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<a name="Bidiagonalization"></a>
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Next: <a href="Givens-Rotations.html#Givens-Rotations" accesskey="n" rel="next">Givens Rotations</a>, Previous: <a href="Hessenberg_002dTriangular-Decomposition-of-Real-Matrices.html#Hessenberg_002dTriangular-Decomposition-of-Real-Matrices" accesskey="p" rel="previous">Hessenberg-Triangular Decomposition of Real Matrices</a>, Up: <a href="Linear-Algebra.html#Linear-Algebra" accesskey="u" rel="up">Linear Algebra</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="Bidiagonalization-1"></a>
<h3 class="section">14.13 Bidiagonalization</h3>
<a name="index-bidiagonalization-of-real-matrices"></a>

<p>A general matrix <em>A</em> can be factorized by similarity
transformations into the form,
</p>
<div class="example">
<pre class="example">A = U B V^T
</pre></div>

<p>where <em>U</em> and <em>V</em> are orthogonal matrices and <em>B</em> is a
<em>N</em>-by-<em>N</em> bidiagonal matrix with non-zero entries only on the
diagonal and superdiagonal.  The size of <var>U</var> is <em>M</em>-by-<em>N</em>
and the size of <var>V</var> is <em>N</em>-by-<em>N</em>.
</p>
<dl>
<dt><a name="index-gsl_005flinalg_005fbidiag_005fdecomp"></a>Function: <em>int</em> <strong>gsl_linalg_bidiag_decomp</strong> <em>(gsl_matrix * <var>A</var>, gsl_vector * <var>tau_U</var>, gsl_vector * <var>tau_V</var>)</em></dt>
<dd><p>This function factorizes the <em>M</em>-by-<em>N</em> matrix <var>A</var> into
bidiagonal form <em>U B V^T</em>.  The diagonal and superdiagonal of the
matrix <em>B</em> are stored in the diagonal and superdiagonal of <var>A</var>.
The orthogonal matrices <em>U</em> and <var>V</var> are stored as compressed
Householder vectors in the remaining elements of <var>A</var>.  The
Householder coefficients are stored in the vectors <var>tau_U</var> and
<var>tau_V</var>.  The length of <var>tau_U</var> must equal the number of
elements in the diagonal of <var>A</var> and the length of <var>tau_V</var> should
be one element shorter.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005flinalg_005fbidiag_005funpack"></a>Function: <em>int</em> <strong>gsl_linalg_bidiag_unpack</strong> <em>(const gsl_matrix * <var>A</var>, const gsl_vector * <var>tau_U</var>, gsl_matrix * <var>U</var>, const gsl_vector * <var>tau_V</var>, gsl_matrix * <var>V</var>, gsl_vector * <var>diag</var>, gsl_vector * <var>superdiag</var>)</em></dt>
<dd><p>This function unpacks the bidiagonal decomposition of <var>A</var> produced by
<code>gsl_linalg_bidiag_decomp</code>, (<var>A</var>, <var>tau_U</var>, <var>tau_V</var>)
into the separate orthogonal matrices <var>U</var>, <var>V</var> and the diagonal
vector <var>diag</var> and superdiagonal <var>superdiag</var>.  Note that <var>U</var>
is stored as a compact <em>M</em>-by-<em>N</em> orthogonal matrix satisfying
<em>U^T U = I</em> for efficiency.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005flinalg_005fbidiag_005funpack2"></a>Function: <em>int</em> <strong>gsl_linalg_bidiag_unpack2</strong> <em>(gsl_matrix * <var>A</var>, gsl_vector * <var>tau_U</var>, gsl_vector * <var>tau_V</var>, gsl_matrix * <var>V</var>)</em></dt>
<dd><p>This function unpacks the bidiagonal decomposition of <var>A</var> produced by
<code>gsl_linalg_bidiag_decomp</code>, (<var>A</var>, <var>tau_U</var>, <var>tau_V</var>)
into the separate orthogonal matrices <var>U</var>, <var>V</var> and the diagonal
vector <var>diag</var> and superdiagonal <var>superdiag</var>.  The matrix <var>U</var>
is stored in-place in <var>A</var>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005flinalg_005fbidiag_005funpack_005fB"></a>Function: <em>int</em> <strong>gsl_linalg_bidiag_unpack_B</strong> <em>(const gsl_matrix * <var>A</var>, gsl_vector * <var>diag</var>, gsl_vector * <var>superdiag</var>)</em></dt>
<dd><p>This function unpacks the diagonal and superdiagonal of the bidiagonal
decomposition of <var>A</var> from <code>gsl_linalg_bidiag_decomp</code>, into
the diagonal vector <var>diag</var> and superdiagonal vector <var>superdiag</var>.
</p></dd></dl>

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Next: <a href="Givens-Rotations.html#Givens-Rotations" accesskey="n" rel="next">Givens Rotations</a>, Previous: <a href="Hessenberg_002dTriangular-Decomposition-of-Real-Matrices.html#Hessenberg_002dTriangular-Decomposition-of-Real-Matrices" accesskey="p" rel="previous">Hessenberg-Triangular Decomposition of Real Matrices</a>, Up: <a href="Linear-Algebra.html#Linear-Algebra" accesskey="u" rel="up">Linear Algebra</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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