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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{\FullPartialFractionExpansionXmpTitle}{FullPartialFractionExpansion}
+\newcommand{\FullPartialFractionExpansionXmpNumber}{9.29}
+%
+% =====================================================================
+\begin{page}{FullPartialFractionExpansionXmpPage}{9.29 FullPartialFractionExpansion}
+% =====================================================================
+\beginscroll
+
+The domain \spadtype{FullPartialFractionExpansion} implements
+factor-free conversion of quotients to full partial fractions.
+
+\xtc{
+Our examples will all involve quotients of univariate polynomials
+with rational number coefficients.
+}{
+\spadpaste{Fx := FRAC UP(x, FRAC INT) \bound{Fx}}
+}
+\xtc{
+Here is a simple-looking rational function.
+}{
+\spadpaste{f : Fx := 36 / (x**5-2*x**4-2*x**3+4*x**2+x-2) \bound{f}\free{Fx}}
+}
+\xtc{
+We use \spadfunFrom{fullPartialFraction}{FullPartialFractionExpansion}
+to convert it to an object of type
+\spadtype{FullPartialFractionExpansion}.
+}{
+\spadpaste{g := fullPartialFraction f \bound{g}\free{f}}
+}
+\xtc{
+Use a coercion to change it back into a quotient.
+}{
+\spadpaste{g :: Fx \free{g}}
+}
+\xtc{
+Full partial fractions differentiate faster than rational
+functions.
+}{
+\spadpaste{g5 := D(g, 5) \free{g}\bound{g5}}
+}
+\xtc{
+}{
+\spadpaste{f5 := D(f, 5) \free{f}\bound{f5}}
+}
+\xtc{
+We can check that the two forms represent the same function.
+}{
+\spadpaste{g5::Fx - f5 \free{Fx g5 f5}}
+}
+\xtc{
+Here are some examples that are more complicated.
+}{
+\spadpaste{f : Fx := (x**5 * (x-1)) / ((x**2 + x + 1)**2 * (x-2)**3) \free{Fx}\bound{f2}}
+}
+\xtc{
+}{
+\spadpaste{g := fullPartialFraction f \free{f2}\bound{g2}}
+}
+\xtc{
+}{
+\spadpaste{g :: Fx - f \free{f2 g2 Fx}}
+}
+\xtc{
+}{
+\spadpaste{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) \free{Fx}\bound{f3}}
+}
+\xtc{
+}{
+\spadpaste{g := fullPartialFraction f \free{f3}\bound{g3}}
+}
+\xtc{
+}{
+\spadpaste{g :: Fx - f \free{f3 g3 Fx}}
+}
+\xtc{
+}{
+\spadpaste{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)\free{Fx}\bound{f4}}
+}
+\xtc{
+}{
+\spadpaste{g := fullPartialFraction f \free{f4}\bound{g4}}
+}
+\xtc{
+This verification takes much longer than the conversion to
+partial fractions.
+}{
+\spadpaste{g :: Fx - f \free{f4 g4 Fx}}
+}
+
+For more information, see the paper:
+Bronstein, M and Salvy, B.
+``Full Partial Fraction Decomposition of Rational Functions,''
+{\it Proceedings of ISSAC'93, Kiev}, ACM Press.
+All see \downlink{`PartialFraction'}{PartialFractionXmpPage}\ignore{PartialFraction} for standard partial fraction
+decompositions.
+\endscroll
+\autobuttons
+\end{page}
+%