ON THE GENERALIZATION OF SOME ALGORITHMS FOR THE SIMULTANEOUS APPROXIMATION OF POLYNOMIAL ROOTS

Abstract

This chapter describes the generalization of some algorithms for the simultaneous approximation of polynomial roots. It presents an interval method of the third order for the simultaneous determination of all zeros of a polynomial. This method is an interval extension of Ehrlichs method. Using the square root iteration, a similar interval method of the fourth order can be defined. These methods are based on the formulas of the same type and they are constructed in a similar way. This suggests that these methods can be generalized. The corresponding algorithms are formulated in interval arithmetic. The chapter presents a theorem that defines the generalized interval method for the simultaneous determination of the polynomial simple zeros. It also presents a proof that the iterative interval process has the order of convergence equal to k + 2. A new generalized interval method for the simultaneous determination of all real simple zeros of the polynomial can be derived.