Geodesic
Improved convergence analysis of the Secant method using restricted convergence domains with real-world applications
Argyros, Ioannis K. 1 ; Magreñán, Alberto2 ; Sarría, Íñigo2 ; Sicilia, Juan Antonio 3
1 Department of Mathematics Sciences, Cameron University, Lawton, OK 73505, USA
2 Escuela Superior de Ingeniería y Tecnología, Universidad Internacional de La Rioja, Av de la Paz, 137, 26002 Logroño, La Rioja, Spain
3 Escuela Superior de Ingeniería y Tecnología, Universidad Internacional de La Rioja, Av de la Paz, 137, 26002 Logroño, Spain, Av de la Paz, 137, 26002 Logroño, La Rioja, Spain
Journal of nonlinear sciences and its applications, Tome 11 (2018) no. 11, p. 1215-1224.

Voir la notice de l'article provenant de la source International Scientific Research Publications

In this paper, we are concerned with the problem of approximating a solution of a nonlinear equations by means of using the Secant method. We present a new semilocal convergence analysis for Secant method using restricted convergence domains. According to this idea we find a more precise domain where the inverses of the operators involved exist than in earlier studies. This way we obtain smaller Lipschitz constants leading to more precise majorizing sequences. Our convergence criteria are weaker and the error bounds are more precise than in earlier studies. Under the same computational cost on the parameters involved our analysis includes the computation of the bounds on the limit points of the majorizing sequences involved. Different real-world applications are also presented to illustrate the theoretical results obtained in this study.
DOI : 10.22436/jnsa.011.11.01
Classification : 65H10, 65G99, 65B05, 65N30, 47H17, 49M15
Keywords: Secant method, Banach space, majorizing sequence, divided difference, local convergence, semilocal convergence

Argyros, Ioannis K.  1 ; Magreñán, Alberto 2 ; Sarría, Íñigo 2 ; Sicilia, Juan Antonio  3

1 Department of Mathematics Sciences, Cameron University, Lawton, OK 73505, USA
2 Escuela Superior de Ingeniería y Tecnología, Universidad Internacional de La Rioja, Av de la Paz, 137, 26002 Logroño, La Rioja, Spain
3 Escuela Superior de Ingeniería y Tecnología, Universidad Internacional de La Rioja, Av de la Paz, 137, 26002 Logroño, Spain, Av de la Paz, 137, 26002 Logroño, La Rioja, Spain 
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     journal = {Journal of nonlinear sciences and its applications},
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Argyros, Ioannis K. ; Magreñán, Alberto; Sarría, Íñigo; Sicilia, Juan Antonio . Improved convergence analysis of the Secant method using restricted convergence domains with real-world applications. Journal of nonlinear sciences and its applications, Tome 11 (2018) no. 11, p. 1215-1224. doi : 10.22436/jnsa.011.11.01. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.011.11.01/

[1] Amat, S.; Busquier, S. On a higher order Secant method, Appl. Math. Comput., Volume 141 (2003), pp. 321-329 | DOI

[2] Amat, S.; Busquier, S.; Negra, M. Adaptive approximation of nonlinear operators, Numer. Funct. Anal. Optim., Volume 25 (2004), pp. 397-405 | DOI

[3] Amat, S.; Ezquerro, J. A.; Hernández-Vern, M. A. Approximation of inverse operators by a new family of high-order iterative methods, Numer. Linear Algebra Appl., Volume 21 (2014), pp. 629-664 | DOI | Zbl

[4] Amat, S.; Hernández-Vern, M. A.; Rubio, M. J. Improving the applicability of the Secant method to solve nonlinear systems of equations, Appl. Math. Comput., Volume 247 (2014), pp. 741-752 | DOI | Zbl

[5] Argyros, I. K. Computational theory of iterative methods, Elsevier B. V., Amsterdam, 2007

[6] Argyros, I. K.; Cho, Y. J.; S. Hilout Numerical method for equations and its applications, CRC Press, Boca Raton, 2012

[7] Argyros, I. K.; González, D.; Á . A. Magreñán A semilocal convergence for a uniparametric family of efficient Secant-like methods , J. Funct. Spaces, Volume 2014 (2014), pp. 1-10 | Zbl

[8] Argyros, I. K.; Hilout, S. Weaker conditions for the convergence of Newton’s method , J. Complexity, Volume 28 (2012), pp. 364-387 | DOI

