Chris@16: //  (C) Copyright John Maddock 2006.
Chris@16: //  Use, modification and distribution are subject to the
Chris@16: //  Boost Software License, Version 1.0. (See accompanying file
Chris@16: //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
Chris@16: 
Chris@16: #ifndef BOOST_MATH_TOOLS_SOLVE_ROOT_HPP
Chris@16: #define BOOST_MATH_TOOLS_SOLVE_ROOT_HPP
Chris@16: 
Chris@16: #ifdef _MSC_VER
Chris@16: #pragma once
Chris@16: #endif
Chris@16: 
Chris@16: #include <boost/math/tools/precision.hpp>
Chris@16: #include <boost/math/policies/error_handling.hpp>
Chris@16: #include <boost/math/tools/config.hpp>
Chris@16: #include <boost/math/special_functions/sign.hpp>
Chris@16: #include <boost/cstdint.hpp>
Chris@16: #include <limits>
Chris@16: 
Chris@101: #ifdef BOOST_MATH_LOG_ROOT_ITERATIONS
Chris@101: #  define BOOST_MATH_LOGGER_INCLUDE <boost/math/tools/iteration_logger.hpp>
Chris@101: #  include BOOST_MATH_LOGGER_INCLUDE
Chris@101: #  undef BOOST_MATH_LOGGER_INCLUDE
Chris@101: #else
Chris@101: #  define BOOST_MATH_LOG_COUNT(count)
Chris@101: #endif
Chris@101: 
Chris@16: namespace boost{ namespace math{ namespace tools{
Chris@16: 
Chris@16: template <class T>
Chris@16: class eps_tolerance
Chris@16: {
Chris@16: public:
Chris@16:    eps_tolerance(unsigned bits)
Chris@16:    {
Chris@16:       BOOST_MATH_STD_USING
Chris@16:       eps = (std::max)(T(ldexp(1.0F, 1-bits)), T(4 * tools::epsilon<T>()));
Chris@16:    }
Chris@16:    bool operator()(const T& a, const T& b)
Chris@16:    {
Chris@16:       BOOST_MATH_STD_USING
Chris@16:       return fabs(a - b) <= (eps * (std::min)(fabs(a), fabs(b)));
Chris@16:    }
Chris@16: private:
Chris@16:    T eps;
Chris@16: };
Chris@16: 
Chris@16: struct equal_floor
Chris@16: {
Chris@16:    equal_floor(){}
Chris@16:    template <class T>
Chris@16:    bool operator()(const T& a, const T& b)
Chris@16:    {
Chris@16:       BOOST_MATH_STD_USING
Chris@16:       return floor(a) == floor(b);
Chris@16:    }
Chris@16: };
Chris@16: 
Chris@16: struct equal_ceil
Chris@16: {
Chris@16:    equal_ceil(){}
Chris@16:    template <class T>
Chris@16:    bool operator()(const T& a, const T& b)
Chris@16:    {
Chris@16:       BOOST_MATH_STD_USING
Chris@16:       return ceil(a) == ceil(b);
Chris@16:    }
Chris@16: };
Chris@16: 
Chris@16: struct equal_nearest_integer
Chris@16: {
Chris@16:    equal_nearest_integer(){}
Chris@16:    template <class T>
Chris@16:    bool operator()(const T& a, const T& b)
Chris@16:    {
Chris@16:       BOOST_MATH_STD_USING
Chris@16:       return floor(a + 0.5f) == floor(b + 0.5f);
Chris@16:    }
Chris@16: };
Chris@16: 
Chris@16: namespace detail{
Chris@16: 
Chris@16: template <class F, class T>
Chris@16: void bracket(F f, T& a, T& b, T c, T& fa, T& fb, T& d, T& fd)
Chris@16: {
Chris@16:    //
Chris@16:    // Given a point c inside the existing enclosing interval
Chris@16:    // [a, b] sets a = c if f(c) == 0, otherwise finds the new 
Chris@16:    // enclosing interval: either [a, c] or [c, b] and sets
Chris@16:    // d and fd to the point that has just been removed from
Chris@16:    // the interval.  In other words d is the third best guess
Chris@16:    // to the root.
