Some efficient derivative free methods with memory for solving nonlinear equations

Based on the family of three-point derivative free methods without memory of eighth order convergence proposed by Džunić et al. [J. Džunić, M.S. Petković, L.D. Petković, Three-point methods with and without memory for solving nonlinear equations, Appl. Math. Comput. 218 (2012) 4917–4927], we present three methods with memory by suitable variation of a free parameter in each iterative step. This free parameter is calculated using Newton’s interpolatory polynomial of the third degree in two ways and Newton’s interpolatory polynomial of the fourth degree. Consequently, the R  -order of convergence is increased from 8 to 11+1372≈11.352, 6+42≈11.657 and 12. The increase in the convergence order is achieved without any additional function evaluations and therefore, the proposed methods possess a very high computational efficiency. Numerical examples are presented and the performance is compared with the existing three-point methods with and without memory of the basic family. Moreover, theoretical order of convergence is verified on the examples.