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Reference for algorithms: https://www.hindawi.com/journals/jam/2014/895036/
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In algebra, the partial fraction decomposition or partial fraction expansion of a rational function (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.[1]
The importance of the partial fraction decomposition lies in the fact that it provides an algorithm for computing the antiderivative of a rational function[2], computing explicitly their Taylor series expansions, computing their inverse Z-transform, their inverse Laplace transform, and many other applications. The concept was discovered in 1702 by both Johann Bernoulli and Gottfried Leibniz independently.[3]
In symbols, one can use partial fraction expansion to change a rational fraction in the form
where and are polynomials, into an expression of the form
where:
- the denominators of each fraction is a power of an irreducible (not factorable into polynomials of positive degree) polynomial and
- the numerators are polynomials of smaller degree than this irreducible polynomial .
As factorization of polynomials may be difficult, a coarser decomposition might be preferred, which consists of replacing factorization by square-free factorization. This amounts to replace "irreducible" by "square-free" in the preceding description of the outcome.
Statement of the theorem[edit]
Theorem — If and are nonzero polynomials over a field , writing as a product of powers of distinct irreducible polynomials
there are (unique) polynomials and with such that
If , then .
When the field is the complex numbers, each must have degree 1 by the fundamental theorem of algebra and the numerators are constant.
When is the real numbers, some of the might be quadratic without real roots, so, in the partial fraction decomposition, a quotient of a linear polynomial by a power of a quadratic might occur.
In the preceding theorem, one may replace "distinct irreducible polynomials" by "pairwise coprime polynomials that are coprime with their derivative". For example, the may be the factors of the square-free factorization of . When is the field of the rational numbers, as it is typically the case in computer algebra, this allows to replace factorization by greatest common divisor to compute the partial fraction decomposition.
Proof and efficient computation[edit]
Given a rational function the polynomial long division allows to write , where . This allows to write
as a sum of a polynomial and a proper fraction . In other words, the fraction satisfies that .
If with and relatively prime, then by Bézout's identity for polynomials there are polynomials and such that
These polynomials can be computed using the extended Euclid's algorithm.
Therefore,
- .
A further application of polynomial long division, dividing by , allows to write with . Therefore,
- ,
where and by degree considerations and .
This allows writing
where the last two fractions are proper and have denominators of smaller degree than .
Repeating the same argument with each of the resulting fractions eventually each denominator is only a power of an irreducible polynomial. In this case the argument above cannot be applied any further because any factorization of a power of an irreducible polynomial into two factors of positive degree would give factors that are not relatively prime.
Finally, for a fraction
with irreducible and , repeated application of the polynomial long division, dividing , and the subsequent quotients obtained, by yields
where and for This allows to write
- ,
which results in the final form
as a sum of simple fractions.
Special cases[edit]
The general algorithm to compute partial fraction decomposition involves multiple uses of the extended Euclid's algorithm and in particular polynomial long division. A number of alternative procedures are available when the denominator has a factorization with few irreducible factors, irreducible factors with small degree, and each irreducible factor raised to small powers.
Heaviside cover-up method[edit]
One such method is Heaviside cover-up method, which is applicable when the denominator is a product of different polynomials of degree 1.
The partial fraction decomposition for a proper fraction would have the form
The cover-up method consists in computing by multiplying the whole equation by , cancelling this, and evaluating the resulting equation at . This yields
- .
Example[edit]
For covering and evaluating at yields
and covering and evaluating at yields
- .
Therefore
- .
Application to symbolic integration[edit]
For the purpose of symbolic integration, the preceding result may be refined into
Theorem — Let f and g be nonzero polynomials over a field K. Write g as a product of powers of pairwise coprime polynomials which have no multiple root in an algebraically closed field:
There are (unique) polynomials b and cij with deg cij < deg pi such that
where denotes the derivative of
This reduces the computation of the antiderivative of a rational function to the integration of the last sum, which is called the logarithmic part, because its antiderivative is a linear combination of logarithms. In fact, we have
There are various methods to compute above decomposition. The one that is the simplest to describe is probably the so-called Hermite's method. As the degree of cij is bounded by the degree of pi, and the degree of b is the difference of the degrees of f and g (if this difference is non negative; otherwise, b=0), one may write these unknowns polynomials as polynomials with unknown coefficients. Reducing the two members of above formula to the same denominator and writing that the coefficients of each power of x are the same in the two numerators, one gets a system of linear equations which can be solved to obtain the desired values for the unknowns coefficients.
