);</div>
<div class="line">cout << <span class="stringliteral">"If we construct a HouseholderSequence from v and h"</span> << endl;</div>
<div class="line">cout << <span class="stringliteral">"and convert it to a matrix, we get:"</span> << endl;</div>
<div class="line">cout << hhSeqAsMatrix << endl;</div>
</div><!-- fragment --><p> Output: </p>
<pre class="fragment">The matrix v is:
0.68 0.597 -0.33
-0.211 0.823 0.536
0.566 -0.605 -0.444
The first Householder vector is: v_0 = 1 -0.211 0.566
The second Householder vector is: v_1 = 0 1 -0.605
The third Householder vector is: v_2 = 0 0 1
The Householder coefficients are: h = 0.108 -0.0452 0.258
The first Householder reflection is represented by H_0 =
0.892 0.0228 -0.0611
0.0228 0.995 0.0129
-0.0611 0.0129 0.965
The second Householder reflection is represented by H_1 =
1 0 0
0 1.05 -0.0273
0 -0.0273 1.02
The third Householder reflection is represented by H_2 =
1 0 0
0 1 0
0 0 0.742
Their product is H_0 H_1 H_2 =
0.892 0.0255 -0.0466
0.0228 1.04 -0.0105
-0.0611 -0.0129 0.728
If we construct a HouseholderSequence from v and h
and convert it to a matrix, we get:
0.892 0.0255 -0.0466
0.0228 1.04 -0.0105
-0.0611 -0.0129 0.728
</pre><dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1HouseholderSequence.html#a11ea4599a61fff6b27734e4dc31f2701" title="Sets the length of the Householder sequence. ">setLength()</a>, <a class="el" href="classEigen_1_1HouseholderSequence.html#af60a8214544812a06d520bd6605c539b" title="Sets the shift of the Householder sequence. ">setShift()</a> </dd></dl>
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<h2 class="groupheader">Member Function Documentation</h2>
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<p>Number of columns of transformation viewed as a matrix. </p>
<dl class="section return"><dt>Returns</dt><dd>Number of columns</dd></dl>
<p>This equals the dimension of the space that the transformation acts on. </p>
<p>References <a class="el" href="classEigen_1_1HouseholderSequence.html#a5552abd83dbd03c85cea6d61fd8875a5">HouseholderSequence< VectorsType, CoeffsType, Side >::rows()</a>.</p>
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<td class="memname"><a class="el" href="classEigen_1_1HouseholderSequence.html">HouseholderSequence</a>< VectorsType, CoeffsType, Side > & derived </td>
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<dl class="section return"><dt>Returns</dt><dd>a reference to the derived object </dd></dl>
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<td class="memname">const <a class="el" href="classEigen_1_1HouseholderSequence.html">HouseholderSequence</a>< VectorsType, CoeffsType, Side > & derived </td>
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<dl class="section return"><dt>Returns</dt><dd>a const reference to the derived object </dd></dl>
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<td class="memname">const <a class="el" href="classEigen_1_1Block.html">EssentialVectorType</a> essentialVector </td>
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<td class="paramname"><em>k</em></td><td>)</td>
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<p>Essential part of a Householder vector. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[in]</td><td class="paramname">k</td><td>Index of Householder reflection </td></tr>
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<dl class="section return"><dt>Returns</dt><dd>Vector containing non-trivial entries of k-th Householder vector</dd></dl>
<p>This function returns the essential part of the Householder vector <img class="formulaInl" alt="$ v_i $" src="form_85.png"/>. This is a vector of length <img class="formulaInl" alt="$ n-i $" src="form_137.png"/> containing the last <img class="formulaInl" alt="$ n-i $" src="form_137.png"/> entries of the vector </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ v_i = [\underbrace{0, \ldots, 0}_{i-1\mbox{ zeros}}, 1, \underbrace{*, \ldots,*}_{n-i\mbox{ arbitrary entries}} ]. \]" src="form_136.png"/>
