Abstract Improved iterative method of Euler’s type for the simultaneous inclusion of polynomial zeros is considered. To accelerate the convergence of the basic method of the fourth order we applied Börsch–Supan’s correction. It is proved that the R-order of convergence of the improved Euler-like method is six. The convergence analysis is derived under computationally verifiable initial conditions. The proposed algorithm possesses great computational efficiency since the increase of the convergence rate from 4 to 6 is obtained with negligible number of additional calculations. In order to demonstrate convergence properties of the suggested method, two numerical examples are given.