| /* 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 Hervé Brönnimann, 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 |