Controlled Runge Kutta Implementation Doesn't Match Documentation

Read the controlled Runge-Kutta implementation in include/boost/numeric/odeint/stepper/controlled_runge_kutta.hpp alongside the linked steppers documentation. Confirm which step-size formula is intended, then update the documentation so it matches the confirmed behavior and verify the rendered formula is accurate.

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.