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2013, 3(3): 549-555. doi: 10.3934/naco.2013.3.549

Another note on some quadrature based three-step iterative methods for non-linear equations

Sanjay Khattri 1,

1. 

Stord Haugesund University College, Department of Engineering, Haugesund, Norway

Received  June 2012 Revised  March 2013 Published  July 2013

The recent paper [H. Ding, Y. Zhang, S. Wang, X. Yang, A note on some quadrature based three-step iterative methods for non-linear equations, Appl. Math. Comput. 215 (1): (2009) 53--57] shows that the Algorithm 2.2 and Algorithm 2.3 in the article [N.A. Mir, T. Zaman, Some quadrature based three-step iterative methods for non-linear equations, Appl. Math. Comput. 193 (2): (2007) 366--373] have twelfth-order and ninth order convergence respectively, not seventh-order as claimed in the later article. In this work; we propose two simple modifications, without increasing the computational cost or functional evaluations, of the Algorithms 2.3 and 2.3. The first modification improves the convergence order of the Algorithm 2.3 from ninth-order to tenth order. In the second modification, we remove evaluation of the second derivative from the Algorithm 2.2 while preserving its twelfth-order convergent nature.
Keywords: tenth order, ninth order, twelfth order, Iterative methods, nonlinear., convergence, Newton.
Mathematics Subject Classification: Primary: 65H05; 41A25; Secondary: 65B1.
Citation: Sanjay Khattri. Another note on some quadrature based three-step iterative methods for non-linear equations. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 549-555. doi: 10.3934/naco.2013.3.549
References:
[1]

ARPREC, C++/Fortran-90 arbitrary precision package,, Available at: \url{http://crd-legacy.lbl.gov/~dhbailey/mpdist/}., ().   Google Scholar

[2]

H. Ding, Y. Zhang, S. Wang and X. Yang, A note on some quadrature based three-step iterative methods for non-linear equations, Appl. Math. Comput., 215 (2009), 53-57. doi: 10.1016/j.amc.2009.04.036.  Google Scholar

[3]

N. A. Mir and T. Zaman, Some quadrature based three-step iterative methods for non-linear equations, Appl. Math. Comput., 193 (2007), 366-373. doi: 10.1016/j.amc.2007.03.071.  Google Scholar

show all references

References:
[1]

ARPREC, C++/Fortran-90 arbitrary precision package,, Available at: \url{http://crd-legacy.lbl.gov/~dhbailey/mpdist/}., ().   Google Scholar

[2]

H. Ding, Y. Zhang, S. Wang and X. Yang, A note on some quadrature based three-step iterative methods for non-linear equations, Appl. Math. Comput., 215 (2009), 53-57. doi: 10.1016/j.amc.2009.04.036.  Google Scholar

[3]

N. A. Mir and T. Zaman, Some quadrature based three-step iterative methods for non-linear equations, Appl. Math. Comput., 193 (2007), 366-373. doi: 10.1016/j.amc.2007.03.071.  Google Scholar

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