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<!DOCTYPE MAN SYSTEM "../../manrev.dtd">
<MAN>
  <LANGUAGE>eng</LANGUAGE>
  <TITLE>sylmX</TITLE>
  <TYPE>Scilab Function</TYPE>
  <DATE>April 1993</DATE>
  <SHORT_DESCRIPTION name="sylm"> Sylvester matrix</SHORT_DESCRIPTION>
  <CALLING_SEQUENCE>
    <CALLING_SEQUENCE_ITEM>[S]=sylm(a,b)  </CALLING_SEQUENCE_ITEM>
  </CALLING_SEQUENCE>
  <PARAM>
    <PARAM_INDENT>
      <PARAM_ITEM>
        <PARAM_NAME>a,b</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: two polynomials</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>S</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: matrix</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
    </PARAM_INDENT>
  </PARAM>
  <DESCRIPTION>
    <P><VERB>sylm(a,b)</VERB> gives the Sylvester matrix associated to polynomials
    <VERB>a</VERB> and <VERB>b</VERB>, i.e. the matrix <VERB>S</VERB> such that:</P>
    <P><VERB>coeff( a*x + b*y )' = S * [coeff(x)';coeff(y)']</VERB>.</P>
    <P>
    Dimension of <VERB>S</VERB> is equal to <VERB>degree(a)+degree(b)</VERB>.</P>
    <P>
    If <VERB>a</VERB> and <VERB>b</VERB> are coprime polynomials then</P>
    <P><VERB>rank(sylm(a,b))=degree(a)+degree(b))</VERB> and the instructions</P>
    <VERBATIM>
<![CDATA[
  u = sylm(a,b) \ eye(na+nb,1)
  x = poly(u(1:nb),'z','coeff')
  y = poly(u(nb+1:na+nb),'z','coeff')
   ]]>
    </VERBATIM>
    <P>
    compute Bezout factors <VERB>x</VERB> and <VERB>y</VERB> of minimal degree 
    such that <VERB>a*x+b*y = 1</VERB></P>
  </DESCRIPTION>
</MAN>
