boost/numeric/interval/arith2.hpp
/* Boost interval/arith2.hpp template implementation file
*
* This header provides some auxiliary arithmetic
* functions: fmod, sqrt, square, pov, inverse and
* a multi-interval division.
*
* Copyright 2002-2003 Herv� Br�nnimann, Guillaume Melquiond, Sylvain Pion
*
* 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)
*/
#ifndef BOOST_NUMERIC_INTERVAL_ARITH2_HPP
#define BOOST_NUMERIC_INTERVAL_ARITH2_HPP
#include <boost/config.hpp>
#include <boost/numeric/interval/detail/interval_prototype.hpp>
#include <boost/numeric/interval/detail/test_input.hpp>
#include <boost/numeric/interval/detail/bugs.hpp>
#include <boost/numeric/interval/detail/division.hpp>
#include <boost/numeric/interval/arith.hpp>
#include <boost/numeric/interval/policies.hpp>
#include <algorithm>
#include <cassert>
#include <boost/config/no_tr1/cmath.hpp>
namespace boost {
namespace numeric {
template<class T, class Policies> inline
interval<T, Policies> fmod(const interval<T, Policies>& x,
const interval<T, Policies>& y)
{
if (interval_lib::detail::test_input(x, y))
return interval<T, Policies>::empty();
typename Policies::rounding rnd;
typedef typename interval_lib::unprotect<interval<T, Policies> >::type I;
T const &yb = interval_lib::user::is_neg(x.lower()) ? y.lower() : y.upper();
T n = rnd.int_down(rnd.div_down(x.lower(), yb));
return (const I&)x - n * (const I&)y;
}
template<class T, class Policies> inline
interval<T, Policies> fmod(const interval<T, Policies>& x, const T& y)
{
if (interval_lib::detail::test_input(x, y))
return interval<T, Policies>::empty();
typename Policies::rounding rnd;
typedef typename interval_lib::unprotect<interval<T, Policies> >::type I;
T n = rnd.int_down(rnd.div_down(x.lower(), y));
return (const I&)x - n * I(y);
}
template<class T, class Policies> inline
interval<T, Policies> fmod(const T& x, const interval<T, Policies>& y)
{
if (interval_lib::detail::test_input(x, y))
return interval<T, Policies>::empty();
typename Policies::rounding rnd;
typedef typename interval_lib::unprotect<interval<T, Policies> >::type I;
T const &yb = interval_lib::user::is_neg(x) ? y.lower() : y.upper();
T n = rnd.int_down(rnd.div_down(x, yb));
return x - n * (const I&)y;
}
namespace interval_lib {
template<class T, class Policies> inline
interval<T, Policies> division_part1(const interval<T, Policies>& x,
const interval<T, Policies>& y, bool& b)
{
typedef interval<T, Policies> I;
b = false;
if (detail::test_input(x, y))
return I::empty();
if (zero_in(y))
if (!user::is_zero(y.lower()))
if (!user::is_zero(y.upper()))
return detail::div_zero_part1(x, y, b);
else
return detail::div_negative(x, y.lower());
else
if (!user::is_zero(y.upper()))
return detail::div_positive(x, y.upper());
else
return I::empty();
else
return detail::div_non_zero(x, y);
}
template<class T, class Policies> inline
interval<T, Policies> division_part2(const interval<T, Policies>& x,
const interval<T, Policies>& y, bool b = true)
{
if (!b) return interval<T, Policies>::empty();
return detail::div_zero_part2(x, y);
}
template<class T, class Policies> inline
interval<T, Policies> multiplicative_inverse(const interval<T, Policies>& x)
{
typedef interval<T, Policies> I;
if (detail::test_input(x))
return I::empty();
T one = static_cast<T>(1);
typename Policies::rounding rnd;
if (zero_in(x)) {
typedef typename Policies::checking checking;
if (!user::is_zero(x.lower()))
if (!user::is_zero(x.upper()))
return I::whole();
else
return I(checking::neg_inf(), rnd.div_up(one, x.lower()), true);
else
if (!user::is_zero(x.upper()))
return I(rnd.div_down(one, x.upper()), checking::pos_inf(), true);
else
return I::empty();
} else
return I(rnd.div_down(one, x.upper()), rnd.div_up(one, x.lower()), true);
}
namespace detail {
template<class T, class Rounding> inline
T pow_dn(const T& x_, int pwr, Rounding& rnd) // x and pwr are positive
{
T x = x_;
T y = (pwr & 1) ? x_ : static_cast<T>(1);
pwr >>= 1;
while (pwr > 0) {
x = rnd.mul_down(x, x);
if (pwr & 1) y = rnd.mul_down(x, y);
pwr >>= 1;
}
return y;
}
template<class T, class Rounding> inline
T pow_up(const T& x_, int pwr, Rounding& rnd) // x and pwr are positive
{
T x = x_;
T y = (pwr & 1) ? x_ : static_cast<T>(1);
pwr >>= 1;
while (pwr > 0) {
x = rnd.mul_up(x, x);
if (pwr & 1) y = rnd.mul_up(x, y);
pwr >>= 1;
}
return y;
}
} // namespace detail
} // namespace interval_lib
template<class T, class Policies> inline
interval<T, Policies> pow(const interval<T, Policies>& x, int pwr)
{
BOOST_USING_STD_MAX();
using interval_lib::detail::pow_dn;
using interval_lib::detail::pow_up;
