Chris@16: /* Boost interval/arith2.hpp template implementation file
Chris@16:  *
Chris@16:  * This header provides some auxiliary arithmetic
Chris@16:  * functions: fmod, sqrt, square, pov, inverse and
Chris@16:  * a multi-interval division.
Chris@16:  *
Chris@16:  * Copyright 2002-2003 Hervé Brönnimann, Guillaume Melquiond, Sylvain Pion
Chris@16:  *
Chris@16:  * Distributed under the Boost Software License, Version 1.0.
Chris@16:  * (See accompanying file LICENSE_1_0.txt or
Chris@16:  * copy at http://www.boost.org/LICENSE_1_0.txt)
Chris@16:  */
Chris@16: 
Chris@16: #ifndef BOOST_NUMERIC_INTERVAL_ARITH2_HPP
Chris@16: #define BOOST_NUMERIC_INTERVAL_ARITH2_HPP
Chris@16: 
Chris@16: #include <boost/config.hpp>
Chris@16: #include <boost/numeric/interval/detail/interval_prototype.hpp>
Chris@16: #include <boost/numeric/interval/detail/test_input.hpp>
Chris@16: #include <boost/numeric/interval/detail/bugs.hpp>
Chris@16: #include <boost/numeric/interval/detail/division.hpp>
Chris@16: #include <boost/numeric/interval/arith.hpp>
Chris@16: #include <boost/numeric/interval/policies.hpp>
Chris@16: #include <algorithm>
Chris@16: #include <cassert>
Chris@16: #include <boost/config/no_tr1/cmath.hpp>
Chris@16: 
Chris@16: namespace boost {
Chris@16: namespace numeric {
Chris@16: 
Chris@16: template<class T, class Policies> inline
Chris@16: interval<T, Policies> fmod(const interval<T, Policies>& x,
Chris@16:                            const interval<T, Policies>& y)
Chris@16: {
Chris@16:   if (interval_lib::detail::test_input(x, y))
Chris@16:     return interval<T, Policies>::empty();
Chris@16:   typename Policies::rounding rnd;
Chris@16:   typedef typename interval_lib::unprotect<interval<T, Policies> >::type I;
Chris@16:   T const &yb = interval_lib::user::is_neg(x.lower()) ? y.lower() : y.upper();
Chris@16:   T n = rnd.int_down(rnd.div_down(x.lower(), yb));
Chris@16:   return (const I&)x - n * (const I&)y;
Chris@16: }
Chris@16: 
Chris@16: template<class T, class Policies> inline
Chris@16: interval<T, Policies> fmod(const interval<T, Policies>& x, const T& y)
Chris@16: {
Chris@16:   if (interval_lib::detail::test_input(x, y))
Chris@16:     return interval<T, Policies>::empty();
Chris@16:   typename Policies::rounding rnd;
Chris@16:   typedef typename interval_lib::unprotect<interval<T, Policies> >::type I;
Chris@16:   T n = rnd.int_down(rnd.div_down(x.lower(), y));
Chris@16:   return (const I&)x - n * I(y);
Chris@16: }
Chris@16: 
Chris@16: template<class T, class Policies> inline
Chris@16: interval<T, Policies> fmod(const T& x, const interval<T, Policies>& y)
Chris@16: {
Chris@16:   if (interval_lib::detail::test_input(x, y))
Chris@16:     return interval<T, Policies>::empty();
Chris@16:   typename Policies::rounding rnd;
Chris@16:   typedef typename interval_lib::unprotect<interval<T, Policies> >::type I;
Chris@16:   T const &yb = interval_lib::user::is_neg(x) ? y.lower() : y.upper();
Chris@16:   T n = rnd.int_down(rnd.div_down(x, yb));
Chris@16:   return x - n * (const I&)y;
Chris@16: }
Chris@16: 
Chris@16: namespace interval_lib {
Chris@16: 
Chris@16: template<class T, class Policies> inline
Chris@16: interval<T, Policies> division_part1(const interval<T, Policies>& x,
Chris@16:                                      const interval<T, Policies>& y, bool& b)
