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partfrac()

Partial fraction decomposition of a rational in the C set
(130 downloads for this version - 310 downloads for all versions)
Details
Version
1.1
Author
Samuel Gougeon
Maintainer
Samuel Gougeon
Category
License
Supported Scilab Version
5.5
Creation Date
August 27, 2015
Description
            --> partfrac

 Partial fraction decomposition of a rational in the C set

 SYNTAX
 ------
            partfrac    // Displays this help
  i       = partfrac(f)
 [i,r]    = partfrac(f)
 [i,r,d]  = partfrac(f)
 [i,r,d,t]= partfrac(f)

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

 d : (3xN) matrix describing the N terms of the r's decomposition:
     d(1,:): real or complex coefficient of numerators.
     d(2,:): real or complex value of the considered poles.
     d(3,:): integer>0: powers m of the denominators (x-pole)^m
     d is such that sum( d(1,:)./ ((x-d(2,:)).^real(d(3,:))) ) - r == 0.
     NOTE: real() must be applied to d(3,:) only because d is complex-encoded.
           See http://bugzilla.scilab.org/14116

 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 f = 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 f, or remainder r of p/q : r = modulo(p,q)/q
    such that degree(numer(r))< degree(q) and  f = i + r
  * the rational decomposition of r, aka partial fraction decomposition of f:
    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 = d(1,:), pole = d(2,:), m = d(3,:)

 In addition, partfrac() may return as text the literal expression t of the
 whole decomposition of f. 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 d or/and t is expected, polyroots() is required (> See also)

 NOTE: pfss(f) 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.

 FEEDBACK
 ---------
  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");
 f = (3-x+x^2-4*x^3+2*x^4) / ((x-1)*(x-2)^2)
 [E, R, F, T] = partfrac(f);
 E, R, F, T
 D = F(1,:)./((x-F(2,:)).^real(F(3,:)))
 clean(R-sum(D))
 // Comparison with pfss():
 list2vec(pfss(f)).'

 RESULTS
 -------

-->x = poly(0,"x");
 
--> f = (3-x+x^2-4*x^3+2*x^4) / ((x-1)*(x-2)^2)
 f  =
             2    3    4  
    3 - x + x - 4x + 2x   
    -------------------   
                 2   3    
    - 4 + 8x - 5x + x     
 
--> [E, R, F, T] = partfrac(f);
 
--> E,R,F,T
 E  =
    6 + 2x   

 R  =
                  2   
    27 - 41x + 15x    
    --------------    
               2   3  
  - 4 + 8x - 5x + x   

 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,:)).^real(F(3,:)))

 D  =
!     14            5             1     !
!  ---------  --------------  --------- !
!                          2            !
!  -2 +  1x    4 - 4x +  1x   -1 +  1x  !
 
--> clean(R-sum(D))
 ans  =                                                       
   0
   -
   1
 
--> // Comparison with pfss():
 
--> list2vec(pfss(f)).'
 ans  =

   - 23 + 14x        1       6 + 2x   
    ----------     -----     ------   
              2                       
    4 - 4x + x   - 1 + x       1      
            
Files (1)
[7.61 kB]
Miscellaneous file
File defining the function partfrac(), help contents and examples

* MAKE partfrac() AVAILABLE in EVERY SESSION: Click here to see HOW TO

* Get help: Enter "partfrac" without parameters

CHANGES 1.0 => 2.0
------------------
 * The format of the returned decomposition was a vector of rationals.
   It is now a 3-row matrix specifying for each term, the numerator 
   (coefficient), the value of the considered pole, and the 'pole exponent'.
 * References to pfss() added.
 * Help updated.

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