On an iterative algorithm of Ulm-type for solving equations
DOI:
https://doi.org/10.33993/jnaat422-986Keywords:
Banach space, iterative algorithm, semilocal convergence, divided difference of operator, Fréchet-derivativeAbstract
We provide a semilocal convergence analysis of an iterative algorithm for solving nonlinear operator equations in a Banach space setting. This algorithm is of order \(1.839\ldots\), and has already been studied in [3, 8, 18, 20]. Using our new idea of recurrent functions we show that a finer analysis is possible with sufficient convergence conditions that can be weaker than before, and under the same computational cost. Numerical examples are also provided in this study.Downloads
References
S. Amat and S. Busquier, A modified secant method for semismooth equations, Appl. Math. Lett., 16 (2003) no. 6, pp. 877-881, https://doi.org/10.1016/S0893-9659(03)90011-5 DOI: https://doi.org/10.1016/S0893-9659(03)90011-5
S. Amat and S. Busquier, On a higher order Secant method, Appl. Math. Comput., 141 (2003), pp. 321-329, https://doi.org/10.1016/S0096-3003(02)00257-6 DOI: https://doi.org/10.1016/S0096-3003(02)00257-6
S. Amat, S.Busquier and V. F. Candela, A class of quasi-Newton generalized Steffensen methods on Banach spaces, J. Comput. Appl. Math., 149 (2002), no. 2, pp. 397-406, https://doi.org/10.1016/S0377-0427(02)00484-3 DOI: https://doi.org/10.1016/S0377-0427(02)00484-3
S. Amat, S. Busquier and J. M. Gutiérrez, On the local convergence of secant-type methods, Int. J. Comput. Math., 81 (2004), pp. 1153-1161. DOI: https://doi.org/10.1080/00207160412331284123
N. Anderson and Å. Björck, A new high order method of regula falsi type for computing a root of an equation, BIT, 13 (1973), pp. 253—264, https://doi.org/10.1007/BF01951936 DOI: https://doi.org/10.1007/BF01951936
I. K. Argyros The secant method in generalized Banach spaces, Appl. Math. Comput., 39 (1990), pp. 111—121, https://doi.org/10.1016/0096-3003(90)90026-Y DOI: https://doi.org/10.1016/0096-3003(90)90026-Y
I. K. Argyros A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space, J. Math. Anal. Appl., 298 (2004), pp. 374-397, https://doi.org/10.1016/j.jmaa.2004.04.008 DOI: https://doi.org/10.1016/j.jmaa.2004.04.008
I. K. Argyros, Newton Methods, Nova Science Publ. Inc., New York, 2005.
R. P. Brent, Algorithms for Minimization without Derivatives, Prentice Hall, Englewood Cliffs, New Jersey, 1973.
E. Cătinaş, On some iterative methods for solving nonlinear equations, Rev. Anal. Numér. Théor. Approx., 23 (1994), pp. 47-53, http://ictp.acad.ro/jnaat/journal/article/view/1994-vol23-no1-art
J. A. Ezquerro and M. A. Hernández, Multipoint super-Halley type approximation algorithms in Banach spaces, Numer. Funct. Anal. Optim., 21 (2000), no. 7-8, pp. 845-858, https://doi.org/10.1080/01630560008816989 DOI: https://doi.org/10.1080/01630560008816989
M. A. Hernández, M. J. Rubio and J. A. Ezquerro, Secant-like methods for solving nonlinear integral equations of the Hammerstein type, J. Comput. Appl. Math., 115 (2000), pp. 245-254, https://doi.org/10.1016/S0377-0427(99)00116-8 DOI: https://doi.org/10.1016/S0377-0427(99)00116-8
M. A. Hernández and M. J.Rubio, Semilocal convergence of the secant method under mild convergence conditions of differentiability, Comput. Math. Appl., 44 (2002), pp. 277-285, https://doi.org/10.1016/S0898-1221(02)00147-5 DOI: https://doi.org/10.1016/S0898-1221(02)00147-5
L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982.
M. A. Mertvecova, An analog of the process of tangent hyperbolas for general functional equations (Russian), Dokl. Akad. Navk. SSSR, 88 (1953), pp. 611-614.
J. M. Ortega, and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York 1970.
I. Păvăloiu, A convergence theorem concerning the method of chord, Rev. Anal. Numér. Théor. Approx., 21 (1972), no.1, pp. 59-65.
F. A. Potra, An iterative algorithm of order 1.839… for solving nonlinear operator equations, Numer. Funct. Anal. Optim., 7 (1984-85), no. 1, pp. 75-106, https://doi.org/10.1080/01630568508816182 DOI: https://doi.org/10.1080/01630568508816182
W. C. Rheinboldt, An adaptive continuation process for solving systems of nonlinear equations, Polish Academy of Science, Banach Ctr. Publ., 3 (1977), pp. 129-142. DOI: https://doi.org/10.4064/-3-1-129-142
S. Uľm, Iteration methods with divided differences of the second order, (Russian), Dokl. Akad. Nauk SSSR, 158 (1964), pp. 55-58, Soviet Math. Dokl., 5, pp. 1187-1190.
J. Vandergraft, Newton's method for convex operators in partially ordered spaces, SIAM J. Numer. Anal., 4 (1967) pp. 406-432, https://doi.org/10.1137/0704037 DOI: https://doi.org/10.1137/0704037
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