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partfrac()

Partial fraction decomposition of a rational in the C set
(1023 downloads for this version - 2852 downloads for all versions)
Details
Version
1.0
A more recent valid version with binaries for Scilab exists: 1.3.1
Author
Samuel Gougeon
Maintainer
Samuel Gougeon
Category
License
Supported Scilab Version
5.5
Creation Date
August 22, 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 : Vector of rationals: rational Decomposition of r, in the complex set C,
     such that sum(d)-r == 0. Denominators are all of degree 1.
 t : 3-rows column of Texts. write(%io(2),t) displays i+d in a comprehensive
     way, expliciting the multiplicity of poles in d. Coefficients format 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:
    vector d of elementary rationals c/(x-pole)^m where c and poles are
    decimal or complex numbers and m are the multiplicities of poles.

 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, while in d
 denominators (x-pole)^(m>1) are developed. write(%io(2), t) may be used to
 display it.

 DEPENDENCY: If d or/and t is expected, polyroots() is required (>See also)

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

 SEE ALSO
 --------
  modulo    : remainder after polynomial division
  polyroots : http://fileexchange.scilab.org/toolboxes/362000
  pdiv_inc  : 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
 write(%io(2),T)
 clean(R-sum(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 +  1x    4 - 4x +  1x   -1 +  1x  !

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

--> write(%io(2),T)
          14       5        1  
6 + 2x + ---- + -------- + ----
         -2+x   (-2+x)^2   -1+x

--> clean(R-sum(F))
                                                          
   0                            
   -
   1
            
Files (2)
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Screenshot
Screenshot of example for thumbnail
[6.69 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
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