Chris@16: {
Chris@16:   typedef interval<T, Policies> I;
Chris@16:   b = false;
Chris@16:   if (detail::test_input(x, y))
Chris@16:     return I::empty();
Chris@16:   if (zero_in(y))
Chris@16:     if (!user::is_zero(y.lower()))
Chris@16:       if (!user::is_zero(y.upper()))
Chris@16:         return detail::div_zero_part1(x, y, b);
Chris@16:       else
Chris@16:         return detail::div_negative(x, y.lower());
Chris@16:     else
Chris@16:       if (!user::is_zero(y.upper()))
Chris@16:         return detail::div_positive(x, y.upper());
Chris@16:       else
Chris@16:         return I::empty();
Chris@16:   else
Chris@16:     return detail::div_non_zero(x, y);
Chris@16: }
Chris@16: 
Chris@16: template<class T, class Policies> inline
Chris@16: interval<T, Policies> division_part2(const interval<T, Policies>& x,
Chris@16:                                      const interval<T, Policies>& y, bool b = true)
Chris@16: {
Chris@16:   if (!b) return interval<T, Policies>::empty();
Chris@16:   return detail::div_zero_part2(x, y);
Chris@16: }
Chris@16: 
Chris@16: template<class T, class Policies> inline
Chris@16: interval<T, Policies> multiplicative_inverse(const interval<T, Policies>& x)
Chris@16: {
Chris@16:   typedef interval<T, Policies> I;
Chris@16:   if (detail::test_input(x))
Chris@16:     return I::empty();
Chris@16:   T one = static_cast<T>(1);
Chris@16:   typename Policies::rounding rnd;
Chris@16:   if (zero_in(x)) {
Chris@16:     typedef typename Policies::checking checking;
Chris@16:     if (!user::is_zero(x.lower()))
Chris@16:       if (!user::is_zero(x.upper()))
Chris@16:         return I::whole();
Chris@16:       else
Chris@16:         return I(checking::neg_inf(), rnd.div_up(one, x.lower()), true);
Chris@16:     else
Chris@16:       if (!user::is_zero(x.upper()))
Chris@16:         return I(rnd.div_down(one, x.upper()), checking::pos_inf(), true);
Chris@16:       else
Chris@16:         return I::empty();
Chris@16:   } else 
Chris@16:     return I(rnd.div_down(one, x.upper()), rnd.div_up(one, x.lower()), true);
Chris@16: }
Chris@16: 
Chris@16: namespace detail {
Chris@16: 
Chris@16: template<class T, class Rounding> inline
Chris@16: T pow_dn(const T& x_, int pwr, Rounding& rnd) // x and pwr are positive
Chris@16: {
Chris@16:   T x = x_;
Chris@16:   T y = (pwr & 1) ? x_ : static_cast<T>(1);
Chris@16:   pwr >>= 1;
Chris@16:   while (pwr > 0) {
Chris@16:     x = rnd.mul_down(x, x);
Chris@16:     if (pwr & 1) y = rnd.mul_down(x, y);
Chris@16:     pwr >>= 1;
Chris@16:   }
Chris@16:   return y;
Chris@16: }
Chris@16: 
Chris@16: template<class T, class Rounding> inline
Chris@16: T pow_up(const T& x_, int pwr, Rounding& rnd) // x and pwr are positive
Chris@16: {
Chris@16:   T x = x_;
Chris@16:   T y = (pwr & 1) ? x_ : static_cast<T>(1);
Chris@16:   pwr >>= 1;
Chris@16:   while (pwr > 0) {
Chris@16:     x = rnd.mul_up(x, x);
Chris@16:     if (pwr & 1) y = rnd.mul_up(x, y);
Chris@16:     pwr >>= 1;
Chris@16:   }
Chris@16:   return y;
Chris@16: }
Chris@16: 
Chris@16: } // namespace detail
Chris@16: } // namespace interval_lib
Chris@16: 
Chris@16: template<class T, class Policies> inline
Chris@16: interval<T, Policies> pow(const interval<T, Policies>& x, int pwr)
Chris@16: {
Chris@16:   BOOST_USING_STD_MAX();