Chris@16:    //
Chris@16:    BOOST_MATH_STD_USING  // For ADL of std math functions
Chris@16:    T tol = tools::epsilon<T>() * 2;
Chris@16:    //
Chris@16:    // If the interval [a,b] is very small, or if c is too close 
Chris@16:    // to one end of the interval then we need to adjust the
Chris@16:    // location of c accordingly:
Chris@16:    //
Chris@16:    if((b - a) < 2 * tol * a)
Chris@16:    {
Chris@16:       c = a + (b - a) / 2;
Chris@16:    }
Chris@16:    else if(c <= a + fabs(a) * tol)
Chris@16:    {
Chris@16:       c = a + fabs(a) * tol;
Chris@16:    }
Chris@16:    else if(c >= b - fabs(b) * tol)
Chris@16:    {
Chris@16:       c = b - fabs(a) * tol;
Chris@16:    }
Chris@16:    //
Chris@16:    // OK, lets invoke f(c):
Chris@16:    //
Chris@16:    T fc = f(c);
Chris@16:    //
Chris@16:    // if we have a zero then we have an exact solution to the root:
Chris@16:    //
Chris@16:    if(fc == 0)
Chris@16:    {
Chris@16:       a = c;
Chris@16:       fa = 0;
Chris@16:       d = 0;
Chris@16:       fd = 0;
Chris@16:       return;
Chris@16:    }
Chris@16:    //
Chris@16:    // Non-zero fc, update the interval:
Chris@16:    //
Chris@16:    if(boost::math::sign(fa) * boost::math::sign(fc) < 0)
Chris@16:    {
Chris@16:       d = b;
Chris@16:       fd = fb;
Chris@16:       b = c;
Chris@16:       fb = fc;
Chris@16:    }
Chris@16:    else
Chris@16:    {
Chris@16:       d = a;
Chris@16:       fd = fa;
Chris@16:       a = c;
Chris@16:       fa= fc;
Chris@16:    }
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: inline T safe_div(T num, T denom, T r)
Chris@16: {
Chris@16:    //
Chris@16:    // return num / denom without overflow,
Chris@16:    // return r if overflow would occur.
Chris@16:    //
Chris@16:    BOOST_MATH_STD_USING  // For ADL of std math functions
Chris@16: 
Chris@16:    if(fabs(denom) < 1)
Chris@16:    {
Chris@16:       if(fabs(denom * tools::max_value<T>()) <= fabs(num))
Chris@16:          return r;
Chris@16:    }
Chris@16:    return num / denom;
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: inline T secant_interpolate(const T& a, const T& b, const T& fa, const T& fb)
Chris@16: {
Chris@16:    //
Chris@16:    // Performs standard secant interpolation of [a,b] given
Chris@16:    // function evaluations f(a) and f(b).  Performs a bisection
Chris@16:    // if secant interpolation would leave us very close to either
Chris@16:    // a or b.  Rationale: we only call this function when at least
Chris@16:    // one other form of interpolation has already failed, so we know
Chris@16:    // that the function is unlikely to be smooth with a root very
Chris@16:    // close to a or b.
Chris@16:    //
Chris@16:    BOOST_MATH_STD_USING  // For ADL of std math functions
Chris@16: 
Chris@16:    T tol = tools::epsilon<T>() * 5;
Chris@16:    T c = a - (fa / (fb - fa)) * (b - a);
Chris@16:    if((c <= a + fabs(a) * tol) || (c >= b - fabs(b) * tol))
Chris@16:       return (a + b) / 2;
Chris@16:    return c;
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: T quadratic_interpolate(const T& a, const T& b, T const& d,
Chris@16:                            const T& fa, const T& fb, T const& fd, 
Chris@16:                            unsigned count)
Chris@16: {
Chris@16:    //
Chris@16:    // Performs quadratic interpolation to determine the next point,
Chris@16:    // takes count Newton steps to find the location of the
Chris@16:    // quadratic polynomial.
