Extending the solvability of equations using secant-type methods in Banach space

Authors

  • Ioannis K. Argyros Cameron University
  • Santhosh George Department of Mathematical and Computational Sciences, NIT Karnataka

Keywords:

nonlinear equations in Banach space, secant method, Secant-type method, Semi-local convergence, restricted convergence region, Lipschitz conditions

Abstract

We extend the solvability of equations dened on a Banach space using numerically ecient secant-type methods. The convergence domain of these methods is enlarged using our new idea of restricted convergence region. By using this approach, we obtain a more precise location where the iterates lie than in earlier studies leading to tighter Lipschitz constants. This way the semi-local convergence produces weaker sucient convergence criteria and tighter error bounds than in earlier works. These improvements are also obtained under the same computational eort, since the new Lipschitz constants are special cases of the old ones.

Downloads

Download data is not yet available.

Author Biography

Ioannis K. Argyros, Cameron University

Full tenured Professor of Mathematics.

References

S. Amat, M.A. Hernandez, and N. Romero, A modified Chebyshev’s iterative method with at least sixth order of convergence, Appl. Math. Comput. 206 (1), (2008), 164–174, https://doi.org/10.1016/j.amc.2008.08.050

S. Amat, M.A. Hernandez, and N. Romero, Semilocal convergence of a sixth order iterative method for quadratic equations, Applied Numerical Mathematics, 62 (2012), 833-841, https://doi.org/10.1016/j.apnum.2012.03.001

I.K. Argyros, A unifying local semi-local convergence analysis and applications fortwo-point Newton-like methods in Banach space, Journal of Mathematical Analysis and Applications, 298 (2) (2004), 374–397, https://doi.org/10.1016/j.jmaa.2004.04.008

I.K. Argyros, and H. Ren, On an improved local convergence analysis for the Secant method, Numer. Algor., 52 (2009), 257–271, https://doi.org/10.1007/s11075-009-9271-6.

I.K. Argyros, and S. Hilout, Weaker conditions for the convergence of Newton’s method, J. Complexity, 28 (2012), 364–387, https://doi.org/10.1016/j.jco.2011.12.003

I.K. Argyros, A. A. Magrenan, Iterative methods and their dynamics with applications, CRC Press, New York, 2017.

W. E. Bosarge and P.L. Falb, A multipoint method of third order, J. Optim. Theory Appl., 4 (1969), 156-166, https://doi.org/10.1007/BF00930576

W.E. Bosarge and P.L. Falb, Infinite dimensional multipoint methods and the solution of two point boundary value problems, Numer. Math., 14 (1970), 264–286, https://doi.org/10.1007/BF02163335

J.E. Dennis, On the Kantorovich hypothesis for Newton’s method, SIAM. J. Numer. Anal., 6 (1969), 493-507, https://doi.org/10.1137/0706045

J.E.Dennis, Toward a unified convergence theory for Newton-like methods, In Nonlinear functional analysis and applications, L. B. Rall, Ed., Academic Press, 1971, New York.

P. Deuflhard and G. Heindl, Affine invariant convergence theorems for Newton’smethod and extensions to related methods, Siam J. Numer. Anal.16(1979) 1–10, https://doi.org/10.1137/0716001.

J.A. Ezquerro, M.A. Hernandez and M. J. Rubio, Secant-like methods for solving nonlinear integral equations of the Hammerstein type, J. Comput. Appl. Math. 115, 1-2 (2000) 245–254, https://doi.org/10.1016/S0377-0427(99)00116-8.

W. B. Gragg and R. A. Tapia, Optimal error bounds for the Newton-Kantorovich theorem, SIAM J. Numer. Anal., 11, 1 (1974), 10-13, https://doi.org/10.1137/0711002

M. A. Hernandez and M. J. Rubio, A uniparameteric family of iterative processes for solving nondifferentiable equations, J. Math. Anal. Appl., 275 (2002) 821–834, https://doi.org/10.1016/S0022-247X(02)00432-8

M. A. Hernandez-Veron, and M. J. Rubio, On the ball of convergence of Secant-like methods for non-differentiable operators, Applied Mathematics and Computation (2015), https://doi.org/10.1016/j.amc.2015.10.007

M. A. Hernandez, M. J. Rubio and J. A. Ezquerro, Solving a special case of conservation problems by Secant-like methods, Appl. Math. Comput., 169, 2 (2005) 926–942, https://doi.org/10.1016/j.amc.2004.09.070.

H. J. Kornstaedt, Ein al lgemeiner Konvergenzsatz fur veracharfte Newton-Verfahren, ISNM 28, Birkhauser Verlag, Basel and Stuttgart, 1975, 53–69.

P. Laasonen, Einuberquadratisch konvergenter iterative algoriyhmus, Ann. Acad. Sci. Fenn. Ser I, 450 (1969), 1-10

A. A. Magrenan, Different anomalies in a Jarratt family of iterative root finding methods, Appl. Math. Comput. 233 (2014), 29-38, https://doi.org/10.1016/j.amc.2014.01.037

A. A. Magrenan, A new tool to study real dynamics: The convergence plane, Appl. Math. Comput. 248 (2014), https://doi.org/10.1016/j.amc.2014.01.037

G. J. Miel,The Kantorovich theorem with optimal error bounds, Amer. Math. Monthly, 86 (1979), 212-215, https://doi.org/10.1080/00029890.1979.11994773.

G. J. Miel, An updated version of the Kantorovich theorem for Newton’s method, Technical summary report, Mathematical research center, University of Wisconsin, Madison,1980.

I. Moret, On a general iterative scheme for Newton-type methods, Numer. Funct. Anal. Optimiz., 9(11-12), (1987-1988), 1115-1137, https://doi.org/10.1080/01630568808816277

J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations inseveral variables, Academic Press, 1970, New York.

F. A. Potra, On the convergence of a class of Newton-like methods, Iterative solutionof nonlinear systems of equations, Lecture notes in Mathematics 953, Springer Verlag,1982, New York.

F. A. Potra and V. Ptak, Nondiscrete Induction and Iterative methods, Pitman Publishing Limited, London, 1984.

F. A. Potra, Sharp error bounds for a class of Newton-like methods, Libertas Mathematica, vol. 5(1985), 71–83.

H. Ren and I.K. Argyros, Local convergence of efficient Secant-type methods for solving nonlinear equations, Appl. Math. Comput., 218 (2012) 7655–7664, https://doi.org/10.1016/j.amc.2012.01.036.

J. F. Traub, Iterative methods for the solutions of equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1964.

P. P. Zabrejko and D. F. Nguen, The majorant method in the theory of Newton-Kantorovich approximations and Ptak error estimates, Numer. Funct. Anal. Optimiz., 9 (1987), 671-684, https://doi.org/10.1080/01630568708816254.

E. J. Zehnder, A remark on Newton’s method, Communus Pure Appl. Math., 27 (1974), 361-366, https://doi.org/10.1002/cpa.3160270305

T. Yamamoto, Error bounds for Newton-like methods under Kantorovich type Assump-tions, MCR Technical Summary Report Nr. 2846, University of Wisconsin, Madison(1985).

Downloads

Published

2021-12-31

How to Cite

Argyros, I. K., & George, S. (2021). Extending the solvability of equations using secant-type methods in Banach space. J. Numer. Anal. Approx. Theory, 50(2), 97–107. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/1134

Issue

Section

Articles