Chris@16:   using interval_lib::detail::pow_dn;
Chris@16:   using interval_lib::detail::pow_up;
Chris@16:   typedef interval<T, Policies> I;
Chris@16: 
Chris@16:   if (interval_lib::detail::test_input(x))
Chris@16:     return I::empty();
Chris@16: 
Chris@16:   if (pwr == 0)
Chris@16:     if (interval_lib::user::is_zero(x.lower())
Chris@16:         && interval_lib::user::is_zero(x.upper()))
Chris@16:       return I::empty();
Chris@16:     else
Chris@16:       return I(static_cast<T>(1));
Chris@16:   else if (pwr < 0)
Chris@16:     return interval_lib::multiplicative_inverse(pow(x, -pwr));
Chris@16: 
Chris@16:   typename Policies::rounding rnd;
Chris@16:   
Chris@16:   if (interval_lib::user::is_neg(x.upper())) {        // [-2,-1]
Chris@16:     T yl = pow_dn(static_cast<T>(-x.upper()), pwr, rnd);
Chris@16:     T yu = pow_up(static_cast<T>(-x.lower()), pwr, rnd);
Chris@16:     if (pwr & 1)     // [-2,-1]^1
Chris@16:       return I(-yu, -yl, true);
Chris@16:     else             // [-2,-1]^2
Chris@16:       return I(yl, yu, true);
Chris@16:   } else if (interval_lib::user::is_neg(x.lower())) { // [-1,1]
Chris@16:     if (pwr & 1) {   // [-1,1]^1
Chris@16:       return I(-pow_up(static_cast<T>(-x.lower()), pwr, rnd), pow_up(x.upper(), pwr, rnd), true);
Chris@16:     } else {         // [-1,1]^2
Chris@16:       return I(static_cast<T>(0), pow_up(max BOOST_PREVENT_MACRO_SUBSTITUTION(static_cast<T>(-x.lower()), x.upper()), pwr, rnd), true);
Chris@16:     }
Chris@16:   } else {                                // [1,2]
Chris@16:     return I(pow_dn(x.lower(), pwr, rnd), pow_up(x.upper(), pwr, rnd), true);
Chris@16:   }
Chris@16: }
Chris@16: 
Chris@16: template<class T, class Policies> inline
Chris@16: interval<T, Policies> sqrt(const interval<T, Policies>& x)
Chris@16: {
Chris@16:   typedef interval<T, Policies> I;
Chris@16:   if (interval_lib::detail::test_input(x) || interval_lib::user::is_neg(x.upper()))
Chris@16:     return I::empty();
Chris@16:   typename Policies::rounding rnd;
Chris@16:   T l = !interval_lib::user::is_pos(x.lower()) ? static_cast<T>(0) : rnd.sqrt_down(x.lower());
Chris@16:   return I(l, rnd.sqrt_up(x.upper()), true);
Chris@16: }
Chris@16: 
Chris@16: template<class T, class Policies> inline
Chris@16: interval<T, Policies> square(const interval<T, Policies>& x)
Chris@16: {
Chris@16:   typedef interval<T, Policies> I;
Chris@16:   if (interval_lib::detail::test_input(x))
Chris@16:     return I::empty();
Chris@16:   typename Policies::rounding rnd;
Chris@16:   const T& xl = x.lower();
Chris@16:   const T& xu = x.upper();
Chris@16:   if (interval_lib::user::is_neg(xu))
Chris@16:     return I(rnd.mul_down(xu, xu), rnd.mul_up(xl, xl), true);
Chris@16:   else if (interval_lib::user::is_pos(x.lower()))
Chris@16:     return I(rnd.mul_down(xl, xl), rnd.mul_up(xu, xu), true);
Chris@16:   else
Chris@16:     return I(static_cast<T>(0), (-xl > xu ? rnd.mul_up(xl, xl) : rnd.mul_up(xu, xu)), true);
Chris@16: }
Chris@16: 
Chris@16: namespace interval_lib {
Chris@16: namespace detail {
Chris@16: 
Chris@16: template< class I > inline
Chris@16: I root_aux(typename I::base_type const &x, int k) // x and k are bigger than one
Chris@16: {
Chris@16:   typedef typename I::base_type T;
Chris@16:   T tk(k);
Chris@16:   I y(static_cast<T>(1), x, true);