Chris@16:    //
Chris@16:    // Point d must lie outside of the interval [a,b], it is the third
Chris@16:    // best approximation to the root, after a and b.
Chris@16:    //
Chris@16:    // Note: this does not guarentee to find a root
Chris@16:    // inside [a, b], so we fall back to a secant step should
Chris@16:    // the result be out of range.
Chris@16:    //
Chris@16:    // Start by obtaining the coefficients of the quadratic polynomial:
Chris@16:    //
Chris@16:    T B = safe_div(T(fb - fa), T(b - a), tools::max_value<T>());
Chris@16:    T A = safe_div(T(fd - fb), T(d - b), tools::max_value<T>());
Chris@16:    A = safe_div(T(A - B), T(d - a), T(0));
Chris@16: 
Chris@16:    if(A == 0)
Chris@16:    {
Chris@16:       // failure to determine coefficients, try a secant step:
Chris@16:       return secant_interpolate(a, b, fa, fb);
Chris@16:    }
Chris@16:    //
Chris@16:    // Determine the starting point of the Newton steps:
Chris@16:    //
Chris@16:    T c;
Chris@16:    if(boost::math::sign(A) * boost::math::sign(fa) > 0)
Chris@16:    {
Chris@16:       c = a;
Chris@16:    }
Chris@16:    else
Chris@16:    {
Chris@16:       c = b;
Chris@16:    }
Chris@16:    //
Chris@16:    // Take the Newton steps:
Chris@16:    //
Chris@16:    for(unsigned i = 1; i <= count; ++i)
Chris@16:    {
Chris@16:       //c -= safe_div(B * c, (B + A * (2 * c - a - b)), 1 + c - a);
Chris@16:       c -= safe_div(T(fa+(B+A*(c-b))*(c-a)), T(B + A * (2 * c - a - b)), T(1 + c - a));
Chris@16:    }
Chris@16:    if((c <= a) || (c >= b))
Chris@16:    {
Chris@16:       // Oops, failure, try a secant step:
Chris@16:       c = secant_interpolate(a, b, fa, fb);
Chris@16:    }
Chris@16:    return c;
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: T cubic_interpolate(const T& a, const T& b, const T& d, 
Chris@16:                     const T& e, const T& fa, const T& fb, 
Chris@16:                     const T& fd, const T& fe)
Chris@16: {
Chris@16:    //
Chris@16:    // Uses inverse cubic interpolation of f(x) at points 
Chris@16:    // [a,b,d,e] to obtain an approximate root of f(x).
Chris@16:    // Points d and e lie outside the interval [a,b]
Chris@16:    // and are the third and forth best approximations
Chris@16:    // to the root that we have found so far.
Chris@16:    //
Chris@16:    // Note: this does not guarentee to find a root
Chris@16:    // inside [a, b], so we fall back to quadratic
Chris@16:    // interpolation in case of an erroneous result.
