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Higher-order simultaneous methods for the determination of polynomial multiple zeros

Starting from Laguerre's method and using Newton's and Halley's corrections for a multiple zero, new simultaneous methods of Laguerre's type for finding multiple (real or complex) zeros of polynomials are constructed. The convergence order of the proposed methods is five and six, respectively. By applying the Gauss–Seidel approach, these methods are further accelerated. The lower bounds of the R-order of convergence of the improved (single-step) methods are derived. Faster convergence of all proposed methods is attained with negligible number of additional operations, which provides a high computational efficiency of these methods. A detailed convergence analysis and numerical results are given.