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305 lines
9.9 KiB
305 lines
9.9 KiB
/* 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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