--> 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 =
2 3 4
3 - x + x - 4x + 2x
-------------------
2 3
- 4 + 8x - 5x + x
--> [I, R, F, T] = partfrac(r);
--> I,R,F,T
I =
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,:)).^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
