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@(@\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }@)@This is cppad-20221105 documentation. Here is a link to its current documentation .
Do One Thread's Work for Multi-Threaded Newton Method

Syntax
multi_newton_worker()

Purpose
This function finds all the zeros in the interval [ low , up ] .

low
This is the value of the multi_newton_common information
    low = work_all_[thread_num]->xlow
up
This is the value of the multi_newton_common information
    up = work_all_[thread_num]->xup
thread_num
This is the number for the current thread; see thread_num .

Source 

namespace {
void multi_newton_worker(void)
{
    // Split [xlow, xup] into num_sub intervales and
    // look for one zero in each sub-interval.
    size_t thread_num    = thread_alloc::thread_num();
    size_t num_threads   = std::max(num_threads_, size_t(1));
    bool   ok            = thread_num < num_threads;
    size_t num_sub       = work_all_[thread_num]->num_sub;
    double xlow          = work_all_[thread_num]->xlow;
    double xup           = work_all_[thread_num]->xup;
    vector<double>& x    = work_all_[thread_num]->x;

    // check arguments
    ok &= max_itr_ > 0;
    ok &= num_sub > 0;
    ok &= xlow < xup;
    ok &= x.size() == 0;

    // check for special case where there is nothing for this thread to do
    if( num_sub == 0 )
    {   work_all_[thread_num]->ok = ok;
        return;
    }

    // check for a zero on each sub-interval
    size_t i;
    double xlast = xlow - 2.0 * sub_length_; // over sub_length_ away from x_low
    double flast = 2.0 * epsilon_;           // any value > epsilon_ would do
    for(i = 0; i < num_sub; i++)
    {
        // note that when i == 0, xlow_i == xlow (exactly)
        double xlow_i = xlow + double(i) * sub_length_;

        // note that when i == num_sub - 1, xup_i = xup (exactly)
        double xup_i  = xup  - double(num_sub - i - 1) * sub_length_;

        // initial point for Newton iterations
        double xcur = (xup_i + xlow_i) / 2.;

        // Newton iterations
        bool more_itr = true;
        size_t itr    = 0;
        // initialize these values to avoid MSC C++ warning
        double fcur=0.0, dfcur=0.0;
        while( more_itr )
        {   fun_(xcur, fcur, dfcur);

            // check end of iterations
            if( fabs(fcur) <= epsilon_ )
                more_itr = false;
            if( (xcur == xlow_i ) & (fcur * dfcur > 0.) )
                more_itr = false;
            if( (xcur == xup_i)   & (fcur * dfcur < 0.) )
                more_itr = false;

            // next Newton iterate
            if( more_itr )
            {   xcur = xcur - fcur / dfcur;
                // keep in bounds
                xcur = std::max(xcur, xlow_i);
                xcur = std::min(xcur, xup_i);

                more_itr = ++itr < max_itr_;
            }
        }
        if( fabs( fcur ) <= epsilon_ )
        {   // check for case where xcur is lower bound for this
            // sub-interval and upper bound for previous sub-interval
            if( fabs(xcur - xlast) >= sub_length_ )
            {   x.push_back( xcur );
                xlast = xcur;
                flast = fcur;
            }
            else if( fabs(fcur) < fabs(flast) )
            {   x[ x.size() - 1] = xcur;
                xlast            = xcur;
                flast            = fcur;
            }
        }
    }
    work_all_[thread_num]->ok = ok;
}
}
Input File: example/multi_thread/multi_newton.cpp