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<p> The index <img class="formulaInl" alt="$ i $" src="form_84.png"/> equals <code>k</code> + <a class="el" href="classEigen_1_1HouseholderSequence.html#a83a0b0137bca9a5429116067ca0a1e0a" title="Returns the shift of the Householder sequence. ">shift()</a>, corresponding to the k-th column of the matrix <code>v</code> passed to the constructor.</p>
<dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1HouseholderSequence.html#af60a8214544812a06d520bd6605c539b" title="Sets the shift of the Householder sequence. ">setShift()</a>, <a class="el" href="classEigen_1_1HouseholderSequence.html#a83a0b0137bca9a5429116067ca0a1e0a" title="Returns the shift of the Householder sequence. ">shift()</a> </dd></dl>
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<td class="memname">internal::matrix_type_times_scalar_type<Scalar, OtherDerived>::Type operator* </td>
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<p>Computes the product of a Householder sequence with a matrix. </p>
<dl class="params"><dt>Parameters</dt><dd>
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<tr><td class="paramdir">[in]</td><td class="paramname">other</td><td>Matrix being multiplied. </td></tr>
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<dl class="section return"><dt>Returns</dt><dd>Expression object representing the product.</dd></dl>
<p>This function computes <img class="formulaInl" alt="$ HM $" src="form_140.png"/> where <img class="formulaInl" alt="$ H $" src="form_74.png"/> is the Householder sequence represented by <code>*this</code> and <img class="formulaInl" alt="$ M $" src="form_139.png"/> is the matrix <code>other</code>. </p>
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<p>Number of rows of transformation viewed as a matrix. </p>
<dl class="section return"><dt>Returns</dt><dd>Number of rows</dd></dl>
<p>This equals the dimension of the space that the transformation acts on. </p>
<p>References <a class="el" href="group__enums.html#gga3f4a95405348aa63ef907e692901011fa02217d2d79111440646e3b00c747745f">Eigen::OnTheLeft</a>.</p>
<p>Referenced by <a class="el" href="classEigen_1_1HouseholderSequence.html#aaca1908a5ec508a25ff0a8bca803e5f3">HouseholderSequence< VectorsType, CoeffsType, Side >::cols()</a>.</p>
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<td class="memname"><a class="el" href="classEigen_1_1HouseholderSequence.html">HouseholderSequence</a>& setLength </td>
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<p>Sets the length of the Householder sequence. </p>
<dl class="params"><dt>Parameters</dt><dd>
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<tr><td class="paramdir">[in]</td><td class="paramname">length</td><td>New value for the length.</td></tr>
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<p>By default, the length <img class="formulaInl" alt="$ n $" src="form_45.png"/> of the Householder sequence <img class="formulaInl" alt="$ H = H_0 H_1 \ldots H_{n-1} $" src="form_141.png"/> is set to the number of columns of the matrix <code>v</code> passed to the constructor, or the number of rows if that is smaller. After this function is called, the length equals <code>length</code>.</p>
<dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1HouseholderSequence.html#ab03433b49fb0a856a189b52c61f4872d" title="Returns the length of the Householder sequence. ">length()</a> </dd></dl>
<p>References <a class="el" href="classEigen_1_1HouseholderSequence.html#ab03433b49fb0a856a189b52c61f4872d">HouseholderSequence< VectorsType, CoeffsType, Side >::length()</a>.</p>
<p>Referenced by <a class="el" href="classEigen_1_1HouseholderSequence.html#aab6c2241712d7faf07019a0d86d3fb79">HouseholderSequence< VectorsType, CoeffsType, Side >::conjugate()</a>.</p>
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<td class="memname"><a class="el" href="classEigen_1_1HouseholderSequence.html">HouseholderSequence</a>& setShift </td>
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<p>Sets the shift of the Householder sequence. </p>
