Summary: A family of seventh-order iterative methods for the solution of nonlinear equations is presented. The new methods are based on R. F. King
’s fourth-order methods [SIAM J. Numer. Anal. 10, 876–879 (1973; Zbl 0266.65040
)] and without using the second derivatives. Per iteration, the new methods require three evaluations of the function and one evaluation of its first derivative. Therefore, this family of methods has efficiency index equal to 1.627. Numerical comparisons are made with several other existing methods to show the performance of the presented methods.