Euler-like Method for the Simultaneous Inclusion of Polynomial Zeros with Weierstrass’ Correction

An 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 fourth order, Carstensen-Petković’s approach [7] using Weierstrass’ correction is applied. It is proved that the R-order of convergence of the improved Euler-like method is (asymptotically) 2 + √ 7 ≈ 4.646 or 5, depending of the type of applied inversion of a disk. The proposed algorithm possesses great computational efficiency since the increase of the convergence rate is obtained without additional calculations. Initial conditions which provide the guaranteed convergence of the considered method are also studied. These conditions are computationally verifiable, which is of practical importance.