Article
MSC:
49M15,
65B05,
65G99,
65H10,
65J15,
65N30 | MR 3232630 | Zbl 06362226 | DOI: 10.1007/s10492-014-0054-6
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Keywords:
semilocal convergence; secant method; Banach space; majorizing sequence; Hölder condition; divided difference; Fréchet-derivative
semilocal convergence; secant method; Banach space; majorizing sequence; Hölder condition; divided difference; Fréchet-derivative
Summary:
We use tighter majorizing sequences than in earlier studies to provide a semilocal convergence analysis for the secant method. Our sufficient convergence conditions are also weaker. Numerical examples are provided where earlier conditions do not hold but for which the new conditions are satisfied.
References:
[1] Argyros, I. K.: A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space. J. Math. Anal. Appl. 298 (2004), 374-397. DOI 10.1016/j.jmaa.2004.04.008 | MR 2086964 | Zbl 1061.47052
[2] Argyros, I. K.: Convergence and Applications of Newton-Type Iterations. Springer New York (2008). MR 2428779 | Zbl 1153.65057
[3] Argyros, I. K.: New sufficient convergence conditions for the secant method. Czech. Math. J. 55 (2005), 175-187. DOI 10.1007/s10587-005-0013-1 | MR 2121665 | Zbl 1081.65043
[4] Argyros, I. K.: On the Newton-Kantorovich hypothesis for solving equations. J. Comput. Appl. Math. 169 (2004), 315-332. DOI 10.1016/j.cam.2004.01.029 | MR 2072881 | Zbl 1055.65066
[5] Argyros, I. K., Cho, Y. J., Hilout, S.: Numerical Methods for Equations and Its Applications. CRC Press Boca Raton (2012). MR 2964315 | Zbl 1254.65068
[6] Argyros, I. K., Hilout, S.: Computational Methods in Nonlinear Analysis. Efficient Algorithms, Fixed Point Theory and Applications. World Scientific Hackensack (2013). MR 3134688 | Zbl 1279.65062
[7] Argyros, I. K., Hilout, S.: Convergence conditions for secant-type methods. Czech. Math. J. 60 (2010), 253-272. DOI 10.1007/s10587-010-0014-6 | MR 2595087 | Zbl 1224.65141
[8] Argyros, I. K., Hilout, S.: Semilocal convergence conditions for the Secant method using recurrent functions. Rev. Anal. Numér. Théor. Approx. 40 (2011), 107-119. MR 3059816 | Zbl 1289.65136
[9] Argyros, I. K., Hilout, S.: Weaker conditions for the convergence of Newton's method. J. Complexity 28 (2012), 364-387. DOI 10.1016/j.jco.2011.12.003 | MR 2914733 | Zbl 1245.65058
[10] Jr., W. E. Bosarge, Falb, P. L.: A multipoint method of third order. J. Optimization Theory Appl. 4 (1969), 155-166. MR 0248581 | Zbl 0172.18703
[11] Jr., J. E. Dennis: Toward a Unified Convergence Theory for Newton-Like Methods. Nonlinear Functional Analysis and Applications, Proc. Adv. Sem. Math. Res. Center, Univ. Wisconsin 1970 L. B. Rall Publication No. 26 of the Mathematics Research Center the University of Wisconsin Academic Press, New York (1971), 425-472. MR 0278556 | Zbl 0276.65029
[12] Hernández, M. A., Rubio, M. J., Ezquerro, J. A.: Secant-like methods for solving nonlinear integral equations of the Hammerstein type. J. Comput. Appl. Math. 115 (2000), 245-254. DOI 10.1016/S0377-0427(99)00116-8 | MR 1747223
[13] Hernández, M. A., Rubio, M. J., Ezquerro, J. A.: Solving a special case of conservative problems by secant-like methods. Appl. Math. Comput. 169 (2005), 926-942. DOI 10.1016/j.amc.2004.09.070 | MR 2174693 | Zbl 1080.65044
[14] Kantorovich, L. V., Akilov, G. P.: Functional Analysis. Pergamon Press. Transl. from the Russian by Howard L. Silcock. 2nd ed Oxford (1982). MR 0664597 | Zbl 0484.46003
[15] Laasonen, P.: Ein überquadratisch konvergenter iterativer Algorithmus. Ann. Acad. Sci. Fenn., Ser. A I 450 (1969), 10 German. MR 0255047 | Zbl 0193.11704
[16] Ortega, J. M., Rheinboldt, W. C.: Iterative Solution of Nonlinear Equations in Several Variables. Computer Science and Applied Mathematics Academic Press, New York (1970). MR 0273810 | Zbl 0241.65046
[17] Potra, F.-A.: On the convergence of a class of Newton-like methods. Iterative Solution of Nonlinear Systems of Equations, Proc. Meeting, Oberwolfach 1982 Lecture Notes in Mathematics 953 Springer, Berlin (1982), 125-137. MR 0678615 | Zbl 0507.65020
[18] Potra, F.-A.: Sharp error bounds for a class of Newton-like methods. Libertas Math. 5 (1985), 71-84. MR 0816258 | Zbl 0581.47050
[19] Potra, F.-A., Pták, V.: Nondiscrete Induction and Iterative Processes. Research Notes in Mathematics 103 Pitman Advanced Publishing Program, Boston (1984). MR 0754338 | Zbl 0549.41001
[20] Potra, F.-A., Pták, V.: Sharp error bounds for Newton's process. Numer. Math. 34 (1980), 63-72. DOI 10.1007/BF01463998 | MR 0560794 | Zbl 0434.65034
[21] Proinov, P. D.: New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems. J. Complexity 26 (2010), 3-42. DOI 10.1016/j.jco.2009.05.001 | MR 2574570 | Zbl 1185.65095
[22] Schmidt, J. W.: Untere Fehlerschranken für Regula-Falsi-Verfahren. Period. Math. Hung. 9 (1978), 241-247 German. DOI 10.1007/BF02018090 | MR 0494896 | Zbl 0401.65036
[23] Wolfe, M. A.: Extended iterative methods for the solution of operator equations. Numer. Math. 31 (1978), 153-174. DOI 10.1007/BF01397473 | MR 0509672 | Zbl 0375.65030
[24] Yamamoto, T.: A convergence theorem for Newton-like methods in Banach spaces. Numer. Math. 51 (1987), 545-557. DOI 10.1007/BF01400355 | MR 0910864 | Zbl 0633.65049
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