boost/math/tools/toms748_solve.hpp
// (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_SOLVE_ROOT_HPP
#define BOOST_MATH_TOOLS_SOLVE_ROOT_HPP
#ifdef _MSC_VER
#pragma once
#endif
#include <boost/math/tools/precision.hpp>
#include <boost/math/policies/error_handling.hpp>
#include <boost/math/tools/config.hpp>
#include <boost/math/special_functions/sign.hpp>
#include <boost/cstdint.hpp>
#include <limits>
#ifdef BOOST_MATH_LOG_ROOT_ITERATIONS
# define BOOST_MATH_LOGGER_INCLUDE <boost/math/tools/iteration_logger.hpp>
# include BOOST_MATH_LOGGER_INCLUDE
# undef BOOST_MATH_LOGGER_INCLUDE
#else
# define BOOST_MATH_LOG_COUNT(count)
#endif
namespace boost{ namespace math{ namespace tools{
template <class T>
class eps_tolerance
{
public:
eps_tolerance()
{
eps = 4 * tools::epsilon<T>();
}
eps_tolerance(unsigned bits)
{
BOOST_MATH_STD_USING
eps = (std::max)(T(ldexp(1.0F, 1-bits)), T(4 * tools::epsilon<T>()));
}
bool operator()(const T& a, const T& b)
{
BOOST_MATH_STD_USING
return fabs(a - b) <= (eps * (std::min)(fabs(a), fabs(b)));
}
private:
T eps;
};
struct equal_floor
{
equal_floor(){}
template <class T>
bool operator()(const T& a, const T& b)
{
BOOST_MATH_STD_USING
return floor(a) == floor(b);
}
};
struct equal_ceil
{
equal_ceil(){}
template <class T>
bool operator()(const T& a, const T& b)
{
BOOST_MATH_STD_USING
return ceil(a) == ceil(b);
}
};
struct equal_nearest_integer
{
equal_nearest_integer(){}
template <class T>
bool operator()(const T& a, const T& b)
{
BOOST_MATH_STD_USING
return floor(a + 0.5f) == floor(b + 0.5f);
}
};
namespace detail{
template <class F, class T>
void bracket(F f, T& a, T& b, T c, T& fa, T& fb, T& d, T& fd)
{
//
// Given a point c inside the existing enclosing interval
// [a, b] sets a = c if f(c) == 0, otherwise finds the new
// enclosing interval: either [a, c] or [c, b] and sets
// d and fd to the point that has just been removed from
// the interval. In other words d is the third best guess
// to the root.
//
BOOST_MATH_STD_USING // For ADL of std math functions
T tol = tools::epsilon<T>() * 2;
//
// If the interval [a,b] is very small, or if c is too close
// to one end of the interval then we need to adjust the
// location of c accordingly:
//
if((b - a) < 2 * tol * a)
{
c = a + (b - a) / 2;
}
else if(c <= a + fabs(a) * tol)
{
c = a + fabs(a) * tol;
}
else if(c >= b - fabs(b) * tol)
{
c = b - fabs(b) * tol;
}
//
// OK, lets invoke f(c):
//
T fc = f(c);
//
// if we have a zero then we have an exact solution to the root:
//
if(fc == 0)
{
a = c;
fa = 0;
d = 0;
fd = 0;
return;
}
//
// Non-zero fc, update the interval:
//
if(boost::math::sign(fa) * boost::math::sign(fc) < 0)
{
d = b;
fd = fb;
b = c;
fb = fc;
}
else
{
d = a;
fd = fa;
a = c;
fa= fc;
}
}
template <class T>
inline T safe_div(T num, T denom, T r)
{
//
// return num / denom without overflow,
// return r if overflow would occur.
