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<!DOCTYPE MAN SYSTEM "../../manrev.dtd">
<MAN>
  <LANGUAGE>eng</LANGUAGE>
  <TITLE>fstair</TITLE>
  <TYPE>Scilab Function</TYPE>
  <DATE>April 1993</DATE>
  <SHORT_DESCRIPTION name="fstair"> computes  pencil  column echelon form by qz transformations</SHORT_DESCRIPTION>
  <CALLING_SEQUENCE>
    <CALLING_SEQUENCE_ITEM>[AE,EE,QE,ZE,blcks,muk,nuk,muk0,nuk0,mnei]=fstair(A,E,Q,Z,stair,rk,tol)  </CALLING_SEQUENCE_ITEM>
  </CALLING_SEQUENCE>
  <PARAM>
    <PARAM_INDENT>
      <PARAM_ITEM>
        <PARAM_NAME>A</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: m x n matrix with real  entries.</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>tol</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: real positive scalar.</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>E</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: column echelon form matrix</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>Q</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: m x m unitary matrix</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>Z</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: n x n unitary matrix</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>stair</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: vector of indexes (see ereduc)</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>rk</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: integer, estimated rank of the matrix</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>AE</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: m x n matrix with real  entries.</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>EE</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: column echelon form matrix</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>QE</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: m x m unitary matrix</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>ZE</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: n x n unitary matrix</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>nblcks</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>:is the number of submatrices having full row rank &gt;= 0  detected in matrix  <VERB>A</VERB>.</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>muk:  </PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>integer array of dimension (n). Contains the column dimensions mu(k)  (k=1,...,nblcks) of the submatrices having full column  rank in the pencil sE(eps)-A(eps)</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>nuk:  </PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>integer array of dimension (m+1). Contains the row dimensions nu(k)  (k=1,...,nblcks) of the submatrices having full row  rank in the pencil sE(eps)-A(eps)</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>muk0:  </PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>integer array of dimension (n). Contains the column dimensions mu(k)  (k=1,...,nblcks) of the submatrices having full column  rank in the pencil sE(eps,inf)-A(eps,inf)</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>nuk:  </PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>integer array of dimension (m+1). Contains the row dimensions nu(k)  (k=1,...,nblcks) of the submatrices having full row  rank in the pencil sE(eps,inf)-A(eps,inf)</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>mnei:  </PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>integer array of dimension (4). mnei(1) = row dimension of sE(eps)-A(eps)</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
    </PARAM_INDENT>
  </PARAM>
  <DESCRIPTION>
    <P>
    Given a pencil <VERB>sE-A</VERB> where matrix <VERB>E</VERB> is in column echelon form the
    function  <VERB>fstair</VERB> computes according to the wishes of the user a
    unitary transformed pencil <VERB>QE(sEE-AE)ZE</VERB> which is more or less similar
    to the generalized Schur form of the pencil <VERB>sE-A</VERB>.
    The function  yields also part of the Kronecker structure of
    the given pencil.</P>
    <P><VERB>Q,Z</VERB> are the unitary matrices used to compute the pencil where E
    is in column echelon form (see ereduc)</P>
  </DESCRIPTION>
  <SEE_ALSO>
    <SEE_ALSO_ITEM>
      <LINK>quaskro</LINK>
    </SEE_ALSO_ITEM>
    <SEE_ALSO_ITEM>
      <LINK>ereduc</LINK>
    </SEE_ALSO_ITEM>
  </SEE_ALSO>
  <AUTHOR>Th.G.J. Beelen (Philips Glass Eindhoven). SLICOT</AUTHOR>
</MAN>
