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A variant of Steffensen’s method of fourth-order convergence and its applications. (English) Zbl 1208.65064
A variant of Steffensen’s method is presented, which uses divided differences instead of derivatives. Fourth-order convergence is proved. Numerical tests are given for nonlinear algebraic and ordinary differential equations.
MSC:
65H05Single nonlinear equations (numerical methods)
34A34Nonlinear ODE and systems, general
65L20Stability and convergence of numerical methods for ODE
65L12Finite difference methods for ODE (numerical methods)
References:
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[2]Traub, J. F.: Iterative methods for the solution of equations, (1964) · Zbl 0121.11204
[3]Amat, S.; Busquier, S.: On a Steffensen’s type method and its behavior for semismooth equations, Appl. math. Comput. 177, 819-823 (2006) · Zbl 1096.65047 · doi:10.1016/j.amc.2005.11.032
[4]Alarcón, V.; Amat, S.; Busquier, S.; López, D. J.: A Steffensen’s type method in Banach spaces with applications on boundary-value problems, J. comput. Appl. math. 216, 243-250 (2008) · Zbl 1139.65040 · doi:10.1016/j.cam.2007.05.008
[5]Jain, P.: Steffensen type methods for solving non-linear equations, Appl. math. Comput. 194, 527-533 (2007) · Zbl 1193.65063 · doi:10.1016/j.amc.2007.04.087
[6]Ren, H.; Wu, Q.; Bi, W.: A class of two-step Steffensen type methods with fourth-order convergence, Appl. math. Comput. 209, 206-210 (2009) · Zbl 1166.65338 · doi:10.1016/j.amc.2008.12.039
[7]Zheng, Q.; Wang, J.; Zhao, P.; Zhang, L.: A Steffensen-like method and its higher-order variants, Appl. math. Comput. 214, 10-16 (2009) · Zbl 1179.65052 · doi:10.1016/j.amc.2009.03.053