[article GCD and LCM
    [quickbook 1.3]
    [copyright 2001-2002 Daryle Walker]
    [license
        Distributed under the Boost Software License, Version 1.0.
        (See accompanying file LICENSE_1_0.txt or copy at
        [@http://www.boost.org/LICENSE_1_0.txt http://www.boost.org/LICENSE_1_0.txt])
    ]
    [authors [Walker, Daryle]]
    [category math]
    [last-revision $Date: 2006-12-29 11:08:32 +0000 (Fri, 29 Dec 2006) $]
]

This manual is also available in 
[@http://svn.boost.org/svn/boost/sandbox/pdf/math/release/math-gcd.pdf 
printer friendly PDF format].


[section:gcd_lcm Greatest Common Divisor and Least Common Multiple]

[section Introduction]

The class and function templates in <boost/math/common_factor.hpp> 
provide run-time and compile-time evaluation of the greatest common divisor 
(GCD) or least common multiple (LCM) of two integers. 
These facilities are useful for many numeric-oriented generic 
programming problems.

[endsect]

[section Synopsis]

   namespace boost
   {
   namespace math
   {

   template < typename IntegerType >
      class gcd_evaluator;
   template < typename IntegerType >
      class lcm_evaluator;

   template < typename IntegerType >
      IntegerType  gcd( IntegerType const &a, IntegerType const &b );
   template < typename IntegerType >
      IntegerType  lcm( IntegerType const &a, IntegerType const &b );
      
   typedef ``['see-below]`` static_gcd_type;

   template < static_gcd_type Value1, static_gcd_type Value2 >
      struct static_gcd;
   template < static_gcd_type Value1, static_gcd_type Value2 >
      struct static_lcm;

   }
   }
   
[endsect]

[section GCD Function Object]

[*Header: ] [@../../../../../boost/math/common_factor_rt.hpp <boost/math/common_factor_rt.hpp>]

   template < typename IntegerType >
   class boost::math::gcd_evaluator
   {
   public:
      // Types
      typedef IntegerType  result_type;
      typedef IntegerType  first_argument_type;
      typedef IntegerType  second_argument_type;

      // Function object interface
      result_type  operator ()( first_argument_type const &a,
      second_argument_type const &b ) const;
   };

The boost::math::gcd_evaluator class template defines a function object 
class to return the greatest common divisor of two integers. 
The template is parameterized by a single type, called IntegerType here. 
This type should be a numeric type that represents integers. 
The result of the function object is always nonnegative, even if either of 
the operator arguments is negative.

This function object class template is used in the corresponding version of 
the GCD function template. If a numeric type wants to customize evaluations 
of its greatest common divisors, then the type should specialize on the 
gcd_evaluator class template.

[endsect]

[section LCM Function Object]

[*Header: ] [@../../../../../boost/math/common_factor_rt.hpp <boost/math/common_factor_rt.hpp>]

   template < typename IntegerType >
   class boost::math::lcm_evaluator
   {
   public:
      // Types
      typedef IntegerType  result_type;
      typedef IntegerType  first_argument_type;
      typedef IntegerType  second_argument_type;

      // Function object interface
      result_type  operator ()( first_argument_type const &a,
      second_argument_type const &b ) const;
   };

The boost::math::lcm_evaluator class template defines a function object 
class to return the least common multiple of two integers. The template 
is parameterized by a single type, called IntegerType here. This type 
should be a numeric type that represents integers. The result of the 
function object is always nonnegative, even if either of the operator 
arguments is negative. If the least common multiple is beyond the range 
of the integer type, the results are undefined.

This function object class template is used in the corresponding version 
of the LCM function template. If a numeric type wants to customize 
evaluations of its least common multiples, then the type should 
specialize on the lcm_evaluator class template.

