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/* 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 #include #include #include #include #include #include #include #include #include namespace boost { namespace numeric { template inline interval fmod(const interval& x, const interval& y) { if (interval_lib::detail::test_input(x, y)) return interval::empty(); typename Policies::rounding rnd; typedef typename interval_lib::unprotect >::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 inline interval fmod(const interval& x, const T& y) { if (interval_lib::detail::test_input(x, y)) return interval::empty(); typename Policies::rounding rnd; typedef typename interval_lib::unprotect >::type I; T n = rnd.int_down(rnd.div_down(x.lower(), y)); return (const I&)x - n * I(y); } template inline interval fmod(const T& x, const interval& y) { if (interval_lib::detail::test_input(x, y)) return interval::empty(); typename Policies::rounding rnd; typedef typename interval_lib::unprotect >::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 inline interval division_part1(const interval& x, const interval& y, bool& b) { typedef interval 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 inline interval division_part2(const interval& x, const interval& y, bool b = true) { if (!b) return interval::empty(); return detail::div_zero_part2(x, y); } template inline interval multiplicative_inverse(const interval& x) { typedef interval I; if (detail::test_input(x)) return I::empty(); T one = static_cast(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 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(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 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(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 inline interval pow(const interval& x, int pwr) { BOOST_USING_STD_MAX(); using interval_lib::detail::pow_dn; using interval_lib::detail::pow_up; typedef interval 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(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(-x.upper()), pwr, rnd); T yu = pow_up(static_cast(-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(static_cast(-x.lower()), pwr, rnd), pow_up(x.upper(), pwr, rnd), true); } else { // [-1,1]^2 return I(static_cast(0), pow_up(max BOOST_PREVENT_MACRO_SUBSTITUTION(static_cast(-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 inline interval sqrt(const interval& x) { typedef interval 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(0) : rnd.sqrt_down(x.lower()); return I(l, rnd.sqrt_up(x.upper()), true); } template inline interval square(const interval& x) { typedef interval 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(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(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(x, k).lower(); if (x == one) return one; return rnd.div_down(one, root_aux(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(x, k).upper(); if (x == one) return one; return rnd.div_up(one, root_aux(rnd.div_down(one, x), k).lower()); } } // namespace detail } // namespace interval_lib template< class T, class Policies > inline interval nth_root(interval const &x, int k) { typedef interval I; if (interval_lib::detail::test_input(x)) return I::empty(); assert(k > 0); if (k == 1) return x; typedef typename interval_lib::unprotect::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(-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(-x.lower(), k), -interval_lib::detail::root_aux_dn(-x.upper(), k), true); } } T u = interval_lib::detail::root_aux_up(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(0), u, true); else // [-1,1]^/3 return I(-interval_lib::detail::root_aux_up(-x.lower(), k), u, true); else // [1,2] return I(interval_lib::detail::root_aux_dn(x.lower(), k), u, true); } } // namespace numeric } // namespace boost #endif // BOOST_NUMERIC_INTERVAL_ARITH2_HPP