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306 lines
9.9 KiB
C++
306 lines
9.9 KiB
C++
/* Boost interval/arith2.hpp template implementation file
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*
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* This header provides some auxiliary arithmetic
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* functions: fmod, sqrt, square, pov, inverse and
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* a multi-interval division.
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*
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* Copyright 2002-2003 Hervé Brönnimann, Guillaume Melquiond, Sylvain Pion
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*
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* Distributed under the Boost Software License, Version 1.0.
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* (See accompanying file LICENSE_1_0.txt or
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* copy at http://www.boost.org/LICENSE_1_0.txt)
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*/
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#ifndef BOOST_NUMERIC_INTERVAL_ARITH2_HPP
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#define BOOST_NUMERIC_INTERVAL_ARITH2_HPP
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#include <boost/config.hpp>
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#include <boost/numeric/interval/detail/interval_prototype.hpp>
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#include <boost/numeric/interval/detail/test_input.hpp>
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#include <boost/numeric/interval/detail/bugs.hpp>
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#include <boost/numeric/interval/detail/division.hpp>
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#include <boost/numeric/interval/arith.hpp>
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#include <boost/numeric/interval/policies.hpp>
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#include <algorithm>
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#include <cassert>
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#include <boost/config/no_tr1/cmath.hpp>
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namespace boost {
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namespace numeric {
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template<class T, class Policies> inline
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interval<T, Policies> fmod(const interval<T, Policies>& x,
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const interval<T, Policies>& y)
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{
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if (interval_lib::detail::test_input(x, y))
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return interval<T, Policies>::empty();
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typename Policies::rounding rnd;
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typedef typename interval_lib::unprotect<interval<T, Policies> >::type I;
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T const &yb = interval_lib::user::is_neg(x.lower()) ? y.lower() : y.upper();
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T n = rnd.int_down(rnd.div_down(x.lower(), yb));
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return (const I&)x - n * (const I&)y;
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}
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template<class T, class Policies> inline
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interval<T, Policies> fmod(const interval<T, Policies>& x, const T& y)
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{
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if (interval_lib::detail::test_input(x, y))
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return interval<T, Policies>::empty();
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typename Policies::rounding rnd;
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typedef typename interval_lib::unprotect<interval<T, Policies> >::type I;
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T n = rnd.int_down(rnd.div_down(x.lower(), y));
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return (const I&)x - n * I(y);
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}
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template<class T, class Policies> inline
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interval<T, Policies> fmod(const T& x, const interval<T, Policies>& y)
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{
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if (interval_lib::detail::test_input(x, y))
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return interval<T, Policies>::empty();
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typename Policies::rounding rnd;
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typedef typename interval_lib::unprotect<interval<T, Policies> >::type I;
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T const &yb = interval_lib::user::is_neg(x) ? y.lower() : y.upper();
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T n = rnd.int_down(rnd.div_down(x, yb));
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return x - n * (const I&)y;
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}
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namespace interval_lib {
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template<class T, class Policies> inline
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interval<T, Policies> division_part1(const interval<T, Policies>& x,
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const interval<T, Policies>& y, bool& b)
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{
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typedef interval<T, Policies> I;
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b = false;
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if (detail::test_input(x, y))
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return I::empty();
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if (zero_in(y))
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if (!user::is_zero(y.lower()))
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if (!user::is_zero(y.upper()))
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return detail::div_zero_part1(x, y, b);
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else
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return detail::div_negative(x, y.lower());
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else
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if (!user::is_zero(y.upper()))
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return detail::div_positive(x, y.upper());
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else
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return I::empty();
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else
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return detail::div_non_zero(x, y);
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}
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template<class T, class Policies> inline
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interval<T, Policies> division_part2(const interval<T, Policies>& x,
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const interval<T, Policies>& y, bool b = true)
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{
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if (!b) return interval<T, Policies>::empty();
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return detail::div_zero_part2(x, y);
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}
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template<class T, class Policies> inline