//
BOOST_MATH_STD_USING // For ADL of std math functions
if(fabs(denom) < 1)
{
if(fabs(denom * tools::max_value<T>()) <= fabs(num))
return r;
}
return num / denom;
}
template <class T>
inline T secant_interpolate(const T& a, const T& b, const T& fa, const T& fb)
{
//
// Performs standard secant interpolation of [a,b] given
// function evaluations f(a) and f(b). Performs a bisection
// if secant interpolation would leave us very close to either
// a or b. Rationale: we only call this function when at least
// one other form of interpolation has already failed, so we know
// that the function is unlikely to be smooth with a root very
// close to a or b.
//
BOOST_MATH_STD_USING // For ADL of std math functions
T tol = tools::epsilon<T>() * 5;
T c = a - (fa / (fb - fa)) * (b - a);
if((c <= a + fabs(a) * tol) || (c >= b - fabs(b) * tol))
return (a + b) / 2;
return c;
}
template <class T>
T quadratic_interpolate(const T& a, const T& b, T const& d,
const T& fa, const T& fb, T const& fd,
unsigned count)
{
//
// Performs quadratic interpolation to determine the next point,
// takes count Newton steps to find the location of the
// quadratic polynomial.
//
// Point d must lie outside of the interval [a,b], it is the third
// best approximation to the root, after a and b.
//
// Note: this does not guarantee to find a root
// inside [a, b], so we fall back to a secant step should
// the result be out of range.
//
// Start by obtaining the coefficients of the quadratic polynomial:
//
T B = safe_div(T(fb - fa), T(b - a), tools::max_value<T>());
T A = safe_div(T(fd - fb), T(d - b), tools::max_value<T>());
A = safe_div(T(A - B), T(d - a), T(0));
if(A == 0)
{
// failure to determine coefficients, try a secant step:
return secant_interpolate(a, b, fa, fb);
}
//
// Determine the starting point of the Newton steps:
//
T c;
if(boost::math::sign(A) * boost::math::sign(fa) > 0)
{
c = a;
}
else
{
c = b;
}
//
// Take the Newton steps:
//
for(unsigned i = 1; i <= count; ++i)
{
//c -= safe_div(B * c, (B + A * (2 * c - a - b)), 1 + c - a);
c -= safe_div(T(fa+(B+A*(c-b))*(c-a)), T(B + A * (2 * c - a - b)), T(1 + c - a));
}
if((c <= a) || (c >= b))
{
// Oops, failure, try a secant step:
c = secant_interpolate(a, b, fa, fb);
}
return c;
}
template <class T>
T cubic_interpolate(const T& a, const T& b, const T& d,
const T& e, const T& fa, const T& fb,
const T& fd, const T& fe)
{
//
// Uses inverse cubic interpolation of f(x) at points
// [a,b,d,e] to obtain an approximate root of f(x).
// Points d and e lie outside the interval [a,b]
// and are the third and forth best approximations
// to the root that we have found so far.
//
// Note: this does not guarantee to find a root
// inside [a, b], so we fall back to quadratic
// interpolation in case of an erroneous result.
//
BOOST_MATH_INSTRUMENT_CODE(" a = " << a << " b = " << b
<< " d = " << d << " e = " << e << " fa = " << fa << " fb = " << fb
<< " fd = " << fd << " fe = " << fe);
T q11 = (d - e) * fd / (fe - fd);
T q21 = (b - d) * fb / (fd - fb);
T q31 = (a - b) * fa / (fb - fa);
T d21 = (b - d) * fd / (fd - fb);
T d31 = (a - b) * fb / (fb - fa);
BOOST_MATH_INSTRUMENT_CODE(
"q11 = " << q11 << " q21 = " << q21 << " q31 = " << q31
<< " d21 = " << d21 << " d31 = " << d31);
T q22 = (d21 - q11) * fb / (fe - fb);
T q32 = (d31 - q21) * fa / (fd - fa);
T d32 = (d31 - q21) * fd / (fd - fa);
T q33 = (d32 - q22) * fa / (fe - fa);
T c = q31 + q32 + q33 + a;
BOOST_MATH_INSTRUMENT_CODE(
"q22 = " << q22 << " q32 = " << q32 << " d32 = " << d32
<< " q33 = " << q33 << " c = " << c);
if((c <= a) || (c >= b))
{
// Out of bounds step, fall back to quadratic interpolation:
c = quadratic_interpolate(a, b, d, fa, fb, fd, 3);
BOOST_MATH_INSTRUMENT_CODE(
"Out of bounds interpolation, falling back to quadratic interpolation. c = " << c);
}
return c;
}
} // namespace detail
template <class F, class T, class Tol, class Policy>
std::pair<T, T> toms748_solve(F f, const T& ax, const T& bx, const T& fax, const T& fbx, Tol tol, boost::uintmax_t& max_iter, const Policy& pol)
{
//
// Main entry point and logic for Toms Algorithm 748
// root finder.