Chris@16:   for(;;) {
Chris@16:     T y0 = median(y);
Chris@16:     I yy = intersect(y, y0 - (pow(I(y0, y0, true), k) - x) / (tk * pow(y, k - 1)));
Chris@16:     if (equal(y, yy)) return y;
Chris@16:     y = yy;
Chris@16:   }
Chris@16: }
Chris@16: 
Chris@16: template< class I > inline // x is positive and k bigger than one
Chris@16: typename I::base_type root_aux_dn(typename I::base_type const &x, int k)
Chris@16: {
Chris@16:   typedef typename I::base_type T;
Chris@16:   typedef typename I::traits_type Policies;
Chris@16:   typename Policies::rounding rnd;
Chris@16:   T one(1);
Chris@16:   if (x > one) return root_aux<I>(x, k).lower();
Chris@16:   if (x == one) return one;
Chris@16:   return rnd.div_down(one, root_aux<I>(rnd.div_up(one, x), k).upper());
Chris@16: }
Chris@16: 
Chris@16: template< class I > inline // x is positive and k bigger than one
Chris@16: typename I::base_type root_aux_up(typename I::base_type const &x, int k)
Chris@16: {
Chris@16:   typedef typename I::base_type T;
Chris@16:   typedef typename I::traits_type Policies;
Chris@16:   typename Policies::rounding rnd;
Chris@16:   T one(1);
Chris@16:   if (x > one) return root_aux<I>(x, k).upper();
Chris@16:   if (x == one) return one;
Chris@16:   return rnd.div_up(one, root_aux<I>(rnd.div_down(one, x), k).lower());
Chris@16: }
Chris@16: 
Chris@16: } // namespace detail
Chris@16: } // namespace interval_lib
Chris@16: 
Chris@16: template< class T, class Policies > inline
Chris@16: interval<T, Policies> nth_root(interval<T, Policies> const &x, int k)
Chris@16: {
Chris@16:   typedef interval<T, Policies> I;
Chris@16:   if (interval_lib::detail::test_input(x)) return I::empty();
Chris@16:   assert(k > 0);
Chris@16:   if (k == 1) return x;
Chris@16:   typename Policies::rounding rnd;
Chris@16:   typedef typename interval_lib::unprotect<I>::type R;
Chris@16:   if (!interval_lib::user::is_pos(x.upper())) {
Chris@16:     if (interval_lib::user::is_zero(x.upper())) {
Chris@16:       T zero(0);
Chris@16:       if (!(k & 1) || interval_lib::user::is_zero(x.lower())) // [-1,0]^/2 or [0,0]
Chris@16:         return I(zero, zero, true);
Chris@16:       else               // [-1,0]^/3
Chris@16:         return I(-interval_lib::detail::root_aux_up<R>(-x.lower(), k), zero, true);
Chris@16:     } else if (!(k & 1)) // [-2,-1]^/2
Chris@16:       return I::empty();
Chris@16:     else {               // [-2,-1]^/3
Chris@16:       return I(-interval_lib::detail::root_aux_up<R>(-x.lower(), k),
Chris@16:                -interval_lib::detail::root_aux_dn<R>(-x.upper(), k), true);
Chris@16:     }
Chris@16:   }
Chris@16:   T u = interval_lib::detail::root_aux_up<R>(x.upper(), k);
Chris@16:   if (!interval_lib::user::is_pos(x.lower()))
Chris@16:     if (!(k & 1) || interval_lib::user::is_zero(x.lower())) // [-1,1]^/2 or [0,1]
Chris@16:       return I(static_cast<T>(0), u, true);
Chris@16:     else                 // [-1,1]^/3
Chris@16:       return I(-interval_lib::detail::root_aux_up<R>(-x.lower(), k), u, true);
Chris@16:   else                   // [1,2]
Chris@16:     return I(interval_lib::detail::root_aux_dn<R>(x.lower(), k), u, true);
Chris@16: }
Chris@16: 
Chris@16: } // namespace numeric
Chris@16: } // namespace boost
Chris@16: 
Chris@16: #endif // BOOST_NUMERIC_INTERVAL_ARITH2_HPP