*Article* **On a Reduced Cost Higher Order Traub-Steffensen-Like Method for Nonlinear Systems**

**Janak Raj Sharma 1, Deepak Kumar 1,\* and Lorentz Jäntschi 2,\***


Received: 30 May 2019; Accepted: 4 July 2019; Published: 8 July 2019

**Abstract:** We propose a derivative-free iterative method with fifth order of convergence for solving systems of nonlinear equations. The scheme is composed of three steps, of which the first two steps are that of third order Traub-Steffensen-type method and the last is derivative-free modification of Chebyshev's method. Computational efficiency is examined and comparison between the efficiencies of presented technique with existing techniques is performed. It is proved that, in general, the new method is more efficient. Numerical problems, including those resulting from practical problems viz. integral equations and boundary value problems, are considered to compare the performance of the proposed method with existing methods. Calculation of computational order of convergence shows that the order of convergence of the new method is preserved in all the numerical examples, which is not so in the case of some of the existing higher order methods. Moreover, the numerical results, including the CPU-time consumed in the execution of program, confirm the accurate and efficient behavior of the new technique.

**Keywords:** nonlinear equations; systems; derivative-free methods; fast algorithms; computational efficiency
