partfrac()

            
--> partfrac
 
 Partial fraction decomposition of a rational in C

 SYNTAX
 ------
               partfrac    // Displays this help
  I          = partfrac(r)
 [I, R]      = partfrac(r)
 [I, R, F]   = partfrac(r)
 [I, R, F, T]= partfrac(r)

 PARAMETERS
 ----------
 r : Single rational = polynomial fraction with real or complex coefficients.
 I : Integer part = non-fractional part of f (single polynomial).
 R : Remainder of r: rational with degree(numerator) < degree(denominator)

 F : (3xN) matrix describing the N terms of the r's decomposition:
     F(1,:): real or complex coefficient of numerators.
     F(2,:): real or complex value of the considered poles.
     F(3,:): integer>0: powers m of the denominators (x-pole)^m
     F is such that sum( F(1,:)./ ((x-F(2,:)).^F(3,:)) ) - r == 0.

 T : 3-rows column of Texts. write(%io(2),t) displays i+d in a comprehensive
     way. The format of coefficients is set with format().

 DESCRIPTION
 -----------
 From a single polynomial fraction r = p/q with coprime polynomials p and q, 
 partfrac(f) extracts 
  * the non fractional part I of f: polynomial such that 
    0 <= degree(I) <= degree(p) - degree(q): I = p - modulo(p,q)
  * the fractional part of r, or remainder R of p/q : R = modulo(p,q)/q
    such that degree(r.num))< degree(q) and  r = I + R
  * the rational decomposition of R, aka partial fraction decomposition of r:
    The 'vector' of elementary rationals c/(x-pole)^m where c and poles are 
    decimal or complex numbers and m are the multiplicities of poles:
    c = F(1,:), pole = F(2,:), m = d(3,:)

 In addition, partfrac() may return as text the literal expression T of the 
 whole decomposition of r. This form shows the factorized forms of denominators
 where poles that are multiple appear with their powers. write(%io(2), T) may 
 be used to display it.

 DEPENDENCY: If F or/and T is expected, polyroots() is required (=>See also)

 NOTE: pfss(r) yields a list mixing the non-fractional part (as last element)
  with the decomposition, and where terms of the decomposition are not always
  elementary: their denominator may be of order 2 even for real poles. See the
  example.

 REFERENCE
 ---------
  Comments, scoring and bug reports are welcome on
  http://fileexchange.scilab.org/toolboxes/451000#new_comment

 SEE ALSO
 --------
  pfss      : Partial (uncomplete) fraction decomposition of a linear system
  modulo    : Remainder after polynomial division
  polyroots : Multiplicities and values of polynomial multiple roots:
              http://fileexchange.scilab.org/toolboxes/362000
  pdiv_inc  : Polynomial division with increasing terms powers:
              http://fileexchange.scilab.org/toolboxes/449000

 EXAMPLE
 -------
 x = poly(0,"x");
 r = (3-x+x^2-4*x^3+2*x^4) / ((x-1)*(x-2)^2)
 [I, R, F, T] = partfrac(r); 
 I, R, F, T
 D = F(1,:)./((x-F(2,:)).^F(3,:))
 clean(R-sum(D))
 // Comparison with pfss():
 list2vec(pfss(r)).' 


 RESULTS
 -------

-->  x = poly(0,"x");

-->  r = (3-x+x^2-4*x^3+2*x^4) / ((x-1)*(x-2)^2)
 r  = 

   3 -x +x² -4x³ +2x⁴  
   ------------------  
    -4 +8x -5x² +x³    

-->  [I, R, F, T] = partfrac(r);

-->  I, R, F, T
 I  = 
  6 +2x

 R  = 
    27 -41x +15x²   
   ---------------  
   -4 +8x -5x² +x³  

 F  = 
   14.   5.   1.
   2.    2.   1.
   1.    2.   1.

 T  = 
  "         14       5        1  "
  "6 +2x + ---- + -------- + ----"
  "        -2+x   (-2+x)^2   -1+x"

-->  D = F(1,:)./((x-F(2,:)).^F(3,:))
 D  =

    14        5        1    
   -----  ---------  -----  
   -2 +x  4 -4x +x²  -1 +x  

-->  clean(R-sum(D))
 ans  =
   0  
   -  
   1  

-->  // Comparison with pfss():

-->  list2vec(pfss(r)).' 
 ans  =

   -23 +14x     1    6 +2x  
   ---------  -----  -----  
   4 -4x +x²  -1 +x    1