On a family of Halley-like methods to find simple roots of nonlinear equations

Halley, 1694, A new, exact and easy method of finding the roots of equations generally and that without any previous reduction, Philos. Trans. R. Soc. London, 18, 136, 10.1098/rstl.1694.0029 Traub, 1964 P. Wynn, On a cubically convergent process for determining the zeros of certain functions. Math. Table & Other Aids Comp. 10, (1956) 97–100; MR 18, p. 418. Candela, 1990, Recurrence relations for rational cubic methods I: The Halley method, Computing, 44, 169, 10.1007/BF02241866 Hernández, 1991, A note on Halley’s method, Numer. Math., 59, 273, 10.1007/BF01385780 Melman, 1997, Geometry and convergence of Euler’s and Halley’s methods, SIAM Rev., 39, 728, 10.1137/S0036144595301140 Scavo, 1995, On the geometry of Halley’s method, Am. Math. Mon., 102, 417, 10.2307/2975033 Hansen, 1977, A family of root finding methods, Numer. Math., 27, 257, 10.1007/BF01396176 Popovski, 1980, A family of one point iteration formulae for finding roots, Int. J. Comput. Math., 8, 85, 10.1080/00207168008803193 Frame, 1944, A variation of Newton’s method, Am. Math. Mon., 51, 36, 10.1080/00029890.1944.11990162 Hartree, 1949, Notes on iterative processes, Proc. Cambridge Philos. Soc., 45, 230, 10.1017/S0305004100024762 Hamilton, 1950, A type of variation on Newton’s method, Am. Math. Mon., 57, 517, 10.2307/2307934 Richmond, 1944, On certain formulae for numerical approximation, J. London Math. Soc., 19, 31, 10.1112/jlms/19.73_Part_1.31 Salehov, 1952, On the convergence of the process of tangent hyperbolas, Dokl. Akad. Nauk. SSSR, 82, 525 Schröder, 1870, Über unendlich viele Algorithmen zur Auflösung der Gleichungen, Math. Ann., 2, 317, 10.1007/BF01444024 Wall, 1948, A modification of Newton’s method, Am. Math. Mon., 55, 90, 10.2307/2305743 Neta, 2006 Petković, 2013 Neta, 2012, Basins of attraction for several methods to find simple roots of nonlinear equations, Appl. Math. Comput., 218, 10548, 10.1016/j.amc.2012.04.017 Householder, 1970 Basto, 2006, A new iterative method to compute nonlinear equations, Appl. Math. Comput., 173, 468, 10.1016/j.amc.2005.04.045 Vrscay, 1988, Extraneous fixed points, basin boundaries and chaotic dynamics for Schröder and König rational iteration functions, Numer. Math., 52, 1, 10.1007/BF01401018 Kneisl, 2001, Julia sets for the super-Newton method Cauchy’s method and Halley’s method, Chaos, 11, 359, 10.1063/1.1368137 M. Basto, V. Semiao, F. Calheiros, Contrasts in the basin of attraction of structurally identical iterative root finding methods, Appl. Math. Comput., in press. Scott, 2011, Basin attractors for various methods, Appl. Math. Comput., 218, 2584, 10.1016/j.amc.2011.07.076 Neta, 2012, Basin attractors for various methods for multiple roots, Appl. Math. Comput., 218, 5043, 10.1016/j.amc.2011.10.071 B.D. Stewart, Attractor Basins of Various Root-Finding Methods, M.S. Thesis, Naval Postgraduate School, Department of Applied Mathematics, Monterey, CA, June 2001. S. Amat, S. Busquier, S. Plaza, Iterative root-finding methods, unpublished report, 2004. Amat, 2004, Review of some iterative root-finding methods from a dynamical point of view, Scientia, 10, 3 Amat, 2004, Dynamics of a family of third-order iterative methods that do not require using second derivatives, Appl. Math. Comput., 154, 735, 10.1016/S0096-3003(03)00747-1 Amat, 2005, Dynamics of the King and Jarratt iterations, Aequationes Math., 69, 212, 10.1007/s00010-004-2733-y Chun, 2012, On optimal fourth-order iterative methods free from second derivative and their dynamics, Appl. Math. Comput., 218, 6427, 10.1016/j.amc.2011.12.013