--> 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