Chris@16:    //
Chris@16:    BOOST_MATH_INSTRUMENT_CODE(" a = " << a << " b = " << b
Chris@16:       << " d = " << d << " e = " << e << " fa = " << fa << " fb = " << fb 
Chris@16:       << " fd = " << fd << " fe = " << fe);
Chris@16:    T q11 = (d - e) * fd / (fe - fd);
Chris@16:    T q21 = (b - d) * fb / (fd - fb);
Chris@16:    T q31 = (a - b) * fa / (fb - fa);
Chris@16:    T d21 = (b - d) * fd / (fd - fb);
Chris@16:    T d31 = (a - b) * fb / (fb - fa);
Chris@16:    BOOST_MATH_INSTRUMENT_CODE(
Chris@16:       "q11 = " << q11 << " q21 = " << q21 << " q31 = " << q31
Chris@16:       << " d21 = " << d21 << " d31 = " << d31);
Chris@16:    T q22 = (d21 - q11) * fb / (fe - fb);
Chris@16:    T q32 = (d31 - q21) * fa / (fd - fa);
Chris@16:    T d32 = (d31 - q21) * fd / (fd - fa);
Chris@16:    T q33 = (d32 - q22) * fa / (fe - fa);
Chris@16:    T c = q31 + q32 + q33 + a;
Chris@16:    BOOST_MATH_INSTRUMENT_CODE(
Chris@16:       "q22 = " << q22 << " q32 = " << q32 << " d32 = " << d32
Chris@16:       << " q33 = " << q33 << " c = " << c);
Chris@16: 
Chris@16:    if((c <= a) || (c >= b))
Chris@16:    {
Chris@16:       // Out of bounds step, fall back to quadratic interpolation:
Chris@16:       c = quadratic_interpolate(a, b, d, fa, fb, fd, 3);
Chris@16:    BOOST_MATH_INSTRUMENT_CODE(
Chris@16:       "Out of bounds interpolation, falling back to quadratic interpolation. c = " << c);
Chris@16:    }
Chris@16: 
Chris@16:    return c;
Chris@16: }
Chris@16: 
Chris@16: } // namespace detail
Chris@16: 
Chris@16: template <class F, class T, class Tol, class Policy>
Chris@16: 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)
Chris@16: {
Chris@16:    //
Chris@16:    // Main entry point and logic for Toms Algorithm 748
Chris@16:    // root finder.
Chris@16:    //
Chris@16:    BOOST_MATH_STD_USING  // For ADL of std math functions
Chris@16: 
Chris@16:    static const char* function = "boost::math::tools::toms748_solve<%1%>";
Chris@16: 
Chris@16:    boost::uintmax_t count = max_iter;
Chris@16:    T a, b, fa, fb, c, u, fu, a0, b0, d, fd, e, fe;
Chris@16:    static const T mu = 0.5f;
Chris@16: 
Chris@16:    // initialise a, b and fa, fb:
Chris@16:    a = ax;
Chris@16:    b = bx;
Chris@16:    if(a >= b)
Chris@101:       return boost::math::detail::pair_from_single(policies::raise_domain_error(
Chris@16:          function, 
Chris@101:          "Parameters a and b out of order: a=%1%", a, pol));
Chris@16:    fa = fax;
Chris@16:    fb = fbx;
Chris@16: 
Chris@16:    if(tol(a, b) || (fa == 0) || (fb == 0))
Chris@16:    {
Chris@16:       max_iter = 0;
Chris@16:       if(fa == 0)
Chris@16:          b = a;
Chris@16:       else if(fb == 0)
Chris@16:          a = b;
Chris@16:       return std::make_pair(a, b);
Chris@16:    }
Chris@16: 
Chris@16:    if(boost::math::sign(fa) * boost::math::sign(fb) > 0)
Chris@101:       return boost::math::detail::pair_from_single(policies::raise_domain_error(
Chris@16:          function, 
Chris@101:          "Parameters a and b do not bracket the root: a=%1%", a, pol));
Chris@16:    // dummy value for fd, e and fe:
Chris@16:    fe = e = fd = 1e5F;
Chris@16: 
Chris@16:    if(fa != 0)
Chris@16:    {
Chris@16:       //
Chris@16:       // On the first step we take a secant step:
Chris@16:       //
Chris@16:       c = detail::secant_interpolate(a, b, fa, fb);
Chris@16:       detail::bracket(f, a, b, c, fa, fb, d, fd);
Chris@16:       --count;
Chris@16:       BOOST_MATH_INSTRUMENT_CODE(" a = " << a << " b = " << b);
Chris@16: 
Chris@16:       if(count && (fa != 0) && !tol(a, b))
Chris@16:       {
Chris@16:          //
Chris@16:          // On the second step we take a quadratic interpolation:
Chris@16:          //
Chris@16:          c = detail::quadratic_interpolate(a, b, d, fa, fb, fd, 2);
Chris@16:          e = d;
Chris@16:          fe = fd;
Chris@16:          detail::bracket(f, a, b, c, fa, fb, d, fd);
Chris@16:          --count;
Chris@16:          BOOST_MATH_INSTRUMENT_CODE(" a = " << a << " b = " << b);
Chris@16:       }
Chris@16:    }
Chris@16: 
Chris@16:    while(count && (fa != 0) && !tol(a, b))
Chris@16:    {
Chris@16:       // save our brackets:
Chris@16:       a0 = a;
Chris@16:       b0 = b;
Chris@16:       //
Chris@16:       // Starting with the third step taken
Chris@16:       // we can use either quadratic or cubic interpolation.
