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<!DOCTYPE MAN SYSTEM "../../manrev.dtd">
<MAN>
  <LANGUAGE>eng</LANGUAGE>
  <TITLE>kroneck</TITLE>
  <TYPE>Scilab Function</TYPE>
  <DATE>April 1993</DATE>
  <SHORT_DESCRIPTION name="kroneck"> Kronecker form of matrix pencil</SHORT_DESCRIPTION>
  <CALLING_SEQUENCE>
    <CALLING_SEQUENCE_ITEM>[Q,Z,Qd,Zd,numbeps,numbeta]=kroneck(F)  </CALLING_SEQUENCE_ITEM>
    <CALLING_SEQUENCE_ITEM>[Q,Z,Qd,Zd,numbeps,numbeta]=kroneck(E,A)  </CALLING_SEQUENCE_ITEM>
  </CALLING_SEQUENCE>
  <PARAM>
    <PARAM_INDENT>
      <PARAM_ITEM>
        <PARAM_NAME>F</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: real matrix pencil <VERB>F=s*E-A</VERB></SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>E,A</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: two real matrices of same dimensions</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>Q,Z</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: two square orthogonal matrices</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>Qd,Zd</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: two vectors of integers</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>numbeps,numeta</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: two vectors of integers</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
    </PARAM_INDENT>
  </PARAM>
  <DESCRIPTION>
    <P>
    Kronecker form of matrix pencil: <VERB>kroneck</VERB> computes two
    orthogonal matrices <VERB>Q, Z</VERB> which put the pencil <VERB>F=s*E -A</VERB> into
    upper-triangular form:</P>
    <VERBATIM>
<![CDATA[

           | sE(eps)-A(eps) |        X       |      X     |      X        |
           |----------------|----------------|------------|---------------|
           |        O       | sE(inf)-A(inf) |      X     |      X        |
Q(sE-A)Z = |---------------------------------|----------------------------|
           |                |                |            |               |
           |        0       |       0        | sE(f)-A(f) |      X        |
           |--------------------------------------------------------------|
           |                |                |            |               |
           |        0       |       0        |      0     | sE(eta)-A(eta)|

   ]]>
    </VERBATIM>
    <P>
    The dimensions of the four blocks are given by:</P>
    <P><VERB>eps=Qd(1) x Zd(1)</VERB>, <VERB>inf=Qd(2) x Zd(2)</VERB>,
    <VERB>f = Qd(3) x Zd(3)</VERB>, <VERB>eta=Qd(4)xZd(4)</VERB></P>
    <P>
    The <VERB>inf</VERB> block contains the infinite modes of
    the pencil.</P>
    <P>
    The <VERB>f</VERB> block contains the finite modes of
    the pencil</P>
    <P>
    The structure of epsilon and eta blocks are given by:</P>
    <P><VERB>numbeps(1)</VERB> = # of eps blocks of size 0 x 1</P>
    <P><VERB>numbeps(2)</VERB> = # of eps blocks of size 1 x 2</P>
    <P><VERB>numbeps(3)</VERB> = # of eps blocks of size 2 x 3     etc...</P>
    <P><VERB>numbeta(1)</VERB> = # of eta blocks of size 1 x 0</P>
    <P><VERB>numbeta(2)</VERB> = # of eta blocks of size 2 x 1</P>
    <P><VERB>numbeta(3)</VERB> = # of eta blocks of size 3 x 2     etc...</P>
    <P>
    The code is taken from T. Beelen (Slicot-WGS group).</P>
  </DESCRIPTION>
  <EXAMPLE>
<![CDATA[
F=randpencil([1,1,2],[2,3],[-1,3,1],[0,3]);
Q=rand(17,17);Z=rand(18,18);F=Q*F*Z;
//random pencil with eps1=1,eps2=1,eps3=1; 2 J-blocks @ infty 
//with dimensions 2 and 3
//3 finite eigenvalues at -1,3,1 and eta1=0,eta2=3
[Q,Z,Qd,Zd,numbeps,numbeta]=kroneck(F);
[Qd(1),Zd(1)]    //eps. part is sum(epsi) x (sum(epsi) + number of epsi) 
[Qd(2),Zd(2)]    //infinity part
[Qd(3),Zd(3)]    //finite part
[Qd(4),Zd(4)]    //eta part is (sum(etai) + number(eta1)) x sum(etai)
numbeps
numbeta
 ]]>
  </EXAMPLE>
  <SEE_ALSO>
    <SEE_ALSO_ITEM>
      <LINK>gschur</LINK>
    </SEE_ALSO_ITEM>
    <SEE_ALSO_ITEM>
      <LINK>gspec</LINK>
    </SEE_ALSO_ITEM>
    <SEE_ALSO_ITEM>
      <LINK>systmat</LINK>
    </SEE_ALSO_ITEM>
    <SEE_ALSO_ITEM>
      <LINK>pencan</LINK>
    </SEE_ALSO_ITEM>
    <SEE_ALSO_ITEM>
      <LINK>randpencil</LINK>
    </SEE_ALSO_ITEM>
    <SEE_ALSO_ITEM>
      <LINK>trzeros</LINK>
    </SEE_ALSO_ITEM>
  </SEE_ALSO>
</MAN>
