Sehrish Umar,Muhammad Mujtaba Shaikh,Abdul Wasim Shaikh,




Cost-efficient,Quadrature,Nonlinear equations,Order of convergence,Efficiency index,


Nonlinear equations and their efficient numerical solution is a fundamental issue in the field of research in mathematics because nature is full of nonlinear models demanding careful solution and consideration. In this work, a new two-step iterative method for solving nonlinear equations has been developed by using quadrature formula so that the cost of evaluations is considerably reduced. The proposed strategy successfully removes the use of an additional derivative in an existing method in literature so that there is no compromise at all on the cubic convergence rate. The developed scheme is cubically convergent and uses a functional and three derivative evaluations only as compared to some other methods in the literature using much higher evaluations. The theorems concerning the derivation of the proposed method and its third order of convergence have been discussed with proofs. Performance evaluation of the new proposed scheme has been discussed with some methods from literature including well-known traditional methods. An exhaustive numerical verification has been done under the same numerical conditions on ten examples from literature. The efficiency index is found to be higher for the new proposed scheme than some schemes with order more than three, and comparable with some methods. The comparison using observed absolute errors, number of iterations, functional and derivative evaluations, and observed convergence reveals that the proposed method finds the solutions quickly and with lesser computational cost as compared to most of the other methods used in the comparison. The results show the encouraging performance of the proposed method.


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