--> 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
