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Greatest Common Divisor and Least Common Multiple

Introduction
Synopsis
GCD Function Object
LCM Function Object
Run-time GCD & LCM Determination
Compile time GCD and LCM determination
Header <boost/math/common_factor.hpp>
Demonstration Program
Rationale
History
Credits

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.

namespace boost
{
namespace math
{

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

template < typename IntegerType >
   constexpr IntegerType  gcd( IntegerType const &a, IntegerType const &b );
template < typename IntegerType >
   constexpr IntegerType  lcm( IntegerType const &a, IntegerType const &b );
template < typename IntegerType, typename... Args >
   constexpr IntegerType  gcd( IntegerType const &a, IntegerType const &b, Args const&... );
template < typename IntegerType, typename... Args >
   constexpr IntegerType  lcm( IntegerType const &a, IntegerType const &b, Args const&... );

template <typename I>
std::pair<typename std::iterator_traits<I>::value_type, I>
   gcd_range(I first, I last);
template <typename I>
std::pair<typename std::iterator_traits<I>::value_type, I>
   lcm_range(I first, I last);

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;

}
}

Header: <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
   constexpr 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.

Note that these function objects are constexpr in C++14 and later only. They are also declared noexcept when appropriate.

Header: <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
   constexpr 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.

Note that these function objects are constexpr in C++14 and later only. They are also declared noexcept when appropriate.

Header: <boost/math/common_factor_rt.hpp>

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

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

template < typename IntegerType, typename... Args >
   constexpr IntegerType  gcd( IntegerType const &a, IntegerType const &b, Args const&... );

template < typename IntegerType, typename... Args >
   constexpr IntegerType  lcm( IntegerType const &a, IntegerType const &b, Args const&... );

template <typename I>
std::pair<typename std::iterator_traits<I>::value_type, I>
   gcd_range(I first, I last);

template <typename I>
std::pair<typename std::iterator_traits<I>::value_type, I>
   lcm_range(I first, I last);

The boost::math::gcd function template returns the greatest common (nonnegative) divisor of the two integers passed to it. boost::math::gcd_range is the iteration of the above gcd algorithm over a range, returning the greatest common divisor of all the elements. The algorithm terminates when the gcd reaches unity or the end of the range. Thus it also returns the iterator after the last element inspected because this may not be equal to the end of the range. The variadic version of gcd behaves similarly but does not indicate which input value caused the gcd to reach unity.

The boost::math::lcm function template returns the least common (nonnegative) multiple of the two integers passed to it. As with gcd, there are range and variadic versions of the function for more than 2 arguments.

Note that these functions are constexpr in C++14 and later only. They are also declared noexcept when appropriate.

[Note] Note

These functions are deprecated in favor of constexpr gcd and lcm on C++14 capable compilers.

Header: <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.

Example

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

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, " ") );
}

This header simply includes the headers <boost/math/common_factor_ct.hpp> and <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).

The program 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.)

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.

  • 24th April 2017 Moved to Jeremy Murphy's improved algorithms, added constexpr and noexcept support, added compiler intrinsic support, added variadic and range based versions of the algorithms.
  • 13 May 2013 Moved into main Boost.Math Quickbook documentation.
  • 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

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.


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