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<title>Householder Transformations - GNU Scientific Library -- Reference Manual</title>
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<h3 class="section">13.11 Householder Transformations</h3>
<p><a name="index-Householder-matrix-1308"></a><a name="index-Householder-transformation-1309"></a><a name="index-transformation_002c-Householder-1310"></a>
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 P takes the form,
<pre class="example"> P = I - \tau v v^T
</pre>
<p class="noindent">where v is a vector (called the <dfn>Householder vector</dfn>) and
\tau = 2/(v^T v). The functions described in this section use the
rank-1 structure of the Householder matrix to create and apply
Householder transformations efficiently.
<div class="defun">
— Function: double <b>gsl_linalg_householder_transform</b> (<var>gsl_vector * v</var>)<var><a name="index-gsl_005flinalg_005fhouseholder_005ftransform-1311"></a></var><br>
— Function: gsl_complex <b>gsl_linalg_complex_householder_transform</b> (<var>gsl_vector_complex * v</var>)<var><a name="index-gsl_005flinalg_005fcomplex_005fhouseholder_005ftransform-1312"></a></var><br>
<blockquote><p>This function prepares a Householder transformation P = I - \tau v
v^T 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 \tau is returned.
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_linalg_householder_hm</b> (<var>double tau, const gsl_vector * v, gsl_matrix * A</var>)<var><a name="index-gsl_005flinalg_005fhouseholder_005fhm-1313"></a></var><br>
— Function: int <b>gsl_linalg_complex_householder_hm</b> (<var>gsl_complex tau, const gsl_vector_complex * v, gsl_matrix_complex * A</var>)<var><a name="index-gsl_005flinalg_005fcomplex_005fhouseholder_005fhm-1314"></a></var><br>
<blockquote><p>This function applies the Householder matrix P 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 P A is stored in <var>A</var>.
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_linalg_householder_mh</b> (<var>double tau, const gsl_vector * v, gsl_matrix * A</var>)<var><a name="index-gsl_005flinalg_005fhouseholder_005fmh-1315"></a></var><br>
— Function: int <b>gsl_linalg_complex_householder_mh</b> (<var>gsl_complex tau, const gsl_vector_complex * v, gsl_matrix_complex * A</var>)<var><a name="index-gsl_005flinalg_005fcomplex_005fhouseholder_005fmh-1316"></a></var><br>
<blockquote><p>This function applies the Householder matrix P 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 A P is stored in <var>A</var>.
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_linalg_householder_hv</b> (<var>double tau, const gsl_vector * v, gsl_vector * w</var>)<var><a name="index-gsl_005flinalg_005fhouseholder_005fhv-1317"></a></var><br>
— Function: int <b>gsl_linalg_complex_householder_hv</b> (<var>gsl_complex tau, const gsl_vector_complex * v, gsl_vector_complex * w</var>)<var><a name="index-gsl_005flinalg_005fcomplex_005fhouseholder_005fhv-1318"></a></var><br>
<blockquote><p>This function applies the Householder transformation P defined by
the scalar <var>tau</var> and the vector <var>v</var> to the vector <var>w</var>. On
output the result P w is stored in <var>w</var>.
</p></blockquote></div>
<!-- @deftypefun int gsl_linalg_householder_hm1 (double tau, gsl_matrix * A) -->
<!-- This function applies the Householder transform, defined by the scalar -->
<!-- @var{tau} and the vector @var{v}, to a matrix being build up from the -->
<!-- identity matrix, using the first column of @var{A} as a householder vector. -->
<!-- @end deftypefun -->
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