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<h3 class="section">14.8 Hessenberg Decomposition of Real Matrices</h3>
<p><a name="index-Hessenberg-decomposition-1362"></a>
A general real matrix A can be decomposed by orthogonal
similarity transformations into the form
where U is orthogonal and H is an upper Hessenberg matrix,
meaning that it has zeros below the first subdiagonal. The
Hessenberg reduction is the first step in the Schur decomposition
for the nonsymmetric eigenvalue problem, but has applications in
other areas as well.
<div class="defun">
— Function: int <b>gsl_linalg_hessenberg_decomp</b> (<var>gsl_matrix * A, gsl_vector * tau</var>)<var><a name="index-gsl_005flinalg_005fhessenberg_005fdecomp-1363"></a></var><br>
<blockquote><p>This function computes the Hessenberg decomposition of the matrix
<var>A</var> by applying the similarity transformation H = U^T A U.
On output, H is stored in the upper portion of <var>A</var>. The
information required to construct the matrix U is stored in
the lower triangular portion of <var>A</var>. U is a product
of N - 2 Householder matrices. The Householder vectors
are stored in the lower portion of <var>A</var> (below the subdiagonal)
and the Householder coefficients are stored in the vector <var>tau</var>.
<var>tau</var> must be of length <var>N</var>.
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_linalg_hessenberg_unpack</b> (<var>gsl_matrix * H, gsl_vector * tau, gsl_matrix * U</var>)<var><a name="index-gsl_005flinalg_005fhessenberg_005funpack-1364"></a></var><br>
<blockquote><p>This function constructs the orthogonal matrix U from the
information stored in the Hessenberg matrix <var>H</var> along with the
vector <var>tau</var>. <var>H</var> and <var>tau</var> are outputs from
<code>gsl_linalg_hessenberg_decomp</code>.
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_linalg_hessenberg_unpack_accum</b> (<var>gsl_matrix * H, gsl_vector * tau, gsl_matrix * V</var>)<var><a name="index-gsl_005flinalg_005fhessenberg_005funpack_005faccum-1365"></a></var><br>
<blockquote><p>This function is similar to <code>gsl_linalg_hessenberg_unpack</code>, except
it accumulates the matrix <var>U</var> into <var>V</var>, so that V' = VU.
The matrix <var>V</var> must be initialized prior to calling this function.
Setting <var>V</var> to the identity matrix provides the same result as
<code>gsl_linalg_hessenberg_unpack</code>. If <var>H</var> is order <var>N</var>, then
<var>V</var> must have <var>N</var> columns but may have any number of rows.
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_linalg_hessenberg_set_zero</b> (<var>gsl_matrix * H</var>)<var><a name="index-gsl_005flinalg_005fhessenberg_005fset_005fzero-1366"></a></var><br>
<blockquote><p>This function sets the lower triangular portion of <var>H</var>, below
the subdiagonal, to zero. It is useful for clearing out the
Householder vectors after calling <code>gsl_linalg_hessenberg_decomp</code>.
</p></blockquote></div>
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