Super-Halley method under majorant conditions in Banach spaces

Shwet  Nisha; P. K. Parida

doi:10.4067/S0719-06462020000100055

DOI:

https://doi.org/10.4067/S0719-06462020000100055

Abstract

In this paper, we have studied local convergence of Super-Halley method in Banach spaces under the assumption of second order majorant conditions. This approach allows us to obtain generalization of earlier convergence analysis under majorizing sequences. Two important special cases of the convergence analysis based on the premises of Kantorovich and Smale type conditions have also been concluded. To show efficacy of our approach we have given three numerical examples.

Keywords

Nonlinear equations , Super-Halley method , Majorant conditions , Local Convergence , Semilocal Convergence , Smale-type conditions , Kantorovich-type conditions

Shwet Nisha Department of Applied Mathematics, Central University of Jharkhand, Ranchi-835205, India.
P. K. Parida Department of Applied Mathematics, Central University of Jharkhand, Ranchi-835205, India. http://orcid.org/0000-0002-8658-7134

Pages: 55–70
Date Published: 2020-04-18
Vol. 22 No. 1 (2020)

I. K. Argyros and H. Ren. Ball convergence theorem for Halley‘s method in banach spaces. J. Appl. Math. Comp. 38(2012), pp. 453-465.

D. Chen, I. K. Argyros and Q. Qian. A local Convergence theorem for the Super-Halley method in Banach space. Appl. Math. 7(1994), pp. 49-52.

P. Deuflhard. Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms. Springer, Berlin Heindelberg, 2004.

P. Deuflhard and G. Heindl. Affine invariant convergent theorems for Newtons method and extensions to related methods. SIAM J. Numer. Anal. 16(1979), pp. 1-10.

J. A. Ezquerro and M. A. Hernández. On a convex acceleration of Newton‘s method, J. Optim. Theory Appl. 100 (1999), pp. 311–326.

O.P. Ferreira. Local convergence of Newton‘s method in Banach space from the viewpoint of the majorant principle. IMA J. Numer. Anal. 29(2009), pp. 746-759.

O.P. Ferreira and B.F. Svaiter. Kantorovich‘s majorants principle for Newton‘s method. Comput. Optim. Appl. 42(2009), pp. 213-229.

W.B. Gragg and R.A. Tapia. Optimal error bounds for the Newton-Kantorovich theorem. SIAM J. Numer. Anal. 11(1974), pp. 10-13.

J. M. Gutiérrez and M. A. Hernández. Recurrence relations for the Super-Halley method, Comput. Math. Appl. 36(1998), pp. 1-8.

J. M. Gutiérrez and M. A. Hernández. Newton‘s method under weak Kantorovich conditions. IMA J. Numer. Anal. 20(2000), pp. 521-532.

J. M. Gutiérrez and M. A. Hernández. An acceleration of Newton‘s method: super-Halley method. Appl. Math. Comput. 117(2001), pp. 223-239.

D. Han, X. Wang. The error estimates of Halley‘s method. Numer. Math. JCU Engl. Ser. 6(1997), pp. 231-240.

M. A. Hernández and N. Romero. On the characterization of some Newton like methods of R-order at least three. J. Comput. Appl. Math. 183(2005), pp. 53-66.

M. A. Hernández and N. Romero. Towards a unified theory for third R-order iterative methods for operators with unbounded second derivative, Appl. Math. Comput. 215(2009), pp. 2248-2261.

L. O. Jay. A note on Q-order of convergence, BIT Numer. Math. 41(2001), pp. 422-429.

L. V. Kantorovich and G. P. Akilov. Functional Analysis. Pergamon Press, Oxford, 1982.

Y. Ling and X. Xu. On the semilocal convergence behaviour of Halley‘s method, Comput. Optim. Appl. 58(2014), pp. 597-61.

F. A. Potra. On Q-order and R-order of convergence, J. Optim. Theory Appl. 63(1989), pp. 415-431.

M. Prashanth and D. K. Gupta. Recurrence relation for Super-Halley‘s method with hölder continuous second derivative in Banach spaces, Kodai Math. J. 36(2013), pp. 119-136.

M. Prashanth, D. K. Gupta and S. Singh. Semilocal convergence for the Super-Halley method. Numer. Anal. Appl., 7(2014), pp. 70-84.

S. Smale. Newton‘s method estimates from data at one point. In: Ewing, R., Gross, K., Martin, C.(eds.) The Merging of Disciplines: New Directions in Pure, Applied and computational Mathematics, 185-196. Springer, New York, 1986.

X. Wang. Convergence of Newton‘s method and inverse functions theorem in Banach space. Math. Comput. 68(1999), pp. 169-186.

X. Wang. Convergence of Newton‘s method and uniqueness of the solution of equations in Banach space, IMA J.Numer. Anal. 20(2000), pp. 123-134.

X. Wang and D. Han. On the dominating sequence method in the point estimates and smale‘s, theorem. Scientia Sinica Ser. A. 33(1990), pp. 135-144.

X. Wang and D. Han. Criterion Î± and Newton‘s method in the weak conditions (in Chinese), Math. Numer. Sinica 19(1997), pp. 103-112.

X. Xu and C. Li. Convergence of Newton‘s method for systems of equations with constant rank derivatives. J. Comput. Math. 25(2007), pp. 705-718.

X. Xu and C. Li. Convergence criterion of Newton‘s method for singular systems of equations with constant rank derivatives. J. Math. Anal. Appl. 245(2008), pp. 689-701.

T. Yamamoto. On the method of tangent hyperbolas in Banach spaces, J. Comput. Appl. Math. 21(1988), pp. 75-86.

T.J. Ypma. Affine invariant convergence results for Newton‘s method. BIT Numer. Math. 22(1982), pp. 108-118.