Procedure[edit]
Given two polynomials and , where the αi are distinct constants and deg P < n, partial fractions are generally obtained by supposing that
and solving for the ci constants, by substitution, by equating the coefficients of terms involving the powers of x, or otherwise. (This is a variant of the method of undetermined coefficients.)
A more direct computation, which is strongly related with Lagrange interpolation consists of writing
where is the derivative of the polynomial .
This approach does not account for several other cases, but can be modified accordingly:
- If then it is necessary to perform the Euclidean division of P by Q, using polynomial long division, giving P(x) = E(x) Q(x) + R(x) with deg R < n. Dividing by Q(x) this gives
- and then seek partial fractions for the remainder fraction (which by definition satisfies deg R < deg Q).
- If Q(x) contains factors which are irreducible over the given field, then the numerator N(x) of each partial fraction with such a factor F(x) in the denominator must be sought as a polynomial with deg N < deg F, rather than as a constant. For example, take the following decomposition over R:
- Suppose Q(x) = (x − α)rS(x) and S(α) ≠ 0. Then Q(x) has a zero α of multiplicity r, and in the partial fraction decomposition, r of the partial fractions will involve the powers of (x − α). For illustration, take S(x) = 1 to get the following decomposition:
Illustration[edit]
In an example application of this procedure, (3x + 5)/(1 − 2x)2 can be decomposed in the form
Clearing denominators shows that 3x + 5 = A + B(1 − 2x). Expanding and equating the coefficients of powers of x gives
- 5 = A + B and 3x = −2Bx
Solving for A and B yields A = 13/2 and B = −3/2. Hence,
Residue method[edit]
Over the complex numbers, suppose f(x) is a rational proper fraction, and can be decomposed into
Let
then according to the uniqueness of Laurent series, aij is the coefficient of the term (x − xi)−1 in the Laurent expansion of gij(x) about the point xi, i.e., its residue
This is given directly by the formula
or in the special case when xi is a simple root,
when
Over the reals[edit]
Partial fractions are used in real-variable integral calculus to find real-valued antiderivatives of rational functions. Partial fraction decomposition of real rational functions is also used to find their Inverse Laplace transforms. For applications of partial fraction decomposition over the reals, see
General result[edit]
Let f(x) be any rational function over the real numbers. In other words, suppose there exist real polynomials functions p(x) and q(x)≠ 0, such that
By dividing both the numerator and the denominator by the leading coefficient of q(x), we may assume without loss of generality that q(x) is monic. By the fundamental theorem of algebra, we can write
where a1,..., am, b1,..., bn, c1,..., cn are real numbers with bi2 − 4ci < 0, and j1,..., jm, k1,..., kn are positive integers. The terms (x − ai) are the linear factors of q(x) which correspond to real roots of q(x), and the terms (xi2 + bix + ci) are the irreducible quadratic factors of q(x) which correspond to pairs of complex conjugate roots of q(x).
Then the partial fraction decomposition of f(x) is the following:
Here, P(x) is a (possibly zero) polynomial, and the Air, Bir, and Cir are real constants. There are a number of ways the constants can be found.
The most straightforward method is to multiply through by the common denominator q(x). We then obtain an equation of polynomials whose left-hand side is simply p(x) and whose right-hand side has coefficients which are linear expressions of the constants Air, Bir, and Cir. Since two polynomials are equal if and only if their corresponding coefficients are equal, we can equate the coefficients of like terms. In this way, a system of linear equations is obtained which always has a unique solution. This solution can be found using any of the standard methods of linear algebra. It can also be found with limits (see Example 5).
Examples[edit]
empty
The role of the Taylor polynomial[edit]
The partial fraction decomposition of a rational function can be related to Taylor's theorem as follows. Let
be real or complex polynomials assume that
satisfies
Also define
Then we have
if, and only if, each polynomial is the Taylor polynomial of of order at the point :
Taylor's theorem (in the real or complex case) then provides a proof of the existence and uniqueness of the partial fraction decomposition, and a characterization of the coefficients.
