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On new iterative method for solving systems of nonlinear equations. (English) Zbl 1197.65048
Author’s abstract: Solving systems of nonlinear equations is a relatively complicated problem for which a number of different approaches have been proposed. In this paper, we employ the Homotopy Analysis Method (HAM) to derive a family of iterative methods for solving systems of nonlinear algebraic equations. Our approach yields second and third order iterative methods which are more efficient than their classical counterparts such as Newton’s, Chebychev’s and Halley’s methods.

65H10 Numerical computation of solutions to systems of equations
65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations
Full Text: DOI
[1] Abbasbandy, S., Tan, Y., Liao, S.J.: Newton-homotopy analysis method for nonlinear equations. Appl. Math. Comput. 188(2), 1794–1800 (2007) · Zbl 1119.65032
[2] Alefeld, G.: On the convergence of Hally’s method. Am. Math. Mon. 88, 530–536 (1981) · Zbl 0486.65035
[3] Amat, S., Busquier, S., Gutiérrez, J.M.: Geometric constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math. 157, 197–205 (2003) · Zbl 1024.65040
[4] Allgower, E., George, K.: Numerical continuation methods: an introduction, 388 pp. QA377.A56 640. Springer, New York (1990)
[5] Broyden, C.G.: A class of of methods for solving nonlinear simultaneous equations. Math. Comput. 19, 577–593 (1965) · Zbl 0131.13905
[6] Broyden, C.G.: Quasi-Newton methods and thier application to function minimization. Math. Comput. 21, 368–381 (1967) · Zbl 0155.46704
[7] Dennis, J.E., More, J.: Quasi-Newton methods: motivation and theory. SIAM Rev. 19, 46–84 (1977) · Zbl 0356.65041
[8] Kaya, D., El-Sayed, S.M.: Adomian’s decomposition method applied to systems of nonlinear algebraic equations. Appl. Math. Comput. 154, 487–493 (2004) · Zbl 1058.65056
[9] Floudas, A., Pardalos, P.M., Adjiman, C., Esposito, W., Gumus, Z., Harding, S., Klepeis, J., Mayer, C., Schweiger, C.: Handbook of Test Problems in Local and Global Optimization. Kluwer Academic, Dordrecht (1999) · Zbl 0943.90001
[10] Floudas, A.: Recent advances in global optimization for process synthesis, design, and control: enclosure of all solutions. Comput. Chem. Eng. S, 963–973 (1999)
[11] Golbabai, A., Javidi, M.: Newton-like iterative methods for solving system of non-linear equations. Appl. Math. Comput. 192, 546–551 (2007) · Zbl 1193.65149
[12] Gutierrez, J.M., Hernández, M.A.: A family of Chebyshev-Hally type methods in Banach spaces. Bull. Aust. Math. Soc. 55, 113–130 (1997) · Zbl 0893.47043
[13] Nedzhibov, G.H.: A family of multi-point iterative methods for solving systems of nonlinear equations. J. Comput. Appl. Math. 222, 244–250 (2008) · Zbl 1154.65037
[14] Homeier, H.H.H.: A modified Newton method with cubic convergence: the multivariate case. J. Comput. Appl. Math. 169, 161–169 (2004) · Zbl 1059.65044
[15] Kubicek, M., Hoffman, H., Hlavacek, V., Sinkule, J.: Multiplicity and stability in a sequence of two nonadiabatic nonisothermal CSTR. Chem. Eng. Sci. 35, 987–996 (1980)
[16] Liao, S.J.: Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman & Hall/CRC, Boca Raton (2003)
[17] Liao, S.J.: On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 47, 499–513 (2004) · Zbl 1086.35005
[18] Liao, S.J.: Notes on the homotopy analysis method: some defnitions and theorems. Commun. Nonlinear Sci. Numer. Simul. (2008). doi: 10.1016/j.cnsns.2008.04.013
[19] Liao, S.J., Tan, Y.: A general approach to obtain series solutions of nonlinear differential equations. Stud. Appl. Math. 119, 297–355 (2007)
[20] Burden, R.L., Faires, J.D.: Numerical Analysis, 8th edn. Thomson Brooks/Cole (2005) · Zbl 0671.65001
[21] Pramanik, S.: Kinematic synthesis of a six-member mechanism for automotive steering. ASME J. Mech. Des. 124, 642–645 (2002)
[22] Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis, 2nd edn. Texts in Applied Mathematics 12. Springer, New York (1992) · Zbl 0771.65002
[23] Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice-Hall, Engle-wood Cliffs (1964) · Zbl 0121.11204
[24] Weerakoon, S., Fernando, T.G.I.: A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13, 87–03 (2000) · Zbl 0973.65037
[25] Werner, W.: Iterative solution of systems of nonlinear equations based upon quadratic approximations. Comput. Math. Appl. 12A(3), 331–343 (1986) · Zbl 0609.65042
[26] Wu, X.Y.: A new continuation Newton-like method and its deformation. Appl. Math. Comput. 112(1), 75–78 (2000) · Zbl 1023.65043
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