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<tr><td class="paramdir">[in]</td><td class="paramname">shift</td><td>New value for the shift.</td></tr>
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<p>By default, a HouseholderSequence object represents <img class="formulaInl" alt="$ H = H_0 H_1 \ldots H_{n-1} $" src="form_141.png"/> and the i-th column of the matrix <code>v</code> passed to the constructor corresponds to the i-th Householder reflection. After this function is called, the object represents <img class="formulaInl" alt="$ H = H_{\mathrm{shift}} H_{\mathrm{shift}+1} \ldots H_{n-1} $" src="form_142.png"/> and the i-th column of <code>v</code> corresponds to the (shift+i)-th Householder reflection.</p>
<dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1HouseholderSequence.html#a83a0b0137bca9a5429116067ca0a1e0a" title="Returns the shift of the Householder sequence. ">shift()</a> </dd></dl>
<p>References <a class="el" href="classEigen_1_1HouseholderSequence.html#a83a0b0137bca9a5429116067ca0a1e0a">HouseholderSequence< VectorsType, CoeffsType, Side >::shift()</a>.</p>
<p>Referenced by <a class="el" href="classEigen_1_1HouseholderSequence.html#aab6c2241712d7faf07019a0d86d3fb79">HouseholderSequence< VectorsType, CoeffsType, Side >::conjugate()</a>.</p>
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<td class="memname"><a class="el" href="classEigen_1_1HouseholderSequence.html">HouseholderSequence</a>& setTrans </td>
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<td class="paramname"><em>trans</em></td><td>)</td>
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<p>Sets the transpose flag. </p>
<dl class="params"><dt>Parameters</dt><dd>
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<tr><td class="paramdir">[in]</td><td class="paramname">trans</td><td>New value of the transpose flag.</td></tr>
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<p>By default, the transpose flag is not set. If the transpose flag is set, then this object represents <img class="formulaInl" alt="$ H^T = H_{n-1}^T \ldots H_1^T H_0^T $" src="form_143.png"/> instead of <img class="formulaInl" alt="$ H = H_0 H_1 \ldots H_{n-1} $" src="form_141.png"/>.</p>
<dl class="section see"><dt>See Also</dt><dd><a class="el" href="classEigen_1_1HouseholderSequence.html#a60112ec072cfadceb8e909715cc33a1c" title="Returns the transpose flag. ">trans()</a> </dd></dl>
<p>References <a class="el" href="classEigen_1_1HouseholderSequence.html#a60112ec072cfadceb8e909715cc33a1c">HouseholderSequence< VectorsType, CoeffsType, Side >::trans()</a>.</p>
<p>Referenced by <a class="el" href="classEigen_1_1HouseholderSequence.html#a5dc0922b29ae5e2a867cff75cbf7ac1a">HouseholderSequence< VectorsType, CoeffsType, Side >::adjoint()</a>, <a class="el" href="classEigen_1_1HouseholderSequence.html#aab6c2241712d7faf07019a0d86d3fb79">HouseholderSequence< VectorsType, CoeffsType, Side >::conjugate()</a>, and <a class="el" href="classEigen_1_1HouseholderSequence.html#a675ccc58e030cd6a2f43864257737ea0">HouseholderSequence< VectorsType, CoeffsType, Side >::transpose()</a>.</p>
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<td class="memname">Index size </td>
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<dl class="section return"><dt>Returns</dt><dd>the number of coefficients, which is <a class="el" href="structEigen_1_1EigenBase.html#a5552abd83dbd03c85cea6d61fd8875a5">rows()</a>*cols(). </dd></dl>
<dl class="section see"><dt>See Also</dt><dd><a class="el" href="structEigen_1_1EigenBase.html#a5552abd83dbd03c85cea6d61fd8875a5">rows()</a>, <a class="el" href="structEigen_1_1EigenBase.html#aaca1908a5ec508a25ff0a8bca803e5f3">cols()</a>, SizeAtCompileTime. </dd></dl>
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<hr/>The documentation for this class was generated from the following files:<ul>
<li><a class="el" href="ForwardDeclarations_8h_source.html">ForwardDeclarations.h</a></li>
<li><a class="el" href="HouseholderSequence_8h_source.html">HouseholderSequence.h</a></li>
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