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interval<T, Policies> multiplicative_inverse(const interval<T, Policies>& x)
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{
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typedef interval<T, Policies> I;
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if (detail::test_input(x))
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return I::empty();
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T one = static_cast<T>(1);
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typename Policies::rounding rnd;
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if (zero_in(x)) {
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typedef typename Policies::checking checking;
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if (!user::is_zero(x.lower()))
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if (!user::is_zero(x.upper()))
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return I::whole();
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else
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return I(checking::neg_inf(), rnd.div_up(one, x.lower()), true);
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else
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if (!user::is_zero(x.upper()))
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return I(rnd.div_down(one, x.upper()), checking::pos_inf(), true);
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else
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return I::empty();
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} else
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return I(rnd.div_down(one, x.upper()), rnd.div_up(one, x.lower()), true);
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}
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namespace detail {
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template<class T, class Rounding> inline
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T pow_dn(const T& x_, int pwr, Rounding& rnd) // x and pwr are positive
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{
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T x = x_;
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T y = (pwr & 1) ? x_ : static_cast<T>(1);
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pwr >>= 1;
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while (pwr > 0) {
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x = rnd.mul_down(x, x);
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if (pwr & 1) y = rnd.mul_down(x, y);
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pwr >>= 1;
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}
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return y;
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}
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template<class T, class Rounding> inline
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T pow_up(const T& x_, int pwr, Rounding& rnd) // x and pwr are positive
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{
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T x = x_;
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T y = (pwr & 1) ? x_ : static_cast<T>(1);
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pwr >>= 1;
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while (pwr > 0) {
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x = rnd.mul_up(x, x);
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if (pwr & 1) y = rnd.mul_up(x, y);
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pwr >>= 1;
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}
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return y;
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}
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} // namespace detail
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} // namespace interval_lib
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template<class T, class Policies> inline
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interval<T, Policies> pow(const interval<T, Policies>& x, int pwr)
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{
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BOOST_USING_STD_MAX();
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using interval_lib::detail::pow_dn;
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using interval_lib::detail::pow_up;
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typedef interval<T, Policies> I;
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if (interval_lib::detail::test_input(x))
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return I::empty();
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if (pwr == 0)
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if (interval_lib::user::is_zero(x.lower())
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&& interval_lib::user::is_zero(x.upper()))
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return I::empty();
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else
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return I(static_cast<T>(1));
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else if (pwr < 0)
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return interval_lib::multiplicative_inverse(pow(x, -pwr));
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typename Policies::rounding rnd;
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if (interval_lib::user::is_neg(x.upper())) { // [-2,-1]
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T yl = pow_dn(static_cast<T>(-x.upper()), pwr, rnd);
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T yu = pow_up(static_cast<T>(-x.lower()), pwr, rnd);
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if (pwr & 1) // [-2,-1]^1
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return I(-yu, -yl, true);
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else // [-2,-1]^2
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return I(yl, yu, true);
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} else if (interval_lib::user::is_neg(x.lower())) { // [-1,1]
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if (pwr & 1) { // [-1,1]^1
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return I(-pow_up(static_cast<T>(-x.lower()), pwr, rnd), pow_up(x.upper(), pwr, rnd), true);
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} else { // [-1,1]^2
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return I(static_cast<T>(0), pow_up(max BOOST_PREVENT_MACRO_SUBSTITUTION(static_cast<T>(-x.lower()), x.upper()), pwr, rnd), true);
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}
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} else { // [1,2]
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return I(pow_dn(x.lower(), pwr, rnd), pow_up(x.upper(), pwr, rnd), true);
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}
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}
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template<class T, class Policies> inline
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interval<T, Policies> sqrt(const interval<T, Policies>& x)
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{
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typedef interval<T, Policies> I;
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if (interval_lib::detail::test_input(x) || interval_lib::user::is_neg(x.upper()))
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return I::empty();
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typename Policies::rounding rnd;
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T l = !interval_lib::user::is_pos(x.lower()) ? static_cast<T>(0) : rnd.sqrt_down(x.lower());
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return I(l, rnd.sqrt_up(x.upper()), true);
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}
