zbMATH — the first resource for mathematics

A multi-step class of iterative methods for nonlinear systems. (English) Zbl 1286.93068
Summary: In this article, the numerical solution of nonlinear systems using iterative methods are dealt with. Toward this goal, a general class of multi-point iteration methods with various orders is constructed. The error analysis is presented to prove the convergence order. Also, a thorough discussion on the computational complexity of the new iterative methods will be given. The analytical discussion of the paper will finally be upheld through solving some application-oriented problems.

93B40 Computational methods in systems theory (MSC2010)
65H10 Numerical computation of solutions to systems of equations
Full Text: DOI
[1] An, H.-B., Bai, Z.-Z.: A globally convergent Newton-GMRES method for large sparse systems of nonlinear equations. Appl. Numer. Math. 57, 235–252 (2007) · Zbl 1123.65040 · doi:10.1016/j.apnum.2006.02.007
[2] An, H.-B., Mo, Z.-Y., Liu, X.-P.: A choice of forcing terms in inexact Newton method. J. Comput. Appl. Math. 200, 47–60 (2007) · Zbl 1112.65044 · doi:10.1016/j.cam.2005.12.030
[3] An, H.-B., Wen, J., Feng, T.: On finite difference approximation of a matrix-vector product in the Jacobian-free Newton–Krylov method. J. Comput. Appl. Math. 236, 1399–1409 (2011) · Zbl 1258.65049 · doi:10.1016/j.cam.2011.09.003
[4] Bailey, D.H., Barrio, R., Borwein, J.M.: High-precision computation: Mathematical physics and dynamics. Appl. Math. Comput. 218, 10106–10121 (2012) · Zbl 1248.65147 · doi:10.1016/j.amc.2012.03.087
[5] Ben-Israel, A., Greville, T.N.E.: Generalized Inverses, 2nd edn. Springer, Berlin (2003) · Zbl 1026.15004
[6] Cordero, A., Hueso, J.L., Martinez, E., Torregrosa, J.R.: A modified Newton–Jarratt’s composition. Numer. Algorithms 55, 87–99 (2010) · Zbl 1251.65074 · doi:10.1007/s11075-009-9359-z
[7] Cruz, W.L., Martinez, J.M., Raydan, M.: Spectral residual method without gradient information for solving large-scale nonlinear systems of equations. Math. Comput. 75, 1429–1448 (2006) · Zbl 1122.65049 · doi:10.1090/S0025-5718-06-01840-0
[8] Dayton, B.H., Li, T.-Y., Zeng, Z.: Multiple zeros of nonlinear systems. Math. Comput. 80, 2143–2168 (2011) · Zbl 1242.65102 · doi:10.1090/S0025-5718-2011-02462-2
[9] Hirsch, M.J., Pardalos, P.M., Resende, M.G.C.: Solving systems of nonlinear equations with continuous GRASP. Nonlinear Anal. Real World Appl. 10, 2000–2006 (2009) · Zbl 1163.90750 · doi:10.1016/j.nonrwa.2008.03.006
[10] http://www.wolfram.com/learningcenter/tutorialcollection/UnconstrainedOptimization/
[11] http://mathematica.stackexchange.com/questions/11364/how-do-i-find-all-the-solutions-of-three-simultaneous-equations-within-a-given-b
[12] Jarratt, P.: Some fourth order multipoint iterative methods for solving equations. Math. Comput. 20, 434–437 (1966) · Zbl 0229.65049 · doi:10.1090/S0025-5718-66-99924-8
[13] Montazeri, H., Soleymani, F., Shateyi, S., Motsa, S.S.: On a new method for computing the numerical solution of systems of nonlinear equations. J. Appl. Math., 2012, Article ID 751975, 15 p, (2012) · Zbl 1268.65075
[14] Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970) · Zbl 0241.65046
[15] Rheinboldt, W.C.: Methods for Solving Systems of Nonlinear Equations, 2nd edn. SIAM, Philadelphia (1998) · Zbl 0906.65051
[16] Sauer, T.: Numerical Analysis, 2nd edn. Pearson (2012) · Zbl 1229.91347
[17] Semenov, V.S.: The method of determining all real nonmultiple roots of systems of nonlinear equations. Comput. Math. Math. Phys. 47, 1428–1434 (2007) · doi:10.1134/S0965542507090047
[18] Shen, C., Chen, X., Liang, Y.: A regularized Newton method for degenerate unconstrained optimization problems. Optim. Lett. 6, 1913–1933 (2012) · Zbl 1258.90106 · doi:10.1007/s11590-011-0386-z
[19] Themistoclakis, W., Vecchio, A.: On the solution of a class of nonlinear systems governed by an $$M$$ -matrix. Discret. Dyn. Nat. Soc., 2012, Article ID 412052, 12 p · Zbl 1248.65052
[20] Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, New York (1964) · Zbl 0121.11204
[21] Tsoulos, I.G., Stavrakoudis, A.: On locating all roots of systems of nonlinear equations inside bounded domain using global optimization methods. Nonlinear Anal. Real World Appl. 11, 2465–2471 (2010) · Zbl 1193.65078 · doi:10.1016/j.nonrwa.2009.08.003
[22] Thukral, R.: Further development of Jarratt method for solving nonlinear equations. Adv. Numer. Anal., 2012, Article ID 493707, 9 p · Zbl 1250.65068
[23] Wagon, S.: Mathematica in Action. Springer, Berlin (2010) · Zbl 1198.65001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.