Modified optimal families of King's and Ostrowski's method for solving nonlinear equations (Q2923266)

The authors present two parameter modified families of King's method [\textit{R. F. King}, SIAM J. Numer. Anal. 10, 876--879 (1973; Zbl 0266.65040)] and Ostrowski's method [\textit{A. M. Ostrowski}, Solution of equations and systems of equations. 2nd ed. Pure and Applied Mathematics, 9. New York-London: Academic Press (1966; Zbl 0222.65070)] having biquartic convergence for computing simple roots of nonlinear equations, permitting \(f'(x) = 0\) in the vicinity of the required root. King's family and Ostrowski's method for simple roots can be seen as special cases of the proposed scheme. The performance of the proposed multipoint methods is compared with King's method, Ostrowski's method and Jarratt's method [\textit{P. Jarratt}, BIT, Nord. Tidskr. Inf.-behandl. 9, 119--124 (1969; Zbl 0188.22101)] in some numerical experiments. All the methods considered are more effective than the similar robust methods available in the literature.