typedef interval<T, Policies> I;
if (interval_lib::detail::test_input(x))
return I::empty();
if (pwr == 0)
if (interval_lib::user::is_zero(x.lower())
&& interval_lib::user::is_zero(x.upper()))
return I::empty();
else
return I(static_cast<T>(1));
else if (pwr < 0)
return interval_lib::multiplicative_inverse(pow(x, -pwr));
typename Policies::rounding rnd;
if (interval_lib::user::is_neg(x.upper())) { // [-2,-1]
T yl = pow_dn(static_cast<T>(-x.upper()), pwr, rnd);
T yu = pow_up(static_cast<T>(-x.lower()), pwr, rnd);
if (pwr & 1) // [-2,-1]^1
return I(-yu, -yl, true);
else // [-2,-1]^2
return I(yl, yu, true);
} else if (interval_lib::user::is_neg(x.lower())) { // [-1,1]
if (pwr & 1) { // [-1,1]^1
return I(-pow_up(-x.lower(), pwr, rnd), pow_up(x.upper(), pwr, rnd), true);
} else { // [-1,1]^2
return I(static_cast<T>(0), pow_up(max BOOST_PREVENT_MACRO_SUBSTITUTION(static_cast<T>(-x.lower()), x.upper()), pwr, rnd), true);
}
} else { // [1,2]
return I(pow_dn(x.lower(), pwr, rnd), pow_up(x.upper(), pwr, rnd), true);
}
}
template<class T, class Policies> inline
interval<T, Policies> sqrt(const interval<T, Policies>& x)
{
typedef interval<T, Policies> I;
if (interval_lib::detail::test_input(x) || interval_lib::user::is_neg(x.upper()))
return I::empty();
typename Policies::rounding rnd;
T l = !interval_lib::user::is_pos(x.lower()) ? static_cast<T>(0) : rnd.sqrt_down(x.lower());
return I(l, rnd.sqrt_up(x.upper()), true);
}
template<class T, class Policies> inline
interval<T, Policies> square(const interval<T, Policies>& x)
{
typedef interval<T, Policies> I;
if (interval_lib::detail::test_input(x))
return I::empty();
typename Policies::rounding rnd;
const T& xl = x.lower();
const T& xu = x.upper();
if (interval_lib::user::is_neg(xu))
return I(rnd.mul_down(xu, xu), rnd.mul_up(xl, xl), true);
else if (interval_lib::user::is_pos(x.lower()))
return I(rnd.mul_down(xl, xl), rnd.mul_up(xu, xu), true);
else
return I(static_cast<T>(0), (-xl > xu ? rnd.mul_up(xl, xl) : rnd.mul_up(xu, xu)), true);
}
namespace interval_lib {
namespace detail {
template< class I > inline
I root_aux(typename I::base_type const &x, int k) // x and k are bigger than one
{
typedef typename I::base_type T;
T tk(k);
I y(static_cast<T>(1), x, true);
for(;;) {
T y0 = median(y);
I yy = intersect(y, y0 - (pow(I(y0, y0, true), k) - x) / (tk * pow(y, k - 1)));
if (equal(y, yy)) return y;
y = yy;
}
}
template< class I > inline // x is positive and k bigger than one
typename I::base_type root_aux_dn(typename I::base_type const &x, int k)
{
typedef typename I::base_type T;
typedef typename I::traits_type Policies;
typename Policies::rounding rnd;
T one(1);
if (x > one) return root_aux<I>(x, k).lower();
if (x == one) return one;
return rnd.div_down(one, root_aux<I>(rnd.div_up(one, x), k).upper());
}
template< class I > inline // x is positive and k bigger than one
typename I::base_type root_aux_up(typename I::base_type const &x, int k)
{
typedef typename I::base_type T;
typedef typename I::traits_type Policies;
typename Policies::rounding rnd;
T one(1);
if (x > one) return root_aux<I>(x, k).upper();
if (x == one) return one;
return rnd.div_up(one, root_aux<I>(rnd.div_down(one, x), k).lower());
}
} // namespace detail
} // namespace interval_lib
template< class T, class Policies > inline
interval<T, Policies> nth_root(interval<T, Policies> const &x, int k)
{
typedef interval<T, Policies> I;
if (interval_lib::detail::test_input(x)) return I::empty();
assert(k > 0);
if (k == 1) return x;
typename Policies::rounding rnd;
typedef typename interval_lib::unprotect<I>::type R;
if (!interval_lib::user::is_pos(x.upper())) {
if (interval_lib::user::is_zero(x.upper())) {
T zero(0);
if (!(k & 1) || interval_lib::user::is_zero(x.lower())) // [-1,0]^/2 or [0,0]
return I(zero, zero, true);
else // [-1,0]^/3
return I(-interval_lib::detail::root_aux_up<R>(-x.lower(), k), zero, true);
} else if (!(k & 1)) // [-2,-1]^/2
return I::empty();
else { // [-2,-1]^/3
return I(-interval_lib::detail::root_aux_up<R>(-x.lower(), k),
-interval_lib::detail::root_aux_dn<R>(-x.upper(), k), true);
}
}
T u = interval_lib::detail::root_aux_up<R>(x.upper(), k);
if (!interval_lib::user::is_pos(x.lower()))
if (!(k & 1) || interval_lib::user::is_zero(x.lower())) // [-1,1]^/2 or [0,1]
return I(static_cast<T>(0), u, true);
else // [-1,1]^/3
return I(-interval_lib::detail::root_aux_up<R>(-x.lower(), k), u, true);
else // [1,2]
return I(interval_lib::detail::root_aux_dn<R>(x.lower(), k), u, true);
}
} // namespace numeric
} // namespace boost
#endif // BOOST_NUMERIC_INTERVAL_ARITH2_HPP