NAME
   Math::Rational::Approx - approximate a number with a rational number

SYNOPSIS
     use Math::Rational::Approx ':all';

     #
     # find rational approximation with maximum denominator
     ($n, $d, $bounds ) = maxD( $x, 10 );

     # refine; note reuse of $bounds from previous call
     ( $n, $d ) = maxD( $x, 20, $bounds );


     #
     # find rational approximation to arbitrary precision using
     # continued fractions

     # one shot, 10 iterations
     ( $numerator, $denominator ) =
                      contfrac_nd( contfrac( 1.234871035, 10 ) );

     # multiple calls on same number; useful for convergence tests
     # keep array containing terms; get fraction and perhaps test it
     ( $terms, $residual ) = contfrac( 1.234871035, 3 );
     ( $n, $d ) = contfrac_nd( $terms  );

     # continue for an additional number of steps; note reuse of $terms;
     # new terms are appended to it
     ( $terms, $residual ) = contfrac( $residual, 3, $terms );

     # new results
     ( $n, $d ) = contfrac_nd( $terms );

DESCRIPTION
   This module and its object oriented companion modules provide various
   means for finding rational number approximations to real numbers. The
   object oriented versions are suitable when repeated refinements are
   required. See Math::Rational::Approx::MaxD and
   Math::Rational::Approx::ContFrac.

INTERFACE
 Maximum denominator
   maxD finds the best rational approximation (n/d) to a fraction with a
   denominator less than a given value. It uses Farey's sequence and is
   based upon the algorithm given at
   <http://www.johndcook.com/blog/2010/10/20/best-rational-approximation/>.

   This is an iterative procedure, searching a given range for the best
   approximation. To enable further refinement, the limiting denominator
   may by adjusted; the approximation will be continued from the last
   calculation.

   maxD
         ( $n, $d, $bounds ) = maxD( $x, $maxD, );
         ( $n, $d, $bounds ) = maxD( $x, $maxD, $bounds );

       Calculate the rational number approximation to $x with denominator
       no greater than $maxD.

       The optional argument, $bounds, is a reference to a four element
       array containing the initial bounds on the region to search. It
       takes the form

         bounds => [ a, b, c, d ]

       where the elements are all non-negative integers and the bounds are
       given by

         a/b < x < c/d
         b < maxD && d < maxD

       The behavior is undefined if the latter condition is not yet, unless
       the bounds are the result of a previous run of maxD.

       By default it searches from float(x) to "float(x)+1".

       maxD returns the determined numerator and denominator as well as an
       arrayref containing the bounds. If the $bounds argument was
       specified this is returned.

 Continued Fractions
   To approximate using continued fractions, one first generates the terms
   in the fraction using contfrac and then calculates the numerator and
   denominator using contfrac_nd.

   contfrac
         ( $terms, $residual ) = contfrac( $x, $n );
         ( $terms, $residual ) = contfrac( $x, $n, $terms );

       Generate $n terms of the continous fraction representing $x.
       Additional terms may be added in subsequent calls to contfrac by
       passing $residual as the number to approximate and supplying the
       previous $terms arrayref:

         ( $terms, $residual ) = contfrac( $x, 10 );
         ( $terms, $residual ) = contfrac( $residual, 3, $terms );

       The arguments are as follows:

       $x  The number to approximate. It must be positive.

       $n  The number of terms to generate.

       $terms
           *Optional*. An arrayref in which to store the terms. The array
           is appended to.

       Returns:

       $terms
           An arrayref which holds the terms. If one was provided in the
           argument list it is passed through.

       $residual
           The residual of the input number as a Math::BigFloat object.
           This is *not* the difference between the input number and the
           rational approximation. It is suitable for reuse in subsequent
           calls to contfrac.

   contfrac_nd
         ( $nominator, $denominator ) = contfrac_nd( $terms );

       Generate the nominator and denominator from the terms created by
       contfrac. They are returned as Math::BigInt objects.

DEPENDENCIES
   Math::BigFloat, POSIX, Params::Validate

INCOMPATIBILITIES
   None reported.

BUGS AND LIMITATIONS
   No bugs have been reported.

   Please report any bugs or feature requests to
   "[email protected]", or through the web interface at
   <http://rt.cpan.org/Public/Dist/Display.html?Name=Math-Rational-Approx>.

SEE ALSO
AUTHOR
   Diab Jerius <[email protected]>

LICENSE AND COPYRIGHT
   Copyright (c) 2012 The Smithsonian Astrophysical Observatory

   Math::Rational::Approx is free software: you can redistribute it and/or
   modify it under the terms of the GNU General Public License as published
   by the Free Software Foundation, either version 3 of the License, or (at
   your option) any later version.

   This program is distributed in the hope that it will be useful, but
   WITHOUT ANY WARRANTY; without even the implied warranty of
   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General
   Public License for more details.

   You should have received a copy of the GNU General Public License along
   with this program. If not, see <http://www.gnu.org/licenses/>.

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