Optimum $L$-decremented rosebrock schemes with complex coefficients for stiff ODE

The extension of coefficients of Rosenbrock type schemes to the complex numbers allows to construct methods with good $L$-decrementation. Therefore such kind of two-stages schemes were investigated theoretically in detail for the case of autonomous system. There were found out several sets of coefficients which lead to the methods of accuracy $O(\tau^4)$ and of highest $L$-decrementation for stability function as well as for internal stability function.