International Journal of Nonlinear Analysis and Applications

A novel two-step iterative method based on real interval arithmetic for finding enclosures of roots of systems of nonlinear equations

Document Type : Research Paper

Authors

1 School of Mathematics, Iran University of Science and Technology (IUST), Narmak, Tehran, Iran

2 Department of Mathematics, Faculty of Sciences, University of Qom, Qom, Iran

Abstract

‎In the present paper‎, ‎a novel two-step iterative method‎, ‎based on real interval arithmetic‎, ‎is produced‎. ‎By using this method‎, ‎we obtain enclosures of roots of systems of nonlinear equations‎. ‎Discussion on the convergence analysis for the produced method is presented‎. ‎The efficiency‎, ‎accuracy‎, ‎and validity of this method are demonstrated by its application to four implemented examples with INTLAB and by comparing our results with the results obtained by other methods available in the literature‎.

Keywords

[1] G. Alefeld, On the convergence of some interval-arithmetic modifications of Newton’s method, SIAM J. Numer. Anal. 21 (1984), 363–372.
[2] G. Alefeld and J. Herzberger, Introduction to interval computations, New York: Academic Press, 1983.
[3] M.A.T. Ansary and G. Panda, New higher order root finding algorithm using interval analysis, Reliable Computing. 21 (2015) 11–24.
[4] R.R. Capdevila, A. Cordero and J.R. Torregrosa, Stability analysis of iterative methods for solving nonlinear algebraic systems, Current Topics Math. Comput. Sci. 9 (2021), 6–24.
[5] R. Chen and A.C. Ward, Introduction to interval matrices in design, AS ME Design Theory Methodol. 42 (1992), 221–227.
[6] A. Cordero, EG. Villalba, JR. Torregrosa and P. Triguero-Navarro, Convergence and stability of a parametric class of iterative schemes for solving nonlinear systems, Math. 9 (2021), no. 1, 86.
[7] A.D. Dimargonoas, Interval analysis of vibrating systems, J. Sound Vib. 183 (1995), 739–749.
[8] T. Eftekhari, On some iterative methods with memory and high efficiency index for solving nonlinear equations, Int. J. Differ. Equ. 2014 (2014), Article ID 495357, 6 pages.
[9] T. Eftekhari, A new sixth-order Steffensen-Type iterative method for solving nonlinear equations, Int. J. Anal. 2014 (2014), Article ID 685796, 5 pages.
[10] T. Eftekhari, A new family of four-step fifteenth-order root-finding methods with high efficiency index, Comput. Methods. Differ. Equ. 3 (2015), no. 1, 51–58.
[11] T. Eftekhari, A new proof of interval extension of the classic Ostrowski’s method and its modified method for computing the enclosure solutions of nonlinear equations, Numer. Algorithms 69 (2015), no. 1, 157–165.
[12] T. Eftekhari, Producing an interval extension of the King method, Appl. Math. Comput. 260 (2015), 288–291.
[13] T. Eftekhari, An Efficient Class of Multipoint Root-Solvers With and Without Memory for Nonlinear Equations, Acta Math. Vietnam. 41 (2016), 299–311.
[14] T. Eftekhari, Interval extension of the three-step Kung and Traub’s method, J. Modern Meth. Numer. Math. 9 (2018), no. 1-2, 42–52.
[15] T. Eftekhari, Interval extension of the Halley method and its modified method for finding the root enclosures of nonlinear equations, Comput. Meth. Differ. Equ. 8 (2020), no. 2, 222–235.
[16] F. Feng, L.S. Shieh and G. Chen, Model conversions of uncertain linear systems using interval multipoint Pade approximation, Appl. Math. Model. 21 (1997), 233–244.
[17] J. Garloff, Interval mathematics: A bibliography, Freiburger Interval- Berichte 85/6 (1985), 1–122.
[18] J. Garloff, Bibliography on interval mathematics, continuation, Freiburger Interval-Berichte 81/2 (1987), 1–50.
[19] J. Garloff and K.P. Schwierz, A Bibliography on Interval-Mathematics, J. Comput. Appl. Math. 6 (1980), 67–79.
[20] D.K. Gupta, Enclosing the solutions of nonlinear systems of equations, Int. J. Comput. Math. 73 (2000), 389–404.
[21] D.K. Gupta and C.N. Kaul, A modification of Krawczyk’s algorithm, Int. J. Comput. Math. 66 (1998), 67–77.
[22] T.C. Henderson, T.M. Sobh, F. Zana, B. Briiderlin and C.Y. Hsu, Sensing strategies based on manufacturing knowledge, ARPA Image Understanding Workshop, University of California Riverside, Monterey, 1994, pp. 1109–1113.
[23] R.B. Kearfott and V. Kreinovich, Applications of interval computations, Dordrecht, Kluwer Academic Publishers, 1996.

