On a generalisation of the root iterations for polynomial complex zeros in circular interval arithmetic

Consider a polynomialP (z) of degreen whose zeros are known to lie inn closed disjoint discs, each disc containing one and only one zero. Starting from the known simultaneous interval processes of the third and fourth order, based on Laguerre iterations, two generalised iterative methods in terms of circular regions are derived in this paper. These interval methods make use of the definition of thek-th root of a disc. The order of convergence of the proposed interval methods isk+2 (k≧1). Both procedures are suitable for simultaneous determination of interval approximations containing real or complex zeros of the considered polynomialP. A criterion for the choice of the appropriatek-th root set is also given. For one of the suggested methods a procedure for accelerating the convergence is proposed. Starting from the expression for interval center, the generalised iterative method of the (k+2)-th order in standard arithmetic is derived.