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.
Highlights
We are concerned with the problem of solving a system of nonlinear equations F ( x ) = 0. (1)This problem can precisely be stated as to find a solution vector α = (α1, α2, ..., αm ) T such that F (α) = 0, where F( x ) : D ⊂ Rm −→ Rm is the given nonlinear vector function F ( x ) =( f 1 ( x ), f 2 ( x ), ..., f m ( x ))T and x = ( x1, x2, ..., xm )T
In the foregoing study, we have developed a fifth order iterative method for approximating solution of systems of nonlinear equations
The methodology is based on third order Traub-Steffensen method and further developed by using derivative free modification of classical Chebyshev’s method
Summary
We are concerned with the problem of solving a system of nonlinear equations F ( x ) = 0. This problem can precisely be stated as to find a solution vector α = (α1 , α2 , ..., αm ) T such that F (α) = 0, where F( x ) : D ⊂ Rm −→ Rm is the given nonlinear vector function F ( x ) =. F ( x ) = 0 of nonlinear equations (see, for example [1,2,3,4,5,6]). Symmetry 2019, 11, 891 in literature for solving such equations. A classical method is cubically convergent Chebyshev’s method (see [7]) x (0) ∈ D,
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