\Header{fractions}{Rational Approximation}
\alias{as.character.fractions}{fractions}
\alias{as.fractions}{fractions}
\alias{is.fractions}{fractions}
\alias{Math.fractions}{fractions}
\alias{Ops.fractions}{fractions}
\alias{print.fractions}{fractions}
\alias{Summary.fractions}{fractions}
\alias{t.fractions}{fractions}
\alias{[.fractions}{fractions}
\alias{[<\Rdash.fractions}{fractions}
\keyword{math}{fractions}
\begin{Description}\relax
Find rational approximations to the components of a real numeric
object using a standard continued fraction method.
\end{Description}
\begin{Usage}
\begin{verbatim}
fractions(x, cycles = 10, max.denominator = 2000, ...)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x}] Any object of mode numeric. Missing values are now allowed.

\item[\code{cycles}] The maximum number of steps to be used in the continued fraction
approximation process.

\item[\code{max.denominator}] An early termination criterion.  If any partial denominator
exceeds \code{max.denominator} the continued fraction stops at that point.

\item[\code{...}] arguments passed to or from other methods.

\end{ldescription}
\end{Arguments}
\begin{Details}\relax
Each component is first expanded in a continued fraction of the
form

\code{x = floor(x) + 1/(p1 + 1/(p2 + ...)))}

where \code{p1}, \code{p2}, \dots are positive integers, terminating either
at \code{cycles} terms or when a \code{pj > max.denominator}.  The
continued fraction is then re-arranged to retrieve the numerator
and denominator as integers.

The numerators and denominators are then combined into a
character vector that becomes the \code{"fracs"} attribute and used in
printed representations.


Arithmetic operations on \code{"fractions"} objects have full floating
point accuracy, but the character representation printed out may
not.
\end{Details}
\begin{Value}
An object of class \code{"fractions"}.  A structure with \code{.Data} component
the same as the input numeric \code{x}, but with the rational
approximations held as a character vector attribute, \code{"fracs"}.
Arithmetic operations on \code{"fractions"} objects are possible.
\end{Value}
\begin{References}\relax
Venables, W. N. and Ripley, B. D. (2002)
\emph{Modern Applied Statistics with S.} Fourth Edition. Springer.
\end{References}
\begin{SeeAlso}\relax
\code{\Link{rational}}
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
X <- matrix(runif(25), 5, 5)
solve(X, X/5)
##              [,1]        [,2]       [,3]        [,4]        [,5]
##  [1,]  2.0000e-01  3.7199e-17 1.2214e-16  5.7887e-17 -8.7841e-17
##  [2,] -1.1473e-16  2.0000e-01 7.0955e-17  2.0300e-17 -1.0566e-16
##  [3,]  2.7975e-16  1.3653e-17 2.0000e-01 -1.3397e-16  1.5577e-16
##  [4,] -2.9196e-16  2.0412e-17 1.5618e-16  2.0000e-01 -2.1921e-16
##  [5,] -3.6476e-17 -3.6430e-17 3.6432e-17  4.7690e-17  2.0000e-01

fractions(solve(X, X/5))
##      [,1] [,2] [,3] [,4] [,5]
## [1,] 1/5    0    0    0    0
## [2,]   0  1/5    0    0    0
## [3,]   0    0  1/5    0    0
## [4,]   0    0    0  1/5    0
## [5,]   0    0    0    0  1/5

fractions(solve(X, X/5)) + 1
##      [,1] [,2] [,3] [,4] [,5]
## [1,] 6/5    1    1    1    1
## [2,]   1  6/5    1    1    1
## [3,]   1    1  6/5    1    1
## [4,]   1    1    1  6/5    1
## [5,]   1    1    1    1  6/5
\end{ExampleCode}
\end{Examples}