Chris@16:       // Cubic interpolation requires that all four function values
Chris@16:       // fa, fb, fd, and fe are distinct, should that not be the case
Chris@16:       // then variable prof will get set to true, and we'll end up
Chris@16:       // taking a quadratic step instead.
Chris@16:       //
Chris@16:       T min_diff = tools::min_value<T>() * 32;
Chris@16:       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);
Chris@16:       if(prof)
Chris@16:       {
Chris@16:          c = detail::quadratic_interpolate(a, b, d, fa, fb, fd, 2);
Chris@16:          BOOST_MATH_INSTRUMENT_CODE("Can't take cubic step!!!!");
Chris@16:       }
Chris@16:       else
Chris@16:       {
Chris@16:          c = detail::cubic_interpolate(a, b, d, e, fa, fb, fd, fe);
Chris@16:       }
Chris@16:       //
Chris@16:       // re-bracket, and check for termination:
Chris@16:       //
Chris@16:       e = d;
Chris@16:       fe = fd;
Chris@16:       detail::bracket(f, a, b, c, fa, fb, d, fd);
Chris@16:       if((0 == --count) || (fa == 0) || tol(a, b))
Chris@16:          break;
Chris@16:       BOOST_MATH_INSTRUMENT_CODE(" a = " << a << " b = " << b);
Chris@16:       //
Chris@16:       // Now another interpolated step:
Chris@16:       //
Chris@16:       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);
Chris@16:       if(prof)
Chris@16:       {
Chris@16:          c = detail::quadratic_interpolate(a, b, d, fa, fb, fd, 3);
Chris@16:          BOOST_MATH_INSTRUMENT_CODE("Can't take cubic step!!!!");
Chris@16:       }
Chris@16:       else
Chris@16:       {
Chris@16:          c = detail::cubic_interpolate(a, b, d, e, fa, fb, fd, fe);
Chris@16:       }
Chris@16:       //
Chris@16:       // Bracket again, and check termination condition, update e:
Chris@16:       //
Chris@16:       detail::bracket(f, a, b, c, fa, fb, d, fd);
Chris@16:       if((0 == --count) || (fa == 0) || tol(a, b))
Chris@16:          break;
Chris@16:       BOOST_MATH_INSTRUMENT_CODE(" a = " << a << " b = " << b);
Chris@16:       //
Chris@16:       // Now we take a double-length secant step:
Chris@16:       //
Chris@16:       if(fabs(fa) < fabs(fb))
Chris@16:       {
Chris@16:          u = a;
Chris@16:          fu = fa;
Chris@16:       }
Chris@16:       else
Chris@16:       {
Chris@16:          u = b;
Chris@16:          fu = fb;
Chris@16:       }
Chris@16:       c = u - 2 * (fu / (fb - fa)) * (b - a);
Chris@16:       if(fabs(c - u) > (b - a) / 2)
Chris@16:       {
Chris@16:          c = a + (b - a) / 2;
Chris@16:       }
Chris@16:       //
Chris@16:       // Bracket again, and check termination condition:
Chris@16:       //
Chris@16:       e = d;
Chris@16:       fe = fd;
Chris@16:       detail::bracket(f, a, b, c, fa, fb, d, fd);
Chris@16:       BOOST_MATH_INSTRUMENT_CODE(" a = " << a << " b = " << b);
Chris@16:       BOOST_MATH_INSTRUMENT_CODE(" tol = " << T((fabs(a) - fabs(b)) / fabs(a)));
Chris@16:       if((0 == --count) || (fa == 0) || tol(a, b))
Chris@16:          break;
Chris@16:       //
Chris@16:       // And finally... check to see if an additional bisection step is 
Chris@16:       // to be taken, we do this if we're not converging fast enough:
Chris@16:       //
Chris@16:       if((b - a) < mu * (b0 - a0))
Chris@16:          continue;
Chris@16:       //
