An iterative method for algebraic equation by Padé approximation
The authors consider iterative formulae with high order of convergence for solving polynomial equations \(f(z)=0\). The numerator of the Padé approximant for \(f(z)/f'(z)\) is derived by combining the algorithm of \textit{B. Viskovatoff} [De la méthode générale pour réduire toutes sortes des quantités en fractions continues. Mémoires de l'Académie Impériale des Sciences de St. Petersburg 1, 226-247 (1803-1806)] with the Euclidean algorithm. The new method has the same order whatever the multiplicity of the zero may be. Numerical examples show that the number of iterations is almost independent of problem. By choosing some parameters and functions, the algorithm can yield the methods of Newton-Raphson, Halley, \textit{I. Kiss} [Z. Angew. Math. Mech. 34, 68-69 (1954; Zbl 0058.337)], \textit{A. W. M. Nourein} [BIT 16, 291-297 (1976; Zbl 0336.65026)] and \textit{T. Pomentale} [Numer. Math. 18, 193-203 (1971; Zbl 0213.418)]. The proposed algorithm is applicable to transcendental equations.
scientific article; zbMATH DE number 4203586
scientific article