FACTA UNIVERSITATIS (NI ˇ S) Ser. Math. Inform. Vol. 22, No. 1 (2007), pp. 91–103 ON THE CONVERGENCE OF THE THIRD ORDER ROOT-SOLVER ∗ M. S. Petkovi´ c, Lj. D. Petkovi´ c and S. M. Ili´ c Abstract. The construction of computationally verifiable initial conditions that provide both the guaranteed and fast convergence of a numerical method for solving nonlinear equations is one of the most important tasks in the field of iterative processes. A suitable convergence procedure, based partially on Smale’s “point estimation theory” from 1981, is applied in this paper to a new cubically convergent derivative free iterative method for the simultaneous approximation of simple zeros of polynomials. We have stated initial conditions which guarantee the convergence of this method. These conditions are of significant practical importance since they depend only on available data: the coefficients of a given polynomial, its degree n and initial approximations to polynomial zeros. 1. Introduction Last years a great attention is paid to state computationally verifiable initial conditions which enable both the guaranteed and fast convergence of the applied iterative method for solving a nonlinear equation f (z )=0. This challenging problem of the theory and practice of iterative processes is often considered in the literature during the last fifty years, but the re- sults were rather of theoretical importance; namely, the established initial conditions depend on unattainable data such as suitable (but unknown) con- stants, “reasonable good initial approximations” (without a proper estimate of their accuracy), or even the sought zeros of an equation to be solved. In general, the construction of computationally verifiable initial condi- tions is a very difficult problem, even in the case of simple functions such as Received February 2, 2007. 2000 Mathematics Subject Classification. 65H05. ∗ This work was supported by the Serbian Ministry of Science under grant 144024. 91