Seventh Convergence Order Solvers Free Of Derivatives For Solving Equations In Banach Space

Seventh convergence order solvers free of derivatives

  • Santhosh George National Institute of Technology Karnataka
  • Ioannis K. Argyros Cameron University
Keywords: Euclidean space, Banach space, Local convergence, Seventh order Solver, Divided Difference, Fr ́echetderivative

Abstract

We study a seventh convergence order solver introduced earlier on the j−dimensional Euclidean space for solving systems of equations. We use hypotheses only on the divided differences of order one in contrast to the earlier study using hypotheses on derivatives reaching up to order eight although these derivatives do not appear on the solver. This way we expand the applicability of the solver, and in the more general setting of Banach space valued operators. Numerical examples complement the theoretical results.

Downloads

Download data is not yet available.

References

A. Amiri, A. Cardero, M. T. Darvishi, J. R. Torregrosa, Stability analysis ofa parametric family of seventh-order iterative methods for solving nonlinearsystems, Appl. Math. Comput., 323, (2018), 43-57.

I. K. Argyros, J. A. Ezquerro, J. M. Guti ́errez, M.A. Her ́nandez, S. Hilout,On the semilocal convergence of efficient Chebyshev-Secant -type solvers,J. Comput. Appl. Math., 235(2011), 3195-3206.

I. K. Argyros, H. Ren, Efficient Steffensen-type algorithms for solving non-linear equations, Int. J. Comput. Math., 90, (2013), 691-704.

I. K. Argyros, S. George, N. Thapa, Mathematical modeling for the solutionof equations and systems of equations with applications, Volume-I, NovaPublishes, NY, 2018.

I. K. Argyros, S. George, N. Thapa, Mathematical modeling for the solutionof equations and systems of equations with applications, Volume-II, NovaPublishes, NY, 2018.

I. K. Argyros and S. Hilout, Weaker conditions for the convergence of New-tons solver. Journal of Complexity, 28(3):364-387, 2012.

A. Cordero, J. L. Hueso, E. Mart ́ınez, J. R. Torregrosa, A modified Newton-Jarratt’s composition, Numer. Algor., 55, (2010), 87-99.

A. Cordero and J. R. Torregrosa, Variants of Newton’s method using fifth-order quadrature formulas, Appl. Math. Comput., 190, (2007), 686-698.9

A. Cordero, J. L. Hueso, E. Mart ́ınez, J. R. Torregrosa, A modified Newton-Jarratt’s composition, Numer. Algor., 55, (2010), 87-99.

M. Grau-S ́anchez, M. Noguera, S. Amat, On the approximation of deriva-tives using divided difference operators preserving the local convergenceorder of iterative solvers, J. Comput. Appl. Math., 237,(2013), 363-372.

T. Lotfi, P. Bakhtiari, A. Cordero, K. Mahdiani, J. R. Torregrosa, Somenew efficient multipoint iterative solvers for solving nonlinear systems ofequations, Int. J. Comput. Math., 92, (2015), 1921-1934.

A. A. Magre ̃n ́an and I. K .Argyros, Improved convergence analysis forNewton-like solvers. Numerical Algorithms, 71(4):811-826, 2016.

A. A. Magre ̃n ́an, A.Cordero, J. M. Guti ́errez, and J. R. Torregrosa, Realqualitative behavior of a fourth-order family of iterative solvers by using theconvergence plane. Mathematics and Computers in Simulation, 105:49-61,2014.

A. A. Magre ̃n ́an and I. K. Argyros, Two-step Newton solvers. Journal ofComplexity, 30(4):533-553, 2014.

A. M. Osrowski, Solution of equations and systems of equations, AcademicPress, New York, 1960.

J. M. Ortega , W. C. Rheinboldt, Iterative Solution of Nonlinear Equationsin Several Variables, Academic Press, New York, 1970.

F.A. Potra and V. Pt ́ak, Nondiscrete induction and iterative processes,volume 103. Pitman Advanced Publishing Program, 1984.

J. F. Steffensen, Remarks on iteration, Skand, Aktuar, 16 (1993), 64-72.

J. R. Sharma, H. Arora, Improved Newton-like solvers for solving systemsof nonlinear equations, SeMA J., 74,2(2017), 147-163.

J. R. Sharma, H. Arora, An efficient derivative free iterative method forsolving systems of nonlinear equations, Appl. Anal. Discrete Math., 7,(2013), 390-403.

J. R. Sharma, H. Arora, A novel derivative free algorithm with seventhorder convergence for solving systems of nonlinear equations, Numer. Algor,67, (2014), 917-933.

J.R. Sharma, P.K. Gupta, An efficient fifth order solver for solving systemsof nonlinear equations, Comput. Math. Appl. 67, (2014), 591–601

Published
2019-08-30
How to Cite
George, S., & Argyros, I. K. (2019). Seventh Convergence Order Solvers Free Of Derivatives For Solving Equations In Banach Space. MathLAB Journal, 3, 118-127. Retrieved from http://purkh.com/index.php/mathlab/article/view/445
Section
Research Articles