[endsect]

[section:run_time Run-time GCD & LCM Determination]

[*Header: ] [@../../../../../boost/math/common_factor_rt.hpp <boost/math/common_factor_rt.hpp>]

   template < typename IntegerType >
   IntegerType  boost::math::gcd( IntegerType const &a, IntegerType const &b );

   template < typename IntegerType >
   IntegerType  boost::math::lcm( IntegerType const &a, IntegerType const &b );

The boost::math::gcd function template returns the greatest common 
(nonnegative) divisor of the two integers passed to it. 
The boost::math::lcm function template returns the least common 
(nonnegative) multiple of the two integers passed to it. 
The function templates are parameterized on the function arguments' 
IntegerType, which is also the return type. Internally, these function 
templates use an object of the corresponding version of the 
gcd_evaluator and lcm_evaluator class templates, respectively.

[endsect]

[section:compile_time Compile time GCD and LCM determination]

[*Header: ] [@../../../../../boost/math/common_factor_ct.hpp <boost/math/common_factor_ct.hpp>]

   typedef ``['unspecified]`` static_gcd_type;

   template < static_gcd_type Value1, static_gcd_type Value2 >
   struct boost::math::static_gcd : public mpl::integral_c<static_gcd_type, implementation_defined>
   {
   };

   template < static_gcd_type Value1, static_gcd_type Value2 >
   struct boost::math::static_lcm : public mpl::integral_c<static_gcd_type, implementation_defined>
   {
   };
   
The type `static_gcd_type` is the widest unsigned-integer-type that is supported 
for use in integral-constant-expressions by the compiler.  Usually this
the same type as `boost::uintmax_t`, but may fall back to being `unsigned long`
for some older compilers.

The boost::math::static_gcd and boost::math::static_lcm class templates 
take two value-based template parameters of the ['static_gcd_type] type 
and inherit from the type `boost::mpl::integral_c`. 
Inherited from the base class, they have a member /value/ 
that is the greatest common factor or least 
common multiple, respectively, of the template arguments. 
A compile-time error will occur if the least common multiple 
is beyond the range of `static_gcd_type`.

[h3 Example]

   #include <boost/math/common_factor.hpp>
   #include <algorithm>
   #include <iterator>


   int main()
   {
      using std::cout;
      using std::endl;

      cout << "The GCD and LCM of 6 and 15 are "
      << boost::math::gcd(6, 15) << " and "
      << boost::math::lcm(6, 15) << ", respectively."
      << endl;

      cout << "The GCD and LCM of 8 and 9 are "
      << boost::math::static_gcd<8, 9>::value
      << " and "
      << boost::math::static_lcm<8, 9>::value
      << ", respectively." << endl;

      int  a[] = { 4, 5, 6 }, b[] = { 7, 8, 9 }, c[3];
      std::transform( a, a + 3, b, c, boost::math::gcd_evaluator<int>() );
      std::copy( c, c + 3, std::ostream_iterator<int>(cout, " ") );
   }

[endsect]

[section:header Header <boost/math/common_factor.hpp>]

This header simply includes the headers 
[@../../../../../boost/math/common_factor_ct.hpp <boost/math/common_factor_ct.hpp>] 
and [@../../../../../boost/math/common_factor_rt.hpp <boost/math/common_factor_rt.hpp>].
 
Note this is a legacy header: it used to contain the actual implementation, 
but the compile-time and run-time facilities 
were moved to separate headers (since they were independent of each other).

[endsect]

[section:demo Demonstration Program]

The program [@../../../../../libs/math/test/common_factor_test.cpp common_factor_test.cpp] is a demonstration of the results from 
instantiating various examples of the run-time GCD and LCM function 
templates and the compile-time GCD and LCM class templates. 
(The run-time GCD and LCM class templates are tested indirectly through 
the run-time function templates.)

[endsect]

[section Rationale]

The greatest common divisor and least common multiple functions are 
greatly used in some numeric contexts, including some of the other 
Boost libraries. Centralizing these functions to one header improves 
code factoring and eases maintainence.

[endsect]

[section History]

* 17 Dec 2005: Converted documentation to Quickbook Format.
* 2 Jul 2002: Compile-time and run-time items separated to new headers. 
* 7 Nov 2001: Initial version 

[endsect]

[section Credits]

The author of the Boost compilation of GCD and LCM computations is 
Daryle Walker. The code was prompted by existing code hiding in the 
implementations of Paul Moore's rational library and Steve Cleary's 
pool library. The code had updates by Helmut Zeisel.

[endsect]

[endsect]