Steffensen-like method in Riemannian manifolds

Chandresh Prasad; P. K. Parida

doi:10.56754/0719-0646.2603.525

DOI:

https://doi.org/10.56754/0719-0646.2603.525

Abstract

In this paper, we present semilocal convergence of Steffensen-like method for approximating zeros of a vector field in Riemannian manifolds. We establish the convergence of Steffensen-like method under Lipschitz continuity condition on first order covariant derivative of a vector field. Finally, two examples are given to show the application of our theorem.

Keywords

Vector fields , Riemannian manifolds , Lipschitz condition , Steffensen-like method

Mathematics Subject Classification:

65H10 , 65D99

Chandresh Prasad Department of Mathematics, Central University of Jharkhand, Ranchi-835222, India. https://orcid.org/0000-0002-0766-8228
P. K. Parida Department of Mathematics, Central University of Jharkhand, Ranchi-835222, India. https://orcid.org/0000-0002-8658-7134

Pages: 525–540
Date Published: 2024-12-12
Vol. 26 No. 3 (2024)

P.-A. Absil, R. Mahony, and R. Sepulchre, Optimization algorithms on matrix manifolds. Princeton University Press, Princeton, NJ, 2008, doi: 10.1515/9781400830244.

R. L. Adler, J.-P. Dedieu, J. Y. Margulies, M. Martens, and M. Shub, “Newton’s method on Riemannian manifolds and a geometric model for the human spine,” IMA J. Numer. Anal., vol. 22, no. 3, pp. 359–390, 2002, doi: 10.1093/imanum/22.3.359.

F. Alvarez, J. Bolte, and J. Munier, “A unifying local convergence result for Newton’s method in Riemannian manifolds,” Found. Comput. Math., vol. 8, no. 2, pp. 197–226, 2008, doi: 10.1007/s10208-006-0221-6.

S. Amat, I. K. Argyros, S. Busquier, R. Castro, S. Hilout, and S. Plaza, “Traub-type high order iterative procedures on Riemannian manifolds,” SeMA J., vol. 63, pp. 27–52, 2014, doi: 10.1007/s40324-014-0010-0.

S. Amat, J. A. Ezquerro, and M. A. Hernández-Verón, “On a Steffensen-like method for solving nonlinear equations,” Calcolo, vol. 53, no. 2, pp. 171–188, 2016, doi: 10.1007/s10092-015-0142- 3.

I. K. Argyros, “An improved unifying convergence analysis of Newton’s method in Riemannian manifolds,” J. Appl. Math. Comput., vol. 25, no. 1-2, pp. 345–351, 2007, doi: 10.1007/BF02832359.

I. K. Argyros, Convergence and applications of Newton-type iterations. Springer, New York, 2008.

I. K. Argyros, Y. J. Cho, and S. Hilout, Numerical Methods for Equations and Variational Inclusions. New York: CRC Press/Taylor and Francis Group, 2012.

I. K. Argyros, Y. J. Cho, and S. Hilout, Numerical methods for equations and its applications. CRC Press, Boca Raton, FL, 2012.

I. K. Argyros, S. Hilout, and M. A. Tabatabai, Mathematical modelling with applications in biosciences and engineering. Nova Science Publishers, Incorporated, 2011.

R. A. Castro, J. C. Rodríguez, W. W. Sierra, G. L. Di Giorgi, and S. J. Gómez, “Chebyshev-Halley’s method on Riemannian manifolds,” J. Comput. Appl. Math., vol. 336, pp. 30–53, 2018, doi: 10.1016/j.cam.2017.12.019.

J.-P. Dedieu and D. Nowicki, “Symplectic methods for the approximation of the exponential map and the Newton iteration on Riemannian submanifolds,” J. Complexity, vol. 21, no. 4, pp. 487–501, 2005, doi: 10.1016/j.jco.2004.09.010.

J.-P. Dedieu, P. Priouret, and G. Malajovich, “Newton’s method on Riemannian manifolds: convariant alpha theory,” IMA J. Numer. Anal., vol. 23, no. 3, pp. 395–419, 2003, doi: 10.1093/imanum/23.3.395.

O. P. Ferreira and B. F. Svaiter, “Kantorovich’s theorem on Newton’s method in Riemannian manifolds,” J. Complexity, vol. 18, no. 1, pp. 304–329, 2002, doi: 10.1006/jcom.2001.0582.

D. Groisser, “Newton’s method, zeroes of vector fields, and the Riemannian center of mass,” Adv. in Appl. Math., vol. 33, no. 1, pp. 95–135, 2004, doi: 10.1016/j.aam.2003.08.003.

S. Lang, Differential and Riemannian manifolds, 3rd ed., ser. Graduate Texts in Mathematics. Springer-Verlag, New York, 1995, vol. 160, doi: 10.1007/978-1-4612-4182-9.

W. Li, F. Szidarovszky, and Y. Kuang, “Notes on the stability of dynamic economic systems,” Appl. Math. Comput., vol. 108, no. 2-3, pp. 85–89, 2000, doi: 10.1016/S0096-3003(98)10140-6.

T. Sakai, Riemannian geometry, ser. Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1996, vol. 149, doi: 10.1090/mmono/149.

M. A. Tabatabai, W. M. Eby, and K. P. Singh, “Hyperbolastic modeling of wound healing,” Math. Comput. Modelling, vol. 53, no. 5-6, pp. 755–768, 2011, doi: 10.1016/j.mcm.2010.10.013.

L. W. Tu, An introduction to manifolds, 2nd ed., ser. Universitext. Springer, New York, 2011, doi: 10.1007/978-1-4419-7400-6.

J. H. Wang, “Convergence of Newton’s method for sections on Riemannian manifolds,” J. Optim. Theory Appl., vol. 148, no. 1, pp. 125–145, 2011, doi: 10.1007/s10957-010-9748-4.