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template<class T, class Policies> inline
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interval<T, Policies> square(const interval<T, Policies>& x)
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{
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typedef interval<T, Policies> I;
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if (interval_lib::detail::test_input(x))
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return I::empty();
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typename Policies::rounding rnd;
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const T& xl = x.lower();
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const T& xu = x.upper();
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if (interval_lib::user::is_neg(xu))
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return I(rnd.mul_down(xu, xu), rnd.mul_up(xl, xl), true);
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else if (interval_lib::user::is_pos(x.lower()))
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return I(rnd.mul_down(xl, xl), rnd.mul_up(xu, xu), true);
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else
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return I(static_cast<T>(0), (-xl > xu ? rnd.mul_up(xl, xl) : rnd.mul_up(xu, xu)), true);
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}
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namespace interval_lib {
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namespace detail {
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template< class I > inline
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I root_aux(typename I::base_type const &x, int k) // x and k are bigger than one
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{
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typedef typename I::base_type T;
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T tk(k);
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I y(static_cast<T>(1), x, true);
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for(;;) {
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T y0 = median(y);
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I yy = intersect(y, y0 - (pow(I(y0, y0, true), k) - x) / (tk * pow(y, k - 1)));
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if (equal(y, yy)) return y;
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y = yy;
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}
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}
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template< class I > inline // x is positive and k bigger than one
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typename I::base_type root_aux_dn(typename I::base_type const &x, int k)
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{
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typedef typename I::base_type T;
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typedef typename I::traits_type Policies;
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typename Policies::rounding rnd;
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T one(1);
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if (x > one) return root_aux<I>(x, k).lower();
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if (x == one) return one;
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return rnd.div_down(one, root_aux<I>(rnd.div_up(one, x), k).upper());
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}
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template< class I > inline // x is positive and k bigger than one
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typename I::base_type root_aux_up(typename I::base_type const &x, int k)
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{
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typedef typename I::base_type T;
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typedef typename I::traits_type Policies;
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typename Policies::rounding rnd;
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T one(1);
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if (x > one) return root_aux<I>(x, k).upper();
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if (x == one) return one;
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return rnd.div_up(one, root_aux<I>(rnd.div_down(one, x), k).lower());
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}
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} // namespace detail
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} // namespace interval_lib
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template< class T, class Policies > inline
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interval<T, Policies> nth_root(interval<T, Policies> const &x, int k)
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{
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typedef interval<T, Policies> I;
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if (interval_lib::detail::test_input(x)) return I::empty();
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assert(k > 0);
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if (k == 1) return x;
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typename Policies::rounding rnd;
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typedef typename interval_lib::unprotect<I>::type R;
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if (!interval_lib::user::is_pos(x.upper())) {
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if (interval_lib::user::is_zero(x.upper())) {
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T zero(0);
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if (!(k & 1) || interval_lib::user::is_zero(x.lower())) // [-1,0]^/2 or [0,0]
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return I(zero, zero, true);
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else // [-1,0]^/3
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return I(-interval_lib::detail::root_aux_up<R>(-x.lower(), k), zero, true);
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} else if (!(k & 1)) // [-2,-1]^/2
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return I::empty();
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else { // [-2,-1]^/3
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return I(-interval_lib::detail::root_aux_up<R>(-x.lower(), k),
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-interval_lib::detail::root_aux_dn<R>(-x.upper(), k), true);
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}
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}
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T u = interval_lib::detail::root_aux_up<R>(x.upper(), k);
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if (!interval_lib::user::is_pos(x.lower()))
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if (!(k & 1) || interval_lib::user::is_zero(x.lower())) // [-1,1]^/2 or [0,1]
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return I(static_cast<T>(0), u, true);
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else // [-1,1]^/3
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return I(-interval_lib::detail::root_aux_up<R>(-x.lower(), k), u, true);
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else // [1,2]
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return I(interval_lib::detail::root_aux_dn<R>(x.lower(), k), u, true);
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}
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} // namespace numeric
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} // namespace boost
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#endif // BOOST_NUMERIC_INTERVAL_ARITH2_HPP
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