5ae334c01443bffb614e3035ab6c56d9e9f82063
[scilab.git] / scilab / modules / linear_algebra / help / en_US / pencil / quaskro.xml
1 <?xml version="1.0" encoding="UTF-8"?>
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16 <refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:svg="http://www.w3.org/2000/svg" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook" xmlns:scilab="http://www.scilab.org" xml:lang="en" xml:id="quaskro">
17     <refnamediv>
18         <refname>quaskro</refname>
19         <refpurpose>quasi-Kronecker form</refpurpose>
20     </refnamediv>
21     <refsynopsisdiv>
22         <title>Syntax</title>
23         <synopsis>[Q,Z,Qd,Zd,numbeps,numbeta]=quaskro(F)
24             [Q,Z,Qd,Zd,numbeps,numbeta]=quaskro(E,A)
25             [Q,Z,Qd,Zd,numbeps,numbeta]=quaskro(F,tol)
26             [Q,Z,Qd,Zd,numbeps,numbeta]=quaskro(E,A,tol)
27         </synopsis>
28     </refsynopsisdiv>
29     <refsection>
30         <title>Arguments</title>
31         <variablelist>
32             <varlistentry>
33                 <term>F</term>
34                 <listitem>
35                     <para>
36                         real matrix pencil <literal>F=s*E-A</literal>  (<literal>s=poly(0,'s')</literal>)
37                     </para>
38                 </listitem>
39             </varlistentry>
40             <varlistentry>
41                 <term>E,A</term>
42                 <listitem>
43                     <para>two real matrices of same dimensions</para>
44                 </listitem>
45             </varlistentry>
46             <varlistentry>
47                 <term>tol</term>
48                 <listitem>
49                     <para>a real number (tolerance, default value=1.d-10)</para>
50                 </listitem>
51             </varlistentry>
52             <varlistentry>
53                 <term>Q,Z</term>
54                 <listitem>
55                     <para>two square orthogonal matrices</para>
56                 </listitem>
57             </varlistentry>
58             <varlistentry>
59                 <term>Qd,Zd</term>
60                 <listitem>
61                     <para>two vectors of integers</para>
62                 </listitem>
63             </varlistentry>
64             <varlistentry>
65                 <term>numbeps</term>
66                 <listitem>
67                     <para>vector of integers</para>
68                 </listitem>
69             </varlistentry>
70         </variablelist>
71     </refsection>
72     <refsection>
73         <title>Description</title>
74         <para>
75             Quasi-Kronecker form of matrix pencil: <literal>quaskro</literal> computes two
76             orthogonal matrices <literal>Q, Z</literal> which put the pencil <literal>F=s*E -A</literal> into
77             upper-triangular form:
78         </para>
79         <programlisting role=""><![CDATA[ 
80            | sE(eps)-A(eps) |        X       |      X     |
81            |----------------|----------------|------------|
82            |        O       | sE(inf)-A(inf) |      X     |
83 Q(sE-A)Z = |=================================|============|
84            |                                 |            |
85            |                O                | sE(r)-A(r) |
86  ]]></programlisting>
87         <para>
88             The dimensions of the blocks are given by:
89         </para>
90         <para>
91             <literal>eps=Qd(1) x Zd(1)</literal>, <literal>inf=Qd(2) x Zd(2)</literal>,
92             <literal>r = Qd(3) x Zd(3)</literal>
93         </para>
94         <para>
95             The <literal>inf</literal> block contains the infinite modes of
96             the pencil.
97         </para>
98         <para>
99             The <literal>f</literal> block contains the finite modes of
100             the pencil
101         </para>
102         <para>
103             The structure of epsilon blocks are given by:
104         </para>
105         <para>
106             <literal>numbeps(1)</literal> = <literal>#</literal> of eps blocks of size 0 x 1
107         </para>
108         <para>
109             <literal>numbeps(2)</literal> = <literal>#</literal> of eps blocks of size 1 x 2
110         </para>
111         <para>
112             <literal>numbeps(3)</literal> = <literal>#</literal> of eps blocks of size 2 x 3     etc...
113         </para>
114         <para>
115             The complete (four blocks) Kronecker form is given by
116             the function <literal>kroneck</literal> which calls <literal>quaskro</literal> on
117             the (pertransposed) pencil <literal>sE(r)-A(r)</literal>.
118         </para>
119         <para>
120             The code is taken from T. Beelen
121         </para>
122     </refsection>
123     <refsection role="see also">
124         <title>See Also</title>
125         <simplelist type="inline">
126             <member>
127                 <link linkend="kroneck">kroneck</link>
128             </member>
129             <member>
130                 <link linkend="gschur">gschur</link>
131             </member>
132             <member>
133                 <link linkend="gspec">gspec</link>
134             </member>
135         </simplelist>
136     </refsection>
137 </refentry>