Local comparison of two sixth order solvers using only the first derivative

Year 2019, Volume: 3 Issue: 4, 220 - 230, 30.12.2019

Santhosh George , İoannis K Argyros

https://doi.org/10.31197/atnaa.581855

Abstract

Two efficient sixth order  solvers are compared involving Banach space valued operators. Earlier papers use hypotheses up to the seventh derivative that do not appear in the solver. But we use hypotheses only on the first derivative. Hence, we expand the applicability of these methods. We use examples to test the older as well as our results.

Keywords

Banach space, Fr\'echet derivative, sixth order of convergence

References

• A. Amiri, A. Cordero, M. Darvishi, J. Torregrosa, Preserving the order of convergence: Low-complexity Jacobian-free iterative schemes for solving nonlinear systems, J. Comput. Appl. Math., 337 (2018), 87-97.
• I.K. Argyros, Computational theory of iterative solvers. Series: Studies in Computational Mathematics, 15, Editors: C.K.Chui and L. Wuytack, Elsevier Publ. Co. New York, U.S.A, 2007.
• I. K.Argyros, A. A. Magr\'e\~nan, A contemporary study of iterative methods, Elsevier (Academic Press), New York, 2018. I. K.Argyros, A. A. Magr\'e\~nan, Iterative methods and their dynamics with applications, CRC Press, New York, USA, 2017. I. K. Argyros, S. George, Mathematical modeling for the solution of equations and systems of equations with applications, Volume-III, Nova Publishes, NY, 2019. A. Cordero, J. L. Hueso, E. Mart\'inez, J. R. Torregrosa, A modified Newton-Jarratt's composition, Numer. Algor., 55, (2010), 87-99. M. Grau-S\'anchez, A. Grau, M. Noguera, Ostrowski-type solvers for solving systems of nonlinear equations, Appl. Math. Comput., 218, (2011), 2377-2385.
• J. Hueso, E. Martinez, C. Teruel, Convergence, efficiency and dynamics of new fourth and sixth order families of iterative solvers for nonlinear systems, Comput. Appl. Math., 275, (2015), 412-420.
• A. A. Magre\~n\'an, A new tool to study real dynamics: The convergence plane, Appl. Math. Comput. 248, (2014), 29-38.
• J.M. Ortega and W.C. Rheinholdt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
• W.C. Rheinboldt,An adaptive continuation process for solving systems of nonlinear equations, In: Mathematical models and numerical solvers (A.N.Tikhonov et al. eds.) pub.3, (1977), 129-142 Banach Center, Warsaw Poland.
• J. Sharma, H. Arora, Efficient Jarratt-like solvers for solving systems of nonlinear equations, Calcolo, 51, (2014), 193--210.
• F. Soleymani, T. Lofti, P. Bakhtiari, A multistep class of iterative solvers for nonlinear systems, Optim. Letters, 8, (2014), 1001-1015.
• J.F.Traub, Iterative solvers for the solution of equations, AMS Chelsea Publishing, 1982.