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% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
\newcommand{\PartialFractionXmpTitle}{PartialFraction}
\newcommand{\PartialFractionXmpNumber}{9.61}
%
% =====================================================================
\begin{page}{PartialFractionXmpPage}{9.61 PartialFraction}
% =====================================================================
\beginscroll
%%
%% pfr.htex
%%
A {\it partial fraction} is a decomposition of a quotient into
%-% \HDindex{partial fraction}{PartialFractionXmpPage}{9.61}{PartialFraction}
a sum
%-% \HDindex{fraction!partial}{PartialFractionXmpPage}{9.61}{PartialFraction}
of quotients where the denominators of the summands are powers of
primes.\footnote{Most people first encounter partial fractions when they
are learning integral calculus.
For a technical discussion of partial fractions, see, for example, Lang's {\it
Algebra.}} For example, the rational number \spad{1/6} is decomposed into
\spad{1/2 -1/3}.
You can compute partial fractions of quotients of objects from domains
belonging to the category \spadtype{EuclideanDomain}.
For example, \spadtype{Integer}, \spadtype{Complex Integer}, and
\spadtype{UnivariatePolynomial(x, Fraction Integer)} all belong to
\spadtype{EuclideanDomain}.
In the examples following, we demonstrate how to decompose quotients of
each of these kinds of object into partial fractions.
Issue the system command
\spadcmd{)show PartialFraction}
to display the full list of operations defined by \spadtype{PartialFraction}.
It is necessary that we know how to factor the denominator when we want to
compute a partial fraction.
Although the interpreter can often do this automatically, it may be
necessary for you to include a call to \spadfun{factor}.
In these examples, it is not necessary to factor the
denominators explicitly.
%
\xtc{
The main operation for computing partial fractions is called
\spadfunFrom{partialFraction}{PartialFraction} and we use this to
compute a decomposition of \spad{1 / 10!}.
The first argument to \spadfunFrom{partialFraction}{PartialFraction} is
the numerator of the quotient and the second argument is the factored
denominator.
}{
\spadpaste{partialFraction(1,factorial 10) \bound{prev1}}
}
\xtc{
Since the denominators are powers of primes, it may be possible
to expand the numerators further with respect to those primes. Use the
operation \spadfunFrom{padicFraction}{PartialFraction} to do this.
}{
\spadpaste{f := padicFraction(\%) \free{prev1}\bound{f}}
}
%
%
\xtc{
The operation \spadfunFrom{compactFraction}{PartialFraction} returns an
expanded fraction into the usual form.
The compacted version is used internally for computational efficiency.
}{
\spadpaste{compactFraction(f) \free{f}}
}
%
\xtc{
You can add, subtract, multiply
and divide partial fractions. In addition, you can extract the parts
of the decomposition.
\spadfunFrom{numberOfFractionalTerms}{PartialFraction} computes
the number of terms in the fractional part.
This does not include the whole part of the fraction,
which you get by calling \spadfunFrom{wholePart}{PartialFraction}.
In this example, the whole part is just \spad{0}.
}{
\spadpaste{numberOfFractionalTerms(f) \free{f}}
}
\xtc{
The operation
\spadfunFrom{nthFractionalTerm}{PartialFraction} returns the individual terms in the
decomposition.
Notice that the object returned is a partial fraction itself.
\spadfunFrom{firstNumer}{PartialFraction} and
\spadfunFrom{firstDenom}{PartialFraction} extract the numerator and
denominator of the first term of the fraction.
}{
\spadpaste{nthFractionalTerm(f,3) \free{f}}
}
%
%
\xtc{
Given two gaussian integers (see \downlink{`Complex'}{ComplexXmpPage}\ignore{Complex}), you can
decompose their quotient into a partial fraction.
}{
\spadpaste{partialFraction(1,- 13 + 14 * \%i) \bound{prev2}}
}
%
\xtc{
To convert back to a quotient, simply use a conversion.
}{
\spadpaste{\% :: Fraction Complex Integer \free{prev2}}
}
To conclude this section, we compute the decomposition of
\texht{\narrowDisplay{1 \over {{(x + 1)}{(x + 2)}^2{(x + 3)}^3{(x + 4)}^4}}}{
\begin{verbatim}
1
-------------------------------
2 3 4
(x + 1)(x + 2) (x + 3) (x + 4)
\end{verbatim}
}
The polynomials in this object have type
\spadtype{UnivariatePolynomial(x, Fraction Integer)}.
%
\xtc{
We use the \spadfunFrom{primeFactor}{Factored} operation (see
\downlink{`Factored'}{FactoredXmpPage}\ignore{Factored}) to create the denominator in factored form directly.
}{
\spadpaste{u : FR UP(x, FRAC INT) := reduce(*,[primeFactor(x+i,i) for i in 1..4]) \bound{u}}
}
%
%
\xtc{
These are the compact and expanded partial fractions for the quotient.
}{
\spadpaste{partialFraction(1,u) \free{u}\bound{prev3}}
}
\xtc{
}{
\spadpaste{padicFraction \% \free{prev3}}
}
All see \downlink{`FullPartialFractionExpansion'}{FullPartialFractionExpansionXmpPage}\ignore{FullPartialFractionExpansion} for examples of
factor-free conversion of quotients to full partial fractions.
\endscroll
\autobuttons
\end{page}
%
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