| Title:
|
Convergence conditions for Secant-type methods (English) |
| Author:
|
Argyros, Ioannis K. |
| Author:
|
Hilout, Said |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
60 |
| Issue:
|
1 |
| Year:
|
2010 |
| Pages:
|
253-272 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We provide new sufficient convergence conditions for the convergence of the secant-type methods to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses recurrent functions, and Lipschitz-type and center-Lipschitz-type instead of just Lipschitz-type conditions on the divided difference of the operator involved. It turns out that this way our error bounds are more precise than earlier ones and under our convergence hypotheses we can cover cases where earlier conditions are violated. Numerical examples are also provided. (English) |
| Keyword:
|
secant method |
| Keyword:
|
Banach space |
| Keyword:
|
majorizing sequence |
| Keyword:
|
divided difference |
| Keyword:
|
Fréchet-derivative |
| MSC:
|
49M15 |
| MSC:
|
65B05 |
| MSC:
|
65G99 |
| MSC:
|
65H10 |
| MSC:
|
65N30 |
| idZBL:
|
Zbl 1224.65141 |
| idMR:
|
MR2595087 |
| . |
| Date available:
|
2010-07-20T16:33:40Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140566 |
| . |
| Reference:
|
[1] Argyros, I. K.: Polynomial operator equations in abstract spaces and applications.St. Lucie/CRC/Lewis Publ. Mathematics series, 1998, Boca Raton, Florida, U.S.A. Zbl 0967.65070, MR 1731346 |
| Reference:
|
[2] Argyros, I. K.: On the Newton-Kantorovich hypothesis for solving equations.J. Comput. Appl. Math. 169 (2004), 315-332. Zbl 1055.65066, MR 2072881, 10.1016/j.cam.2004.01.029 |
| Reference:
|
[3] 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. Zbl 1061.47052, MR 2086964, 10.1016/j.jmaa.2004.04.008 |
| Reference:
|
[4] Argyros, I. K.: New sufficient convergence conditions for the Secant method.Chechoslovak Math. J. 55 (2005), 175-187. Zbl 1081.65043, MR 2121665, 10.1007/s10587-005-0013-1 |
| Reference:
|
[5] Argyros, I. K.: Convergence and Applications of Newton-Type Iterations.Springer-Verlag Publ., New-York (2008). Zbl 1153.65057, MR 2428779 |
| Reference:
|
[6] Argyros, I. K., Hilout, S.: Efficient Methods for Solving Equations and Variational Inequalities.Polimetrica Publisher (2009). MR 2424657 |
| Reference:
|
[7] Bosarge, W. E., Falb, P. L.: A multipoint method of third order.J. Optimiz. Th. Appl. 4 (1969), 156-166. Zbl 0172.18703, MR 0248581, 10.1007/BF00930576 |
| Reference:
|
[8] Chandrasekhar, S.: Radiative Transfer.Dover Publ., New-York (1960). MR 0111583 |
| Reference:
|
[9] Dennis, J. E.: Toward a unified convergence theory for Newton-like methods.In Nonlinear Functional Analysis and Applications (L.B. Rall, ed.), Academic Press, New York (1971), 425-472. Zbl 0276.65029, MR 0278556 |
| Reference:
|
[10] Hernández, M. A., Rubio, M. J., Ezquerro, J. A.: Solving a special case of conservative problems by Secant-like method.Appl. Math. Cmput. 169 (2005), 926-942. MR 2174693, 10.1016/j.amc.2004.09.070 |
| Reference:
|
[11] 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. MR 1747223, 10.1016/S0377-0427(99)00116-8 |
| Reference:
|
[12] Huang, Z.: A note of Kantorovich theorem for Newton iteration.J. Comput. Appl. Math. 47 (1993), 211-217. MR 1237313, 10.1016/0377-0427(93)90004-U |
| Reference:
|
[13] Kantorovich, L. V., Akilov, G. P.: Functional Analysis.Pergamon Press, Oxford (1982). Zbl 0484.46003, MR 0664597 |
| Reference:
|
[14] Laasonen, P.: Ein überquadratisch konvergenter iterativer Algorithmus.Ann. Acad. Sci. Fenn. Ser I 450 (1969), 1-10. Zbl 0193.11704, MR 0255047 |
| Reference:
|
[15] Ortega, J. M., Rheinboldt, W. C.: Iterative Solution of Nonlinear Equations in Several Variables.Academic Press, New York (1970). Zbl 0241.65046, MR 0273810 |
| Reference:
|
[16] Potra, F. A.: Sharp error bounds for a class of Newton-like methods.Libertas Mathematica 5 (1985), 71-84. Zbl 0581.47050, MR 0816258 |
| Reference:
|
[17] Schmidt, J. W.: Untere Fehlerschranken fur Regula-Falsi Verfahren.Period. Hungar. 9 (1978), 241-247. MR 0494896, 10.1007/BF02018090 |
| Reference:
|
[18] Yamamoto, T.: A convergence theorem for Newton-like methods in Banach spaces.Numer. Math. 51 (1987), 545-557. Zbl 0633.65049, MR 0910864, 10.1007/BF01400355 |
| Reference:
|
[19] Wolfe, M. A.: Extended iterative methods for the solution of operator equations.Numer. Math. 31 (1978), 153-174. Zbl 0375.65030, MR 0509672, 10.1007/BF01397473 |
| . |
