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<div class="section">
<div class="titlepage"><div><div><h5 class="title">
<a name="boost_numeric_bindings.reference.lapack.computational_routines.hgeqz"></a><a class="link" href="hgeqz.html" title="hgeqz">hgeqz</a>
</h5></div></div></div>
<a name="boost_numeric_bindings.reference.lapack.computational_routines.hgeqz.prototype"></a><h6>
<a name="id1062874"></a>
<a class="link" href="hgeqz.html#boost_numeric_bindings.reference.lapack.computational_routines.hgeqz.prototype">Prototype</a>
</h6>
<p>
There are two prototypes of <code class="computeroutput"><span class="identifier">hgeqz</span></code>
available, please see below.
</p>
<pre class="programlisting"><span class="identifier">hgeqz</span><span class="special">(</span> <span class="keyword">const</span> <span class="keyword">char</span> <span class="identifier">job</span><span class="special">,</span> <span class="keyword">const</span> <span class="keyword">char</span> <span class="identifier">compq</span><span class="special">,</span> <span class="keyword">const</span> <span class="keyword">char</span> <span class="identifier">compz</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">int_t</span> <span class="identifier">ilo</span><span class="special">,</span> <span class="identifier">MatrixH</span><span class="special">&</span> <span class="identifier">h</span><span class="special">,</span> <span class="identifier">MatrixT</span><span class="special">&</span> <span class="identifier">t</span><span class="special">,</span>
<span class="identifier">VectorALPHAR</span><span class="special">&</span> <span class="identifier">alphar</span><span class="special">,</span> <span class="identifier">VectorALPHAI</span><span class="special">&</span> <span class="identifier">alphai</span><span class="special">,</span> <span class="identifier">VectorBETA</span><span class="special">&</span> <span class="identifier">beta</span><span class="special">,</span>
<span class="identifier">MatrixQ</span><span class="special">&</span> <span class="identifier">q</span><span class="special">,</span> <span class="identifier">MatrixZ</span><span class="special">&</span> <span class="identifier">z</span> <span class="special">);</span>
</pre>
<p>
</p>
<p>
</p>
<pre class="programlisting"><span class="identifier">hgeqz</span><span class="special">(</span> <span class="keyword">const</span> <span class="keyword">char</span> <span class="identifier">job</span><span class="special">,</span> <span class="keyword">const</span> <span class="keyword">char</span> <span class="identifier">compq</span><span class="special">,</span> <span class="keyword">const</span> <span class="keyword">char</span> <span class="identifier">compz</span><span class="special">,</span>
<span class="keyword">const</span> <span class="identifier">int_t</span> <span class="identifier">ilo</span><span class="special">,</span> <span class="identifier">MatrixH</span><span class="special">&</span> <span class="identifier">h</span><span class="special">,</span> <span class="identifier">MatrixT</span><span class="special">&</span> <span class="identifier">t</span><span class="special">,</span>
<span class="identifier">VectorALPHA</span><span class="special">&</span> <span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">VectorBETA</span><span class="special">&</span> <span class="identifier">beta</span><span class="special">,</span> <span class="identifier">MatrixQ</span><span class="special">&</span> <span class="identifier">q</span><span class="special">,</span> <span class="identifier">MatrixZ</span><span class="special">&</span> <span class="identifier">z</span> <span class="special">);</span>
</pre>
<p>
</p>
<a name="boost_numeric_bindings.reference.lapack.computational_routines.hgeqz.description"></a><h6>
<a name="id1063442"></a>
<a class="link" href="hgeqz.html#boost_numeric_bindings.reference.lapack.computational_routines.hgeqz.description">Description</a>
</h6>
<p>
<code class="computeroutput"><span class="identifier">hgeqz</span></code> (short for $FRIENDLY_NAME)
provides a C++ interface to LAPACK routines SHGEQZ, DHGEQZ, CHGEQZ, and
ZHGEQZ. <code class="computeroutput"><span class="identifier">hgeqz</span></code> computes
the eigenvalues of a complex matrix pair (H,T), where H is an upper Hessenberg
matrix and T is upper triangular, using the single-shift QZ method. Matrix
pairs of this type are produced by the reduction to generalized upper
Hessenberg form of a complex matrix pair (A,B):
</p>
<p>
A = Q1<span class="bold"><strong>H*Z1</strong></span>*H, B = Q1<span class="bold"><strong>T*Z1</strong></span>*H,
</p>
<p>
as computed by ZGGHRD.
