and 1 contributors

# 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

None reported.

# BUGS AND LIMITATIONS

No bugs have been reported.

Please report any bugs or feature requests to `bug-math-rational-approx@rt.cpan.org`, or through the web interface at http://rt.cpan.org/Public/Dist/Display.html?Name=Math-Rational-Approx.

# AUTHOR

Diab Jerius <djerius@cpan.org>

# 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/>.