Efficient Halley-like Methods for the Inclusion of Multiple Zeros of Polynomials

Abstract Starting from suitable zero-relation, we derive higher-order iterative methods for the simultaneous inclusion of polynomial multiple zeros in circular complex interval arithmetic. The convergence rate is increased using a family of two-point methods of the fourth order for solving nonlinear equations as a predictor. The methods are more efficient compared to existing inclusion methods for multiple zeros, based on fixed point relations. Using the concept of the R-order of convergence of mutually dependent sequences, we present the convergence analysis of the total-step and the single-step methods. The proposed self-validated methods possess a great computational efficiency since the acceleration of the convergence rate from four to seven is achieved only by a few additional calculations. To demonstrate convergence behavior of the presented methods, two numerical examples are given.

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