</p>
<p>
If JOB='S', then the Hessenberg-triangular pair (H,T) is also reduced
to generalized Schur form,
</p>
<p>
H = Q<span class="bold"><strong>S*Z</strong></span>*H, T = Q<span class="bold"><strong>P*Z</strong></span>*H,
</p>
<p>
where Q and Z are unitary matrices and S and P are upper triangular.
</p>
<p>
Optionally, the unitary matrix Q from the generalized Schur factorization
may be postmultiplied into an input matrix Q1, and the unitary matrix
Z may be postmultiplied into an input matrix Z1. If Q1 and Z1 are the
unitary matrices from ZGGHRD that reduced the matrix pair (A,B) to generalized
Hessenberg form, then the output matrices Q1*Q and Z1*Z are the unitary
factors from the generalized Schur factorization of (A,B):
</p>
<p>
A = (Q1*Q)<span class="bold"><strong>S</strong></span>(Z1<span class="bold"><strong>Z)</strong></span>*H,
B = (Q1*Q)<span class="bold"><strong>P</strong></span>(Z1<span class="bold"><strong>Z)</strong></span>*H.
</p>
<p>
To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,
of (A,B)) are computed as a pair of complex values (alpha,beta). If beta
is nonzero, lambda = alpha / beta is an eigenvalue of the generalized
nonsymmetric eigenvalue problem (GNEP) A*x = lambda*B*x and if alpha
is nonzero, mu = beta / alpha is an eigenvalue of the alternate form
of the GNEP mu*A*y = B*y. The values of alpha and beta for the i-th eigenvalue
can be read directly from the generalized Schur form: alpha = S(i,i),
beta = P(i,i).
</p>
<p>
Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized
Matrix Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973), pp.
241--256.
</p>
<p>
The selection of the LAPACK routine is done during compile-time, and
is determined by the type of values contained in type <code class="computeroutput"><span class="identifier">MatrixH</span></code>.
The type of values is obtained through the <code class="computeroutput"><span class="identifier">value_type</span></code>
meta-function <code class="computeroutput"><span class="keyword">typename</span> <span class="identifier">value_type</span><span class="special"><</span><span class="identifier">MatrixH</span><span class="special">>::</span><span class="identifier">type</span></code>. The dispatching table below illustrates
to which specific routine the code path will be generated.
</p>
<div class="table">
<a name="boost_numeric_bindings.reference.lapack.computational_routines.hgeqz.dispatching_of_hgeqz"></a><p class="title"><b>Table 1.333. Dispatching of hgeqz</b></p>
<div class="table-contents"><table class="table" summary="Dispatching of hgeqz">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Value type of MatrixH
</p>
</th>
<th>
<p>
LAPACK routine
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
<code class="computeroutput"><span class="keyword">float</span></code>
</p>
</td>
<td>
<p>
SHGEQZ
</p>
</td>
</tr>
<tr>
<td>
<p>
<code class="computeroutput"><span class="keyword">double</span></code>
</p>
</td>
<td>
<p>
DHGEQZ
</p>
</td>
</tr>
<tr>
<td>
<p>
<code class="computeroutput"><span class="identifier">complex</span><span class="special"><</span><span class="keyword">float</span><span class="special">></span></code>
</p>
</td>
<td>
<p>
CHGEQZ
</p>
</td>
</tr>
<tr>
<td>
<p>
<code class="computeroutput"><span class="identifier">complex</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span></code>
</p>
</td>
<td>
<p>
ZHGEQZ
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><a name="boost_numeric_bindings.reference.lapack.computational_routines.hgeqz.definition"></a><h6>
<a name="id1063894"></a>
<a class="link" href="hgeqz.html#boost_numeric_bindings.reference.lapack.computational_routines.hgeqz.definition">Definition</a>
</h6>
<p>
Defined in header <code class="computeroutput">boost/numeric/bindings/lapack/computational/hgeqz.hpp</code>.
