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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}
+%