Higher order multi-step interval iterative methods for solving nonlinear equations in \(R^n\)
In this paper, higher order multi-step interval iterative methods are proposed for solving nonlinear equations in \(R^n\). Each method leads to an an interval vector enclosing the approximate solution along with the rigorous error bounds automatically. These methods require solving linear interval systems of equations. Interval extension of Gaussian elimination algorithm is described and used for solving them. The convergence analysis of both the methods is established to show their third and fourth order of convergence. A number of numerical examples are worked out and the performance in terms of iterations count and diameters of resulting interval vectors are measured.
KeywordsNonlinear equations Convergence analysis Rigorous error bounds Multi-step methods Boundary value problems Computational efficiency
Mathematics Subject Classification65G49 65H05
The authors thank the referees for their valuable comments which have improved the presentation of the paper. The authors thankfully acknowledge the financial assistance provided by Council of Scientific and Industrial Research (CSIR), New Delhi, India.
- 1.Alefeld, G., Herzberger, J.: Introduction to Interval Computation. Acadamic Press, New York (1983)Google Scholar
- 18.Ortega, J.M., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables. Academic Press, New York (1970)Google Scholar
- 19.Ostrowski, A.M.: Solution of equations in Euclidean and Banach spaces. Academic Press, New-York (1977)Google Scholar
- 20.Petković, M.S., Neta, B., Petković, L.D., Dunić, J.: Multipoint methods for solving nonlinear equations: a survey. Appl. Math. Comput. 226, 635–660 (2014)Google Scholar
- 24.Rump, S.: INTLABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing. Kluwer Academic Publishers, Dordrecht. 77–104 (1999)Google Scholar