</p>
<a name="boost_numeric_bindings.reference.lapack.computational_routines.hgeqz.parameters_or_requirements_on_types"></a><h6>
<a name="id1063934"></a>
<a class="link" href="hgeqz.html#boost_numeric_bindings.reference.lapack.computational_routines.hgeqz.parameters_or_requirements_on_types">Parameters
or Requirements on Types</a>
</h6>
<div class="variablelist">
<p class="title"><b>Parameters</b></p>
<dl>
<dt><span class="term">MatrixA</span></dt>
<dd><p>
The definition of term 1
</p></dd>
<dt><span class="term">MatrixB</span></dt>
<dd><p>
The definition of term 2
</p></dd>
<dt><span class="term">MatrixC</span></dt>
<dd>
<p>
The definition of term 3.
</p>
<p>
Definitions may contain paragraphs.
</p>
</dd>
</dl>
</div>
<a name="boost_numeric_bindings.reference.lapack.computational_routines.hgeqz.complexity"></a><h6>
<a name="id1064022"></a>
<a class="link" href="hgeqz.html#boost_numeric_bindings.reference.lapack.computational_routines.hgeqz.complexity">Complexity</a>
</h6>
<a name="boost_numeric_bindings.reference.lapack.computational_routines.hgeqz.example"></a><h6>
<a name="id1064047"></a>
<a class="link" href="hgeqz.html#boost_numeric_bindings.reference.lapack.computational_routines.hgeqz.example">Example</a>
</h6>
<p>
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">numeric</span><span class="special">/</span><span class="identifier">bindings</span><span class="special">/</span><span class="identifier">lapack</span><span class="special">/</span><span class="identifier">computational</span><span class="special">/</span><span class="identifier">hgeqz</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">bindings</span><span class="special">;</span>
<span class="identifier">lapack</span><span class="special">::</span><span class="identifier">hgeqz</span><span class="special">(</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">z</span> <span class="special">);</span>
</pre>
<p>
</p>
<p>
this will output
</p>
<p>
</p>
<pre class="programlisting"><span class="special">[</span><span class="number">5</span><span class="special">]</span> <span class="number">0</span> <span class="number">1</span> <span class="number">2</span> <span class="number">3</span> <span class="number">4</span> <span class="number">5</span>
</pre>
<p>
</p>
<a name="boost_numeric_bindings.reference.lapack.computational_routines.hgeqz.notes"></a><h6>
<a name="id1064332"></a>
<a class="link" href="hgeqz.html#boost_numeric_bindings.reference.lapack.computational_routines.hgeqz.notes">Notes</a>
</h6>
<a name="boost_numeric_bindings.reference.lapack.computational_routines.hgeqz.see_also"></a><h6>
<a name="id1064358"></a>
<a class="link" href="hgeqz.html#boost_numeric_bindings.reference.lapack.computational_routines.hgeqz.see_also">See
Also</a>
</h6>
<div class="itemizedlist"><ul class="itemizedlist" type="disc"><li class="listitem">
Originating Fortran source files <a href="http://www.netlib.org/lapack/single/shgeqz.f" target="_top">shgeqz.f</a>,
<a href="http://www.netlib.org/lapack/double/dhgeqz.f" target="_top">dhgeqz.f</a>,
<a href="http://www.netlib.org/lapack/complex/chgeqz.f" target="_top">chgeqz.f</a>,
and <a href="http://www.netlib.org/lapack/complex16/zhgeqz.f" target="_top">zhgeqz.f</a>
at Netlib.
</li></ul></div>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright © 2002 -2009 Rutger ter Borg, Krešimir Fresl, Thomas Klimpel,
Toon Knapen, Karl Meerbergen<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
</p>
</div></td>
</tr></table>
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