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\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/input fparfrac.input}
\author{The Axiom Team}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{License}
<<license>>=
--Copyright The Numerical Algorithms Group Limited 1994.
@
<<*>>=
<<license>>
---------------------------- fparfrac.input ---------------------------------
-- This file demonstrates the factor-free conversion to full partial fractions.
)cl all
Q := FRAC INT
Px := UP(x, Q)
Fx := FRAC Px
-- here is a simple-looking function
f:Fx := 36 / (x**5-2*x**4-2*x**3+4*x**2+x-2)
-- use fullPartialFraction to convert to the type FPARFRAC
g := fullPartialFraction f
-- use :: to convert back to a rational function
g::Fx
-- Full partial fractions differentiate faster than rational functions:
g5 := D(g, 5)
f5 := D(f, 5)
-- check that the two forms represent the same function
g5::Fx - f5
-- Here are more complicated examples:
f:Fx := (x**5 * (x-1)) / ((x**2 + x + 1)**2 * (x-2)**3)
g := fullPartialFraction f
g::Fx - f
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)
g := fullPartialFraction f
g::Fx - f
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)
g := fullPartialFraction f
-- this verification takes much longer than the conversion to partial fractions
g::Fx - f
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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