//
BOOST_MATH_STD_USING // For ADL of std math functions
static const char* function = "boost::math::tools::toms748_solve<%1%>";
//
// Sanity check - are we allowed to iterate at all?
//
if (max_iter == 0)
return std::make_pair(ax, bx);
boost::uintmax_t count = max_iter;
T a, b, fa, fb, c, u, fu, a0, b0, d, fd, e, fe;
static const T mu = 0.5f;
// initialise a, b and fa, fb:
a = ax;
b = bx;
if(a >= b)
return boost::math::detail::pair_from_single(policies::raise_domain_error(
function,
"Parameters a and b out of order: a=%1%", a, pol));
fa = fax;
fb = fbx;
if(tol(a, b) || (fa == 0) || (fb == 0))
{
max_iter = 0;
if(fa == 0)
b = a;
else if(fb == 0)
a = b;
return std::make_pair(a, b);
}
if(boost::math::sign(fa) * boost::math::sign(fb) > 0)
return boost::math::detail::pair_from_single(policies::raise_domain_error(
function,
"Parameters a and b do not bracket the root: a=%1%", a, pol));
// dummy value for fd, e and fe:
fe = e = fd = 1e5F;
if(fa != 0)
{
//
// On the first step we take a secant step:
//
c = detail::secant_interpolate(a, b, fa, fb);
detail::bracket(f, a, b, c, fa, fb, d, fd);
--count;
BOOST_MATH_INSTRUMENT_CODE(" a = " << a << " b = " << b);
if(count && (fa != 0) && !tol(a, b))
{
//
// On the second step we take a quadratic interpolation:
//
c = detail::quadratic_interpolate(a, b, d, fa, fb, fd, 2);
e = d;
fe = fd;
detail::bracket(f, a, b, c, fa, fb, d, fd);
--count;
BOOST_MATH_INSTRUMENT_CODE(" a = " << a << " b = " << b);
}
}
while(count && (fa != 0) && !tol(a, b))
{
// save our brackets:
a0 = a;
b0 = b;
//
// Starting with the third step taken
// we can use either quadratic or cubic interpolation.
// Cubic interpolation requires that all four function values
// fa, fb, fd, and fe are distinct, should that not be the case
// then variable prof will get set to true, and we'll end up
// taking a quadratic step instead.