Sketch of the proof[edit]
The above partial fraction decomposition implies, for each 1 ≤ i ≤ r, a polynomial expansion
so is the Taylor polynomial of , because of the unicity of the polynomial expansion of order , and by assumption .
Conversely, if the are the Taylor polynomials, the above expansions at each hold, therefore we also have
which implies that the polynomial is divisible by
For is also divisible by , so
is divisible by . Since
we then have
and we find the partial fraction decomposition dividing by .
Fractions of integers[edit]
The idea of partial fractions can be generalized to other integral domains, say the ring of integers where prime numbers take the role of irreducible denominators. For example:
Notes[edit]
- ^ Larson, Ron (2016). Algebra & Trigonometry. Cengage Learning. ISBN 9781337271172.
- ^ Horowitz, Ellis. "Algorithms for partial fraction decomposition and rational function integration." Proceedings of the second ACM symposium on Symbolic and algebraic manipulation. ACM, 1971.
- ^ Grosholz, Emily (2000). The Growth of Mathematical Knowledge. Kluwer Academic Publilshers. p. 179. ISBN 978-90-481-5391-6.
References[edit]
- Rao, K. R.; Ahmed, N. (1968). "Recursive techniques for obtaining the partial fraction expansion of a rational function". IEEE Trans. Educ. Vol. 11, no. 2. pp. 152–154. doi:10.1109/TE.1968.4320370.
- Henrici, Peter (1971). "An algorithm for the incomplete decomposition of a rational function into partial fractions". Z. Angew. Math. Phys. Vol. 22, no. 4. pp. 751–755. doi:10.1007/BF01587772.
- Chang, Feng-Cheng (1973). "Recursive formulas for the partial fraction expansion of a rational function with multiple poles". Proc. IEEE. Vol. 61, no. 8. pp. 1139–1140. doi:10.1109/PROC.1973.9216.
- Kung, H. T.; Tong, D. M. (1977). "Fast Algorithms for Partial Fraction Decomposition". SIAM Journal on Computing. 6 (3): 582. doi:10.1137/0206042.
- Eustice, Dan; Klamkin, M. S. (1979). "On the coefficients of a partial fraction decomposition". American Mathematical Monthly. Vol. 86, no. 6. pp. 478–480. JSTOR 2320421.
- Mahoney, J. J.; Sivazlian, B. D. (1983). "Partial fractions expansion: a review of computational methodology and efficiency". J. Comp. Appl. Math. Vol. 9. pp. 247–269. doi:10.1016/0377-0427(83)90018-3.
- Miller, Charles D.; Lial, Margaret L.; Schneider, David I. (1990). Fundamentals of College Algebra (3rd ed.). Addison-Wesley Educational Publishers, Inc. pp. 364–370. ISBN 0-673-38638-4.
- Westreich, David (1991). "partial fraction expansion without derivative evaluation". IEEE Trans. Circ. Syst. Vol. 38, no. 6. pp. 658–660. doi:10.1109/31.81863.
- Kudryavtsev, L. D. (2001) [1994], "Undetermined coefficients, method of", Encyclopedia of Mathematics, EMS Press
- Velleman, Daniel J. (2002). "Partial fractions, binomial coefficients and the integral of an odd power of sec theta". Amer. Math. Monthly. Vol. 109, no. 8. pp. 746–749. JSTOR 3072399.
- Slota, Damian; Witula, Roman (2005). "Three brick method of the partial fraction decomposition of some type of rational expression". Lect. Not. Computer Sci. Vol. 33516. pp. 659–662. doi:10.1007/11428862_89.
- Kung, Sidney H. (2006). "Partial fraction decomposition by division". Coll. Math. J. 37 (2): 132–134. JSTOR 27646303.
- Witula, Roman; Slota, Damian (2008). "Partial fractions decompositions of some rational functions". Appl. Math. Comput. Vol. 197. pp. 328–336. doi:10.1016/j.amc.2007.07.048. MR 2396331.
External links[edit]
- Weisstein, Eric W. "Partial Fraction Decomposition". MathWorld.
- Blake, Sam. "Step-by-Step Partial Fractions".
- [1] Make partial fraction decompositions with Scilab.
[[Category:Algebra]] [[Category:Elementary algebra]] [[Category:Partial fractions]]