[9] Argyros, I. K.; Á . A. Magreñán A unified convergence analysis for secant-type methods , J. Korean Math. Soc., Volume 51 (2014), pp. 1155-1175 | DOI | Zbl

[10] Argyros, I. K.; Magreñán, Á . A. Expanding the applicability of the secant method under weaker conditions , Appl. Math. Comput., Volume 266 (2015), pp. 1000-1012 | DOI

[11] Argyros, I. K.; Magreñán, Á . A. Relaxed Secant-type methods, Nonlinear Stud., Volume 21 (2014), pp. 485-503 | Zbl

[12] E. Cătinaş The inexact, inexact perturbed, and quasi-Newton methods are equivalent models, Math. Comp., Volume 74 (2005), pp. 291-301 | Zbl | DOI

[13] Chandrasekhar, S. Radiative transfer, Dover Publ., New York, 1960

[14] Dennis, J. E. Toward a unified convergence theory for Newton-like methods, Nonlinear Functional Anal. and Appl., Volume 1970 (1970), pp. 425-472 | DOI | Zbl

[15] Ezquerro, J. A.; Gutiérrez, J. M.; Hernández, M. A.; Romero, N.; Rubio, M. J. The Newton method: from Newton to Kantorovich (Spanish), Gac. R. Soc. Mat. Esp., Volume 13 (2010), pp. 53-76 | Zbl

[16] Farhane, N.; Boumhidi, I.; Boumhidi, J. Smart Algorithms to Control a Variable Speed Wind Turbine, Int. J. Interact. Multimed. Artif. Intell., Volume 4 (2017), pp. 88-95 | DOI

[17] Gragg, W. B.; Tapia, R. A. Optimal error bounds for the Newton-Kantorovich theorem, SIAM J. Numer. Anal., Volume 11 (1974), pp. 10-13 | Zbl | DOI

[18] Hernández, M. A.; Rubio, M. J.; J. A. Ezquerro Secant-like methods for solving nonlinear integral equations of the Hammerstein type, J. Comput. Appl. Math., Volume 115 (2000), pp. 245-254 | DOI | Zbl

[19] Jain, D. Families of Newton-like methods with fourth-order convergence, Int. J. Comput. Math., Volume 90 (2013), pp. 1072-1082 | DOI | Zbl

[20] Kantorovich, L. V.; P., G.; Akilov Functional Analysis (Translated from the Russian by Howard L. Silcock), Pergamon Press, Oxford-Elmsford, 1982

[21] Kaur, R.; Arora, S. Nature Inspired Range BasedWireless Sensor Node Localization Algorithms, Int. J. Interact. Multimed. Artif. Intell., Volume 4 (2017), pp. 7-17

[22] Magreñán, Á . A. A new tool to study real dynamics: The convergence plane, Appl. Math. Comput., Volume 248 (2014), pp. 215-224 | Zbl | DOI

[23] Ortega, J. M.; Rheinboldt, W. C. Iterative Solution of Nonlinear Equations in Several Variables, Academic press , , 1970

[24] Potra, F. A.; Pták, V. Nondiscrete induction and iterative processes, Pitman (Advanced Publishing Program), Boston, 1984

[25] Proinov, P. D. General local convergence theory for a class of iterative processes and its applications to Newton’s method , J. Complexity, Volume 25 (2009), pp. 38-62 | DOI | Zbl

[26] Rheinboldt, W. C. An adaptative continuation process for solving systems of nonlinear equations, Banach Ctz. Publ., Volume 3 (1975), pp. 129-142

[27] Schmidt, J. W. Untere fehlerschranken fr regula-falsi-verfahren, Period. Hungar., Volume 9 (1978), pp. 241-247 | Zbl | DOI

[28] J. F. Traub Iterative method for solutions of equations, Prentice-Hall, Englewood Cliffs, 1964

[29] Yamamoto, T. A convergence theorem for Newton-like methods in Banach spaces, Numer. Math., Volume 51 (1987), pp. 545-557 | DOI

[30] Zabrejko, P. P.; Nguen, D. F. The majorant method in the theory of Newton-Kantorovich approximations and the Pták error estimates, Numer. Funct. Anal. Optim., Volume 9 (1987), pp. 671-684 | Zbl | DOI

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