Abstract <p>When explicit stabilized Runge–Kutta methods are used to solve stiff systems of ordinary differential equations, it is necessary to know an estimate for the spectral radius of the Jacobian matrix. Such an estimate can be obtained by applying the Gershgorin circle theorem or the power method. This article explores spectral radius estimation procedures based on the nonlinear power method without calculating the Jacobian matrix. The proposed procedures are incorporated into the integration method and are able to estimate the spectral radius when it changes during the solution process. Examples of solving test problems are given.</p>