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\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/input contfrac.input}
\author{The Axiom Team}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{License}
<<license>>=
--Copyright The Numerical Algorithms Group Limited 1991.
@
<<*>>=
<<license>>
)clear all
r1 := 3/4
r2 := 314159/100000
c1 := r1 :: ContinuedFraction Integer
c2 := r2 :: ContinuedFraction Integer
-- We can view these in the list notation
partialQuotients c1
partialQuotients c2
-- These are algebraic objects, so we can manipulate them accordingly
c1 + c2
c1 * c2
1 / c2
c1 - c2
c2 - c1
-- and can convert them back to rational numbers.
convergents %
)clear all
-- Continued fractions over other Euclidean domains
a0 := ((-122 + 597* %i)/(4 - 4*%i))
b0 := ((-595 - %i)/(3 - 4*%i))
a := continuedFraction(a0)
b := continuedFraction(b0)
a + b
convergents %
last % - (a0 + b0)
a / b
convergents %
last % - (a0/b0)
(a = b)::Boolean
c := continuedFraction(3 + 4*%i, repeating [1 + %i], repeating [5 - %i])
a/c
-- (a = c)::Boolean -- should give error
d := complete continuedFraction(3+4*%i, repeating [1+%i],[i-%i for i in 1..5])
(a = d)::Boolean
q : Fraction UnivariatePolynomial('x, Fraction Integer)
q := (2*x**2 - x + 1) / (3*x**3 - x + 8)
c := continuedFraction q
d := continuedFraction differentiate q
c/d
convergents %
q/differentiate q
)clear all
-- This file illustrates continued fractions.
)set streams calculate 7
-- Use the notation Phi(ai/bi, i = 1..n) for continued fractions
-- a1/(b1 + (a2/b2 + ... (an/bn) ...))
-- 1/(e-1) may be written Phi(i/i, i = 1..)
s := continuedFraction(0, expand [1..], expand [1..])
-- Euler discovered the relation (e-1)/(e+1) = Phi(1/(4i-2), i = 1..)
t := reducedContinuedFraction(0, [4*i-2 for i in 1..])
-- Arithmetic on infinite continued fractions is supported.
-- The results are given in reduced form. We illustrate by using the
-- values s = 1/(e-1) and t = (e-1)/(e+1) to recover the expansion for e.
e := 1/(s*t) - 1
c := convergents e
for i in 1..15 repeat
output numeric c.i
(s = t)::Boolean
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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