Diagnostics of singularities of exact solutions in computations with error control

Diagnostics of singularities of exact solutions to ordinary differential equations numerically solved using conventional schemes are considered. It is shown that the one-stage Rosenbrock scheme with complex coefficients, which was initially proposed for stiff problems, prevents overflowing when solving ill-conditioned problems. An algorithm for the diagnostics of the types of singularities and other specific features of the exact solutions obtained using this scheme on embedded grids is theoretically justified and illustrated by way of examples.