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// (C) Copyright John Maddock 2006.
// Use, modification and distribution are subject to 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_MATH_TOOLS_MINIMA_HPP
#define BOOST_MATH_TOOLS_MINIMA_HPP
#ifdef _MSC_VER
#pragma once
#endif
#include <utility>
#include <boost/config/no_tr1/cmath.hpp>
#include <boost/math/tools/precision.hpp>
#include <boost/math/policies/policy.hpp>
#include <boost/cstdint.hpp>
namespace boost{ namespace math{ namespace tools{
template <class F, class T>
std::pair<T, T> brent_find_minima(F f, T min, T max, int bits, boost::uintmax_t& max_iter)
{
BOOST_MATH_STD_USING
bits = (std::min)(policies::digits<T, policies::policy<> >() / 2, bits);
T tolerance = static_cast<T>(ldexp(1.0, 1-bits));
T x; // minima so far
T w; // second best point
T v; // previous value of w
T u; // most recent evaluation point
T delta; // The distance moved in the last step
T delta2; // The distance moved in the step before last
T fu, fv, fw, fx; // function evaluations at u, v, w, x
T mid; // midpoint of min and max
T fract1, fract2; // minimal relative movement in x
static const T golden = 0.3819660f; // golden ratio, don't need too much precision here!
x = w = v = max;
fw = fv = fx = f(x);
delta2 = delta = 0;
uintmax_t count = max_iter;
do{
// get midpoint
mid = (min + max) / 2;
// work out if we're done already:
fract1 = tolerance * fabs(x) + tolerance / 4;
fract2 = 2 * fract1;
if(fabs(x - mid) <= (fract2 - (max - min) / 2))
break;
if(fabs(delta2) > fract1)
{
// try and construct a parabolic fit:
T r = (x - w) * (fx - fv);
T q = (x - v) * (fx - fw);
T p = (x - v) * q - (x - w) * r;
q = 2 * (q - r);
if(q > 0)
p = -p;
q = fabs(q);
T td = delta2;
delta2 = delta;
// determine whether a parabolic step is acceptible or not:
if((fabs(p) >= fabs(q * td / 2)) || (p <= q * (min - x)) || (p >= q * (max - x)))
{
// nope, try golden section instead
delta2 = (x >= mid) ? min - x : max - x;
delta = golden * delta2;
}
else
{
// whew, parabolic fit:
delta = p / q;
u = x + delta;
if(((u - min) < fract2) || ((max- u) < fract2))
delta = (mid - x) < 0 ? (T)-fabs(fract1) : (T)fabs(fract1);
}
}
else
{
// golden section:
delta2 = (x >= mid) ? min - x : max - x;
delta = golden * delta2;
}
// update current position:
u = (fabs(delta) >= fract1) ? T(x + delta) : (delta > 0 ? T(x + fabs(fract1)) : T(x - fabs(fract1)));
fu = f(u);
if(fu <= fx)
{
// good new point is an improvement!
// update brackets:
if(u >= x)
min = x;
else
max = x;
// update control points:
v = w;
w = x;
x = u;
fv = fw;
fw = fx;
fx = fu;
}
else
{
// Oh dear, point u is worse than what we have already,
// even so it *must* be better than one of our endpoints:
if(u < x)
min = u;
else
max = u;
if((fu <= fw) || (w == x))
{
// however it is at least second best:
v = w;
w = u;
fv = fw;
fw = fu;
}
else if((fu <= fv) || (v == x) || (v == w))
{
// third best:
v = u;
fv = fu;
}
}
}while(--count);
max_iter -= count;
return std::make_pair(x, fx);
}
template <class F, class T>
inline std::pair<T, T> brent_find_minima(F f, T min, T max, int digits)
{
boost::uintmax_t m = (std::numeric_limits<boost::uintmax_t>::max)();
return brent_find_minima(f, min, max, digits, m);
}
}}} // namespaces
#endif
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