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<title>Bidiagonalization - GNU Scientific Library -- Reference Manual</title>
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<a name="Bidiagonalization"></a>
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<h3 class="section">13.10 Bidiagonalization</h3>
<p><a name="index-bidiagonalization-of-real-matrices-1303"></a>
A general matrix A can be factorized by similarity
transformations into the form,
<pre class="example"> A = U B V^T
</pre>
<p class="noindent">where U and V are orthogonal matrices and B is a
N-by-N bidiagonal matrix with non-zero entries only on the
diagonal and superdiagonal. The size of <var>U</var> is M-by-N
and the size of <var>V</var> is N-by-N.
<div class="defun">
— Function: int <b>gsl_linalg_bidiag_decomp</b> (<var>gsl_matrix * A, gsl_vector * tau_U, gsl_vector * tau_V</var>)<var><a name="index-gsl_005flinalg_005fbidiag_005fdecomp-1304"></a></var><br>
<blockquote><p>This function factorizes the M-by-N matrix <var>A</var> into
bidiagonal form U B V^T. The diagonal and superdiagonal of the
matrix B are stored in the diagonal and superdiagonal of <var>A</var>.
The orthogonal matrices U 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></blockquote></div>
<div class="defun">
— Function: int <b>gsl_linalg_bidiag_unpack</b> (<var>const gsl_matrix * A, const gsl_vector * tau_U, gsl_matrix * U, const gsl_vector * tau_V, gsl_matrix * V, gsl_vector * diag, gsl_vector * superdiag</var>)<var><a name="index-gsl_005flinalg_005fbidiag_005funpack-1305"></a></var><br>
<blockquote><p>This function unpacks the bidiagonal decomposition of <var>A</var> given 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 M-by-N orthogonal matrix satisfying
U^T U = I for efficiency.
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_linalg_bidiag_unpack2</b> (<var>gsl_matrix * A, gsl_vector * tau_U, gsl_vector * tau_V, gsl_matrix * V</var>)<var><a name="index-gsl_005flinalg_005fbidiag_005funpack2-1306"></a></var><br>
<blockquote><p>This function unpacks the bidiagonal decomposition of <var>A</var> given 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></blockquote></div>
<div class="defun">
— Function: int <b>gsl_linalg_bidiag_unpack_B</b> (<var>const gsl_matrix * A, gsl_vector * diag, gsl_vector * superdiag</var>)<var><a name="index-gsl_005flinalg_005fbidiag_005funpack_005fB-1307"></a></var><br>
<blockquote><p>This function unpacks the diagonal and superdiagonal of the bidiagonal
decomposition of <var>A</var> given by <code>gsl_linalg_bidiag_decomp</code>, into
the diagonal vector <var>diag</var> and superdiagonal vector <var>superdiag</var>.
</p></blockquote></div>
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