On Some Highly Efficient Derivative Free Methods With and Without Memory for Solving Nonlinear Equations

Abstract

We present derivative free multipoint methods of optimal eighth and sixteenth order convergence for solving nonlinear equations. The schemes are based on derivative free two-point methods proposed by Petković et al. [Petković, M. S., Džunić, J. and Petković, L. D. [2011] "A family of two-point methods with memory for solving nonlinear equations," Appl. Anal. Discrete Math.5, 298–317], which further developed by using rational approximations. Extending the work further, we explore four-point methods with memory with increasing order of convergence from the basic four-point scheme without memory. The order is increased from 16 of the basic method to 20, 22, 23, 23.662, and 24 by suitable variation of a free parameter in each iterative step. This increase in the convergence order is achieved without any additional function evaluations and therefore, the methods with memory possess better computational efficiency than the methods without memory. Numerical examples are presented and the performance is compared with the existing optimal three and four-point methods. Computational results and comparison with the existing methods confirm efficient and robust character of present methods.