Controlled Runge Kutta Implementation Doesn't Match Documentation

[mczek]

In the case where the time step is increased the function increase_step is used from include/boost/numeric/odeint/stepper/controlled_runge_kutta.hpp which I've copied below:

    time_type increase_step(time_type dt, value_type error, const int stepper_order) const
    {
        // returns the increased time step
        BOOST_USING_STD_MIN();
        BOOST_USING_STD_MAX();
        using std::pow;

        // adjust the size if dt is smaller than max_dt (providede max_dt is not zero)
        if(error < 0.5)
        {
            // error should be > 0
            error = max BOOST_PREVENT_MACRO_SUBSTITUTION (
                    static_cast<value_type>( pow( static_cast<value_type>(5.0) , -static_cast<value_type>(stepper_order) ) ) ,
                    error);
            // time_type dt_old = dt;   unused variable warning 
            //error too small - increase dt and keep the evolution and limit scaling factor to 5.0
            dt *= static_cast<value_type>(9)/static_cast<value_type>(10) *
                  pow(error, static_cast<value_type>(-1) / stepper_order);
            if(m_max_dt != static_cast<time_type >(0))
                // limit to maximal stepsize
                dt = detail::min_abs(dt, m_max_dt);
        }
        return dt;
    }

Inside of the control flow, $error = \max(5^{-S}, error)$ and then dt is multiplied by $0.9 * error^{-1/S} = 0.9*\min(5, error^{-1/S})$ which limits the maximum step size change to a factor of 4.5.

The documentation here states that if val < 0.5 : dt​_new = dt​_current min( 0.9 pow( val , -1 / O​S ) , 5 ) which has the factor of 0.9 inside the minimum. In this case the maximum step size change is instead limited to a factor of 5.

I'll offer to attempt a PR for the documentation if someone confirms my understanding is correct.