Chris@16:       // bracket again on a bisection:
Chris@16:       //
Chris@16:       e = d;
Chris@16:       fe = fd;
Chris@16:       detail::bracket(f, a, b, T(a + (b - a) / 2), fa, fb, d, fd);
Chris@16:       --count;
Chris@16:       BOOST_MATH_INSTRUMENT_CODE("Not converging: Taking a bisection!!!!");
Chris@16:       BOOST_MATH_INSTRUMENT_CODE(" a = " << a << " b = " << b);
Chris@16:    } // while loop
Chris@16: 
Chris@16:    max_iter -= count;
Chris@16:    if(fa == 0)
Chris@16:    {
Chris@16:       b = a;
Chris@16:    }
Chris@16:    else if(fb == 0)
Chris@16:    {
Chris@16:       a = b;
Chris@16:    }
Chris@101:    BOOST_MATH_LOG_COUNT(max_iter)
Chris@16:    return std::make_pair(a, b);
Chris@16: }
Chris@16: 
Chris@16: template <class F, class T, class Tol>
Chris@16: 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)
Chris@16: {
Chris@16:    return toms748_solve(f, ax, bx, fax, fbx, tol, max_iter, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: template <class F, class T, class Tol, class Policy>
Chris@16: 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)
Chris@16: {
Chris@16:    max_iter -= 2;
Chris@16:    std::pair<T, T> r = toms748_solve(f, ax, bx, f(ax), f(bx), tol, max_iter, pol);
Chris@16:    max_iter += 2;
Chris@16:    return r;
Chris@16: }
Chris@16: 
Chris@16: template <class F, class T, class Tol>
Chris@16: inline std::pair<T, T> toms748_solve(F f, const T& ax, const T& bx, Tol tol, boost::uintmax_t& max_iter)
Chris@16: {
Chris@16:    return toms748_solve(f, ax, bx, tol, max_iter, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: template <class F, class T, class Tol, class Policy>
Chris@16: 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)
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    static const char* function = "boost::math::tools::bracket_and_solve_root<%1%>";
Chris@16:    //
Chris@16:    // Set up inital brackets:
Chris@16:    //
Chris@16:    T a = guess;
Chris@16:    T b = a;
Chris@16:    T fa = f(a);
Chris@16:    T fb = fa;
Chris@16:    //
Chris@16:    // Set up invocation count:
Chris@16:    //
Chris@16:    boost::uintmax_t count = max_iter - 1;
Chris@16: 
Chris@101:    int step = 32;
Chris@101: 
Chris@16:    if((fa < 0) == (guess < 0 ? !rising : rising))
Chris@16:    {
Chris@16:       //
Chris@16:       // Zero is to the right of b, so walk upwards
Chris@16:       // until we find it:
Chris@16:       //
Chris@16:       while((boost::math::sign)(fb) == (boost::math::sign)(fa))
Chris@16:       {
Chris@16:          if(count == 0)
Chris@101:             return boost::math::detail::pair_from_single(policies::raise_evaluation_error(function, "Unable to bracket root, last nearest value was %1%", b, pol));
Chris@16:          //
Chris@101:          // Heuristic: normally it's best not to increase the step sizes as we'll just end up
Chris@101:          // with a really wide range to search for the root.  However, if the initial guess was *really*
Chris@101:          // bad then we need to speed up the search otherwise we'll take forever if we're orders of
Chris@101:          // magnitude out.  This happens most often if the guess is a small value (say 1) and the result
Chris@101:          // we're looking for is close to std::numeric_limits<T>::min().