//
T min_diff = tools::min_value<T>() * 32;
bool prof = (fabs(fa - fb) < min_diff) || (fabs(fa - fd) < min_diff) || (fabs(fa - fe) < min_diff) || (fabs(fb - fd) < min_diff) || (fabs(fb - fe) < min_diff) || (fabs(fd - fe) < min_diff);
if(prof)
{
c = detail::quadratic_interpolate(a, b, d, fa, fb, fd, 2);
BOOST_MATH_INSTRUMENT_CODE("Can't take cubic step!!!!");
}
else
{
c = detail::cubic_interpolate(a, b, d, e, fa, fb, fd, fe);
}
//
// re-bracket, and check for termination:
//
e = d;
fe = fd;
detail::bracket(f, a, b, c, fa, fb, d, fd);
if((0 == --count) || (fa == 0) || tol(a, b))
break;
BOOST_MATH_INSTRUMENT_CODE(" a = " << a << " b = " << b);
//
// Now another interpolated step:
//
prof = (fabs(fa - fb) < min_diff) || (fabs(fa - fd) < min_diff) || (fabs(fa - fe) < min_diff) || (fabs(fb - fd) < min_diff) || (fabs(fb - fe) < min_diff) || (fabs(fd - fe) < min_diff);
if(prof)
{
c = detail::quadratic_interpolate(a, b, d, fa, fb, fd, 3);
BOOST_MATH_INSTRUMENT_CODE("Can't take cubic step!!!!");
}
else
{
c = detail::cubic_interpolate(a, b, d, e, fa, fb, fd, fe);
}
//
// Bracket again, and check termination condition, update e:
//
detail::bracket(f, a, b, c, fa, fb, d, fd);
if((0 == --count) || (fa == 0) || tol(a, b))
break;
BOOST_MATH_INSTRUMENT_CODE(" a = " << a << " b = " << b);
//
// Now we take a double-length secant step:
//
if(fabs(fa) < fabs(fb))
{
u = a;
fu = fa;
}
else
{
u = b;
fu = fb;
}
c = u - 2 * (fu / (fb - fa)) * (b - a);
if(fabs(c - u) > (b - a) / 2)
{
c = a + (b - a) / 2;
}
//
// Bracket again, and check termination condition:
//
e = d;
fe = fd;
detail::bracket(f, a, b, c, fa, fb, d, fd);
BOOST_MATH_INSTRUMENT_CODE(" a = " << a << " b = " << b);
BOOST_MATH_INSTRUMENT_CODE(" tol = " << T((fabs(a) - fabs(b)) / fabs(a)));
if((0 == --count) || (fa == 0) || tol(a, b))
break;
//
// And finally... check to see if an additional bisection step is
// to be taken, we do this if we're not converging fast enough:
//
if((b - a) < mu * (b0 - a0))
continue;
//
// bracket again on a bisection:
//
e = d;
fe = fd;
detail::bracket(f, a, b, T(a + (b - a) / 2), fa, fb, d, fd);
--count;
BOOST_MATH_INSTRUMENT_CODE("Not converging: Taking a bisection!!!!");
BOOST_MATH_INSTRUMENT_CODE(" a = " << a << " b = " << b);
} // while loop
max_iter -= count;
if(fa == 0)
{
b = a;
}
else if(fb == 0)
{
a = b;
}
BOOST_MATH_LOG_COUNT(max_iter)
return std::make_pair(a, b);
}
template <class F, class T, class Tol>
inline std::pair<T, T> toms748_solve(F f, const T& ax, const T& bx, const T& fax, const T& fbx, Tol tol, boost::uintmax_t& max_iter)
{
return toms748_solve(f, ax, bx, fax, fbx, tol, max_iter, policies::policy<>());
}
template <class F, class T, class Tol, class Policy>
inline std::pair<T, T> toms748_solve(F f, const T& ax, const T& bx, Tol tol, boost::uintmax_t& max_iter, const Policy& pol)
{
if (max_iter <= 2)
return std::make_pair(ax, bx);
max_iter -= 2;
std::pair<T, T> r = toms748_solve(f, ax, bx, f(ax), f(bx), tol, max_iter, pol);
max_iter += 2;
return r;
}
template <class F, class T, class Tol>
inline std::pair<T, T> toms748_solve(F f, const T& ax, const T& bx, Tol tol, boost::uintmax_t& max_iter)
{
return toms748_solve(f, ax, bx, tol, max_iter, policies::policy<>());
}
template <class F, class T, class Tol, class Policy>
std::pair<T, T> bracket_and_solve_root(F f, const T& guess, T factor, bool rising, Tol tol, boost::uintmax_t& max_iter, const Policy& pol)