[24] T. Lotfi and T. Eftekhari, A new optimal eighth-order Ostrowski-Type family of iterative methods for solving nonlinear equations, Chinese J. Math. 2014 (2014), Article ID 369713, 7 pages.
[25] T. Lotfi and M. Momenzadeh, Constructing an efficient multi-step iterative scheme for nonlinear system of equations, Comput. Meth. Differ. Equ. 9 (2021), no. 3, 710–721.
[26] S. Malan, M. Milanese and T. Taragna, Robust analysis and design of control systems using interval arithmetic, Automatica 33 (1997), 1363–1372.
[27] M. Milanese, J. Norton, N. Piet-Lahanier and E. Walter (eds.). Bounding approaches to system identification, New York, Plenum Press, 1996.
[28] M. Moccari and T. Lotfi, Using majorizing sequences for the semi-local convergence of a high-order and multipoint iterative method along with stability analysis, J. Math. Eext. 15 (2020), no. 2, 1–32.
[29] M. Mohamadizade, T. Lotfi and M. Amirfakhriyan, Some Improvements of The Cordero-Torregrosa Method for The Solution of Nonlinear Equations, Int. J. Ind. Math. 13 (2021), no. 3, 333–341.
[30] R.E. Moore, Interval arithmetic and automatic error analysis in digital computing, Ph.D. Dissertation, Stanford University, 1962.
[31] R.E. Moore, Interval analysis, Prentice-Hall, Englewood Cliff, New Jersey, 1966.
[32] R.E. Moore, R.B. Kearfott, and M.J. Cloud, Introduction to interval analysis, SIAM, Philadelphia, 2009.
[33] K. Nickel, Interval mathematics, Lecture Notes in Computer Science 29, Berlin: Springer-Verlag, 1975.
[34] K. Nickel, Interval mathematics, New York: Academic Press, 1980.
[35] K. Nickel, Interval mathematics, Lecture Notes in Computer Science 212. Berlin: Springer-Verlag, 1985.
[36] EP. Oppenheimer and AN. Michel, Application of interval analysis techniques to linear systems: Part IFundamental Results, IEEE Trans. Circ. Syst. 35 (1988), 1129–1138.
[37] EP. Oppenheimer and AN. Michel, Application of interval analysis techniques to linear systems: Part II-The interval matrix exponential function, IEEE Trans. Circ. Syst. 35 (1988), 1230–1242.
[38] E.P. Oppenheimer and A.N. Michel, Application of interval analysis techniques to linear systems: Part III-Initial value problems, IEEE Trans. Circ. Syst. 35 (1988), 1243–1256.
[39] W. Pedrycz, Granular computing: An emerging paradigm, Springer-Verlag Berlin and Heidelberg GmbH, 2013.
[40] M.S. Petkovic, Multi-step root solvers of Traub’s type in real interval Arithmetic, Appl. Math. Comput. 248 (2014), 430–440.
[41] MS. Petkovi´c, LD. Petkovi´c and B. Neta, On generalized Halley-like methods for solving nonlinear equations, Appl. Anal. Discrete Math. 13 (2019), no. 2, 399–422.
[42] A. Piazzi and G. Marro, Robust stability using interval analysis, Int. J. Syst. Sci. 21 (1996), 1381–1390.
[43] A. Piazzi and A. Visoli, An interval algorithm for minimum-jerk trajectory planning of robot manipulators, Proc. 36th Conf. Decision Control, San Diego Calif. U.S.A. 1997, pp. 1924–1927.
[44] A. Piazzi and A. Visoli, Global minimum-time trajectory planning of mechanical manipulators using interval analysis, Int. J. Control 71 (1998), 631–652.
[45] P.D. Proinov, S.I. Ivanov and M.S. Petkovi´c, On the convergence of Gander´s type family of iterative methods for simultaneous approximation of polynomial zeros, Appl. Math. Comput. 349 (2019), 168–183.
[46] S.S. Rao and L. Berke, Analysis of uncertain structural systems using interval analysis, J. Amer. Inst. Aeron. Astron. 35 (1997), 727–735.
[47] H. Ratschek and J.G. Rokne, Computer methods for the range of functions, Chichester: Ellis Horwood, 1984.
[48] J.G. Rokne, Interval arithmetic and interval analysis: An introduction, Pedrycz W. (eds) Granular Computing. Studies in Fuzziness and Soft Computing, Physica, Heidelberg, 2001.
[49] S.M. Rump, INTLAB - INTerval LABoratory, Tibor Csendes, Developments in Reliable Computing, Kluwer Academic Publishers, Dordrecht, 1999.
[50] Y. Seif and T. Lotfi, An efficient multistep iteration scheme for systems of nonlinear algebraic equations associated with integral equations, Math. Meth. Appl. Sci. 43 (2020), no. 14, 8105–8115.
[51] S. Singh and D.K. Gupta, Higher order interval iterative methods for nonlinear equations, J. Appl. Math. Inf. 33 (2015), no. 1-2, 61–76.
[52] S. Singh, D.K. Gupta and F. Roy, Higher order multi-step interval iterative methods for solving nonlinear equations in Rn, SeMA. 74 (2017), no. 2, 133–146.
[53] J.R. Sharma, R.K. Guha and R. Sharma, An efficient fourth order weighted-Newton method for systems of nonlinear equations, Numer. Algorithms 62 (2013), 307–323.
[54] V. Torkashvand, A general class of one-parametric with memory method for solving nonlinear equations, Caspian J. Math. Sci. 10 (2021), no. 2, 309–335.
[55] V. Torkashvand, M.A. Fariborzi Araghi, Construction of iterative adaptive methods with memory with 100% improvement of convergence order, J. Math. Ext. 15 (2020), no. 3, 1–32.
[56] V. Torkashvand and R. Ezzati, On the efficient of adaptive methods to solve nonlinear equations, Int. J. Nonlinear Anal. Appl. 12 (2021), no. 1, 301–316.
[57] V. Torkashvand and M. Kazemi, On an Efficient Family with Memory with High Order of Convergence for Solving Nonlinear Equations, Int. J. Ind. Math. 12 (2020), no. 2, 209–224.
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 2
July 2022
Pages 2685-2695
  • Receive Date: 18 September 2021
  • Revise Date: 25 October 2021
  • Accept Date: 18 April 2022