Chris@16:          //
Chris@101:          if((max_iter - count) % step == 0)
Chris@101:          {
Chris@16:             factor *= 2;
Chris@101:             if(step > 1) step /= 2;
Chris@101:          }
Chris@16:          //
Chris@16:          // Now go ahead and move our guess by "factor":
Chris@16:          //
Chris@16:          a = b;
Chris@16:          fa = fb;
Chris@16:          b *= factor;
Chris@16:          fb = f(b);
Chris@16:          --count;
Chris@16:          BOOST_MATH_INSTRUMENT_CODE("a = " << a << " b = " << b << " fa = " << fa << " fb = " << fb << " count = " << count);
Chris@16:       }
Chris@16:    }
Chris@16:    else
Chris@16:    {
Chris@16:       //
Chris@16:       // Zero is to the left of a, so walk downwards
Chris@16:       // until we find it:
Chris@16:       //
Chris@16:       while((boost::math::sign)(fb) == (boost::math::sign)(fa))
Chris@16:       {
Chris@16:          if(fabs(a) < tools::min_value<T>())
Chris@16:          {
Chris@16:             // Escape route just in case the answer is zero!
Chris@16:             max_iter -= count;
Chris@16:             max_iter += 1;
Chris@16:             return a > 0 ? std::make_pair(T(0), T(a)) : std::make_pair(T(a), T(0)); 
Chris@16:          }
Chris@16:          if(count == 0)
Chris@101:             return boost::math::detail::pair_from_single(policies::raise_evaluation_error(function, "Unable to bracket root, last nearest value was %1%", a, pol));
Chris@16:          //
Chris@101:          // Heuristic: normally it's best not to increase the step sizes as we'll just end up
Chris@101:          // with a really wide range to search for the root.  However, if the initial guess was *really*
Chris@101:          // bad then we need to speed up the search otherwise we'll take forever if we're orders of
Chris@101:          // magnitude out.  This happens most often if the guess is a small value (say 1) and the result
Chris@101:          // we're looking for is close to std::numeric_limits<T>::min().
Chris@16:          //
Chris@101:          if((max_iter - count) % step == 0)
Chris@101:          {
Chris@16:             factor *= 2;
Chris@101:             if(step > 1) step /= 2;
Chris@101:          }
Chris@16:          //
Chris@16:          // Now go ahead and move are guess by "factor":
Chris@16:          //
Chris@16:          b = a;
Chris@16:          fb = fa;
Chris@16:          a /= factor;
Chris@16:          fa = f(a);
Chris@16:          --count;
Chris@16:          BOOST_MATH_INSTRUMENT_CODE("a = " << a << " b = " << b << " fa = " << fa << " fb = " << fb << " count = " << count);
Chris@16:       }
Chris@16:    }
Chris@16:    max_iter -= count;
Chris@16:    max_iter += 1;
Chris@16:    std::pair<T, T> r = toms748_solve(
Chris@16:       f, 
Chris@16:       (a < 0 ? b : a), 
Chris@16:       (a < 0 ? a : b), 
Chris@16:       (a < 0 ? fb : fa), 
Chris@16:       (a < 0 ? fa : fb), 
Chris@16:       tol, 
Chris@16:       count, 
Chris@16:       pol);
Chris@16:    max_iter += count;
Chris@16:    BOOST_MATH_INSTRUMENT_CODE("max_iter = " << max_iter << " count = " << count);
Chris@101:    BOOST_MATH_LOG_COUNT(max_iter)
Chris@16:    return r;
Chris@16: }
Chris@16: 
Chris@16: template <class F, class T, class Tol>
Chris@16: 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)
Chris@16: {
Chris@16:    return bracket_and_solve_root(f, guess, factor, rising, tol, max_iter, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: } // namespace tools
Chris@16: } // namespace math
Chris@16: } // namespace boost
Chris@16: 
Chris@16: 
Chris@16: #endif // BOOST_MATH_TOOLS_SOLVE_ROOT_HPP
Chris@16: