The domain FullPartialFractionExpansion implements factor-free conversion of quotients to full partial fractions.
Our examples will all involve quotients of univariate polynomials with rational number coefficients.
Fx := FRAC UP(x, FRAC INT)
Fraction UnivariatePolynomial(x,Fraction Integer)
Type: Domain
Here is a simple-looking rational function.
f : Fx := 36 / (x**5-2*x**4-2*x**3+4*x**2+x-2)
36
----------------------------
5 4 3 2
x - 2x - 2x + 4x + x - 2
Type: Fraction UnivariatePolynomial(x,Fraction Integer)
We use fullPartialFraction to convert it to an object of type FullPartialFractionExpansion.
g := fullPartialFraction f
4 4 --+ - 3%A - 6
----- - ----- + > ---------
x - 2 x + 1 --+ 2
2 (x - %A)
%A - 1= 0
Type: FullPartialFractionExpansion(Fraction Integer,
UnivariatePolynomial(x,Fraction Integer))
Use a coercion to change it back into a quotient.
g :: Fx
36
----------------------------
5 4 3 2
x - 2x - 2x + 4x + x - 2
Type: Fraction UnivariatePolynomial(x,Fraction Integer)
Full partial fractions differentiate faster than rational functions.
g5 := D(g, 5)
480 480 --+ 2160%A + 4320
- -------- + -------- + > -------------
6 6 --+ 7
(x - 2) (x + 1) 2 (x - %A)
%A - 1= 0
Type: FullPartialFractionExpansion(Fraction Integer,
UnivariatePolynomial(x,Fraction Integer))
f5 := D(f, 5)
10 9 8 7 6
- 544320x + 4354560x - 14696640x + 28615680x - 40085280x
+
5 4 3 2
46656000x - 39411360x + 18247680x - 5870880x + 3317760x + 246240
/
20 19 18 17 16 15 14 13
x - 12x + 53x - 76x - 159x + 676x - 391x - 1596x
+
12 11 10 9 8 7 6 5
2527x + 1148x - 4977x + 1372x + 4907x - 3444x - 2381x + 2924x
+
4 3 2
276x - 1184x + 208x + 192x - 64
Type: Fraction UnivariatePolynomial(x,Fraction Integer)
We can check that the two forms represent the same function.
g5::Fx - f5
0
Type: Fraction UnivariatePolynomial(x,Fraction Integer)
Here are some examples that are more complicated.
f : Fx := (x**5 * (x-1)) / ((x**2 + x + 1)**2 * (x-2)**3)
6 5
x - x
-----------------------------------
7 6 5 3 2
x - 4x + 3x + 9x - 6x - 4x - 8
Type: Fraction UnivariatePolynomial(x,Fraction Integer)
g := fullPartialFraction f
1952 464 32 179 135
---- --- -- - ---- %A + ----
2401 343 49 --+ 2401 2401
------ + -------- + -------- + > ----------------
x - 2 2 3 --+ x - %A
(x - 2) (x - 2) 2
%A + %A + 1= 0
+
37 20
---- %A + ----
--+ 1029 1029
> --------------
--+ 2
2 (x - %A)
%A + %A + 1= 0
Type: FullPartialFractionExpansion(Fraction Integer,
UnivariatePolynomial(x,Fraction Integer))
g :: Fx - f
0
Type: Fraction UnivariatePolynomial(x,Fraction Integer)
f : Fx := (2*x**7-7*x**5+26*x**3+8*x) / (x**8-5*x**6+6*x**4+4*x**2-8)
7 5 3
2x - 7x + 26x + 8x
------------------------
8 6 4 2
x - 5x + 6x + 4x - 8
Type: Fraction UnivariatePolynomial(x,Fraction Integer)
g := fullPartialFraction f
1 1
- -
--+ 2 --+ 1 --+ 2
> ------ + > --------- + > ------
--+ x - %A --+ 3 --+ x - %A
2 2 (x - %A) 2
%A - 2= 0 %A - 2= 0 %A + 1= 0
Type: FullPartialFractionExpansion(Fraction Integer,
UnivariatePolynomial(x,Fraction Integer))
g :: Fx - f
0
Type: Fraction UnivariatePolynomial(x,Fraction Integer)
f:Fx := x**3 / (x**21 + 2*x**20 + 4*x**19 + 7*x**18 + 10*x**17 + 17*x**16 + 22*x**15 + 30*x**14 + 36*x**13 + 40*x**12 + 47*x**11 + 46*x**10 + 49*x**9 + 43*x**8 + 38*x**7 + 32*x**6 + 23*x**5 + 19*x**4 + 10*x**3 + 7*x**2 + 2*x + 1)
3
x
/
21 20 19 18 17 16 15 14 13 12
x + 2x + 4x + 7x + 10x + 17x + 22x + 30x + 36x + 40x
+
11 10 9 8 7 6 5 4 3 2
47x + 46x + 49x + 43x + 38x + 32x + 23x + 19x + 10x + 7x + 2x
+
1
Type: Fraction UnivariatePolynomial(x,Fraction Integer)
g := fullPartialFraction f
1 1 19
- %A - %A - --
--+ 2 --+ 9 27
> ------ + > ---------
--+ x - %A --+ x - %A
2 2
%A + 1= 0 %A + %A + 1= 0
+
1 1
-- %A - --
--+ 27 27
> ----------
--+ 2
2 (x - %A)
%A + %A + 1= 0
+
SIGMA
5 2
%A + %A + 1= 0
,
96556567040 4 420961732891 3 59101056149 2
- ------------ %A + ------------ %A - ------------ %A
912390759099 912390759099 912390759099
+
373545875923 529673492498
- ------------ %A + ------------
912390759099 912390759099
/
x - %A
+
SIGMA
5 2
%A + %A + 1= 0
,
5580868 4 2024443 3 4321919 2 84614 5070620
- -------- %A - -------- %A + -------- %A - ------- %A - --------
94070601 94070601 94070601 1542141 94070601
--------------------------------------------------------------------
2
(x - %A)
+
SIGMA
5 2
%A + %A + 1= 0
,
1610957 4 2763014 3 2016775 2 266953 4529359
-------- %A + -------- %A - -------- %A + -------- %A + --------
94070601 94070601 94070601 94070601 94070601
-------------------------------------------------------------------
3
(x - %A)
Type: FullPartialFractionExpansion(Fraction Integer,UnivariatePolynomial(x,Fraction Integer))
This verification takes much longer than the conversion to partial fractions.
g :: Fx - f
0
Type: Fraction UnivariatePolynomial(x,Fraction Integer)
Use PartialFraction for standard partial fraction decompositions.
For more information, see the paper [BS].
| [BS] | Bronstein, M and Salvy, B., Full Partial Fraction Decomposition of Rational Functions, Proceedings of ISSAC‘93, Kiev, ACM Press. |
See Also: