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<!DOCTYPE MAN SYSTEM "../../manrev.dtd">
<MAN>
  <LANGUAGE>eng</LANGUAGE>
  <TITLE>SFACT</TITLE>
  <TYPE>Scilab Function</TYPE>
  <DATE>April 1993</DATE>
  <SHORT_DESCRIPTION name="sfact"> discrete time spectral factorization</SHORT_DESCRIPTION>
  <CALLING_SEQUENCE>
    <CALLING_SEQUENCE_ITEM>F=sfact(P)  </CALLING_SEQUENCE_ITEM>
  </CALLING_SEQUENCE>
  <PARAM>
    <PARAM_INDENT>
      <PARAM_ITEM>
        <PARAM_NAME>P</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: real polynomial matrix</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
    </PARAM_INDENT>
  </PARAM>
  <DESCRIPTION>
    <P>
    Finds <VERB>F</VERB>, a spectral factor of
    <VERB>P</VERB>. <VERB>P</VERB> is a polynomial matrix such that
    each root of <VERB>P</VERB> has a mirror image w.r.t the unit
    circle. Problem is singular if a root is on the unit circle.</P>
    <P><VERB>sfact(P)</VERB> returns a polynomial matrix
    <VERB>F(z)</VERB> which is antistable and such that</P>
    <P>
      <VERB>P = F(z)* F(1/z) *z^n</VERB>
    </P>
    <P>
    For scalar polynomials a specific algorithm is implemented.
    Algorithms are adapted from Kucera's book.</P>
  </DESCRIPTION>
  <EXAMPLE>
<![CDATA[
//Simple polynomial example
z=poly(0,'z');
p=(z-1/2)*(2-z)
w=sfact(p);
w*numer(horner(w,1/z))
//matrix example
F1=[z-1/2,z+1/2,z^2+2;1,z,-z;z^3+2*z,z,1/2-z];
P=F1*gtild(F1,'d');  //P is symmetric
F=sfact(P)    
roots(det(P))  
roots(det(gtild(F,'d')))  //The stable roots
roots(det(F))             //The antistable roots
clean(P-F*gtild(F,'d'))
//Example of continuous time use
s=poly(0,'s');
p=-3*(s+(1+%i))*(s+(1-%i))*(s+0.5)*(s-0.5)*(s-(1+%i))*(s-(1-%i));p=real(p);
//p(s) = polynomial in s^2 , looks for stable f such that p=f(s)*f(-s) 
w=horner(p,(1-s)/(1+s));  // bilinear transform w=p((1-s)/(1+s))
wn=numer(w);              //take the numerator
fn=sfact(wn);f=numer(horner(fn,(1-s)/(s+1))); //Factor and back transform
f=f/sqrt(horner(f*gtild(f,'c'),0));f=f*sqrt(horner(p,0));      //normalization
roots(f)    //f is stable
clean(f*gtild(f,'c')-p)    //f(s)*f(-s) is p(s) 
 ]]>
  </EXAMPLE>
  <SEE_ALSO>
    <SEE_ALSO_ITEM>
      <LINK>gtild</LINK>
    </SEE_ALSO_ITEM>
    <SEE_ALSO_ITEM>
      <LINK>fspecg</LINK>
    </SEE_ALSO_ITEM>
  </SEE_ALSO>
</MAN>