{
BOOST_MATH_STD_USING
static const char* function = "boost::math::tools::bracket_and_solve_root<%1%>";
//
// Set up initial brackets:
//
T a = guess;
T b = a;
T fa = f(a);
T fb = fa;
//
// Set up invocation count:
//
boost::uintmax_t count = max_iter - 1;
int step = 32;
if((fa < 0) == (guess < 0 ? !rising : rising))
{
//
// Zero is to the right of b, so walk upwards
// until we find it:
//
while((boost::math::sign)(fb) == (boost::math::sign)(fa))
{
if(count == 0)
return boost::math::detail::pair_from_single(policies::raise_evaluation_error(function, "Unable to bracket root, last nearest value was %1%", b, pol));
//
// Heuristic: normally it's best not to increase the step sizes as we'll just end up
// with a really wide range to search for the root. However, if the initial guess was *really*
// bad then we need to speed up the search otherwise we'll take forever if we're orders of
// magnitude out. This happens most often if the guess is a small value (say 1) and the result
// we're looking for is close to std::numeric_limits<T>::min().
//
if((max_iter - count) % step == 0)
{
factor *= 2;
if(step > 1) step /= 2;
}
//
// Now go ahead and move our guess by "factor":
//
a = b;
fa = fb;
b *= factor;
fb = f(b);
--count;
BOOST_MATH_INSTRUMENT_CODE("a = " << a << " b = " << b << " fa = " << fa << " fb = " << fb << " count = " << count);
}
}
else
{
//
// Zero is to the left of a, so walk downwards
// until we find it:
//
while((boost::math::sign)(fb) == (boost::math::sign)(fa))
{
if(fabs(a) < tools::min_value<T>())
{
// Escape route just in case the answer is zero!
max_iter -= count;
max_iter += 1;
return a > 0 ? std::make_pair(T(0), T(a)) : std::make_pair(T(a), T(0));
}
if(count == 0)
return boost::math::detail::pair_from_single(policies::raise_evaluation_error(function, "Unable to bracket root, last nearest value was %1%", a, pol));
//
// Heuristic: normally it's best not to increase the step sizes as we'll just end up
// with a really wide range to search for the root. However, if the initial guess was *really*
// bad then we need to speed up the search otherwise we'll take forever if we're orders of
// magnitude out. This happens most often if the guess is a small value (say 1) and the result
// we're looking for is close to std::numeric_limits<T>::min().
//
if((max_iter - count) % step == 0)
{
factor *= 2;
if(step > 1) step /= 2;
}
//
// Now go ahead and move are guess by "factor":
//
b = a;
fb = fa;
a /= factor;
fa = f(a);
--count;
BOOST_MATH_INSTRUMENT_CODE("a = " << a << " b = " << b << " fa = " << fa << " fb = " << fb << " count = " << count);
}
}
max_iter -= count;
max_iter += 1;
std::pair<T, T> r = toms748_solve(
f,
(a < 0 ? b : a),
(a < 0 ? a : b),
(a < 0 ? fb : fa),
(a < 0 ? fa : fb),
tol,
count,
pol);
max_iter += count;
BOOST_MATH_INSTRUMENT_CODE("max_iter = " << max_iter << " count = " << count);
BOOST_MATH_LOG_COUNT(max_iter)
return r;
}
template <class F, class T, class Tol>
inline std::pair<T, T> bracket_and_solve_root(F f, const T& guess, const T& factor, bool rising, Tol tol, boost::uintmax_t& max_iter)
{
return bracket_and_solve_root(f, guess, factor, rising, tol, max_iter, policies::policy<>());
}
} // namespace tools
} // namespace math
} // namespace boost
#endif // BOOST_MATH_TOOLS_SOLVE_ROOT_HPP