Two new classes of optimal Jarratt-type fourth-order methods

A Family of Combined Iterative Methods for Solving Nonlinear Equations

“…Recently, a number of authors including [1][2][3][4][5][6] have derived new variants of Newton's methods that offer higher order convergence. These methods are frequently composed of more than two formulas and derived in different ways.…”

Modified Halley's Method for Solving Nonlinear Functions With Convergence of Order Six and Efficiency Index 1.8171

“…Jarrat [11] developed a 4th order method that requires only one function evaluation and two derivative evaluations, and similar 4th order method have been described by Soleymani et al [15]. Jarrat's method is similar to those of Kuo's methods in that if the ratio of derivatives at the predictor and corrector steps exceeds a factor of three, the method gives an infinite change in x. Jarratt's methods is similar to those of Kou in that if the ratio of the derivatives at the predictor and corrector steps exceeds a factor of three, the method gives an infinite change in x.…”

“…During the last century, the numerical techniques for solving nonlinear equations have been successfully applied (see, e.g., [1,2,3,4,5,6,7,8,9,11,12,13,14,15,16,17] and the references therein). McDougall and Wotherspoon [13] modified the Newton's method and their modified Newton's method have convergence of order 1 + √ 2.…”

Construction of fourth-order optimal families of iterative methods and their dynamics

“…With the advancement of computer algebra, many researchers like Chun [8], Chun and Ham [9], Cordero et al [10], Sharma and Ghua [11], Kanwar et al [12], Sharifi et al [13], Soleymani et al [14] and Behl et al [15], among others, proposed various optimal schemes or families of methods of order four. But Ostrowski', Jarratt' and King's methods are between the most efficient fourth-order methods known to date.…”