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<title>GNU Scientific Library – Reference Manual: Householder Transformations</title>
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<a name="Householder-Transformations"></a>
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<p>
Next: <a href="Householder-solver-for-linear-systems.html#Householder-solver-for-linear-systems" accesskey="n" rel="next">Householder solver for linear systems</a>, Previous: <a href="Bidiagonalization.html#Bidiagonalization" accesskey="p" rel="previous">Bidiagonalization</a>, Up: <a href="Linear-Algebra.html#Linear-Algebra" accesskey="u" rel="up">Linear Algebra</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<a name="Householder-Transformations-1"></a>
<h3 class="section">14.11 Householder Transformations</h3>
<a name="index-Householder-matrix"></a>
<a name="index-Householder-transformation"></a>
<a name="index-transformation_002c-Householder"></a>
<p>A Householder transformation is a rank-1 modification of the identity
matrix which can be used to zero out selected elements of a vector. A
Householder matrix <em>P</em> takes the form,
where <em>v</em> is a vector (called the <em>Householder vector</em>) and
<em>\tau = 2/(v^T v)</em>. The functions described in this section use the
rank-1 structure of the Householder matrix to create and apply
Householder transformations efficiently.
</p>
<dl>
<dt><a name="index-gsl_005flinalg_005fhouseholder_005ftransform"></a>Function: <em>double</em> <strong>gsl_linalg_householder_transform</strong> <em>(gsl_vector * <var>v</var>)</em></dt>
<dt><a name="index-gsl_005flinalg_005fcomplex_005fhouseholder_005ftransform"></a>Function: <em>gsl_complex</em> <strong>gsl_linalg_complex_householder_transform</strong> <em>(gsl_vector_complex * <var>v</var>)</em></dt>
<dd><p>This function prepares a Householder transformation <em>P = I - \tau v
v^T</em> which can be used to zero all the elements of the input vector except
the first. On output the transformation is stored in the vector <var>v</var>
and the scalar <em>\tau</em> is returned.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005flinalg_005fhouseholder_005fhm"></a>Function: <em>int</em> <strong>gsl_linalg_householder_hm</strong> <em>(double <var>tau</var>, const gsl_vector * <var>v</var>, gsl_matrix * <var>A</var>)</em></dt>
<dt><a name="index-gsl_005flinalg_005fcomplex_005fhouseholder_005fhm"></a>Function: <em>int</em> <strong>gsl_linalg_complex_householder_hm</strong> <em>(gsl_complex <var>tau</var>, const gsl_vector_complex * <var>v</var>, gsl_matrix_complex * <var>A</var>)</em></dt>
<dd><p>This function applies the Householder matrix <em>P</em> defined by the
scalar <var>tau</var> and the vector <var>v</var> to the left-hand side of the
matrix <var>A</var>. On output the result <em>P A</em> is stored in <var>A</var>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005flinalg_005fhouseholder_005fmh"></a>Function: <em>int</em> <strong>gsl_linalg_householder_mh</strong> <em>(double <var>tau</var>, const gsl_vector * <var>v</var>, gsl_matrix * <var>A</var>)</em></dt>
<dt><a name="index-gsl_005flinalg_005fcomplex_005fhouseholder_005fmh"></a>Function: <em>int</em> <strong>gsl_linalg_complex_householder_mh</strong> <em>(gsl_complex <var>tau</var>, const gsl_vector_complex * <var>v</var>, gsl_matrix_complex * <var>A</var>)</em></dt>
<dd><p>This function applies the Householder matrix <em>P</em> defined by the
scalar <var>tau</var> and the vector <var>v</var> to the right-hand side of the
matrix <var>A</var>. On output the result <em>A P</em> is stored in <var>A</var>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005flinalg_005fhouseholder_005fhv"></a>Function: <em>int</em> <strong>gsl_linalg_householder_hv</strong> <em>(double <var>tau</var>, const gsl_vector * <var>v</var>, gsl_vector * <var>w</var>)</em></dt>
<dt><a name="index-gsl_005flinalg_005fcomplex_005fhouseholder_005fhv"></a>Function: <em>int</em> <strong>gsl_linalg_complex_householder_hv</strong> <em>(gsl_complex <var>tau</var>, const gsl_vector_complex * <var>v</var>, gsl_vector_complex * <var>w</var>)</em></dt>
<dd><p>This function applies the Householder transformation <em>P</em> defined by
the scalar <var>tau</var> and the vector <var>v</var> to the vector <var>w</var>. On
output the result <em>P w</em> is stored in <var>w</var>.
</p></dd></dl>
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