| Title:
|
New sufficient convergence conditions for the secant method (English) |
| Author:
|
Argyros, Ioannis K. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
55 |
| Issue:
|
1 |
| Year:
|
2005 |
| Pages:
|
175-187 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We provide new sufficient conditions for the convergence of the secant method to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses “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 the earlier ones and under our convergence hypotheses we can cover cases where the earlier conditions are violated. (English) |
| Keyword:
|
secant method |
| Keyword:
|
Banach space |
| Keyword:
|
majorizing sequence |
| Keyword:
|
divided difference |
| Keyword:
|
Fréchet-derivative |
| MSC:
|
47H17 |
| MSC:
|
47J25 |
| MSC:
|
49M15 |
| MSC:
|
65B05 |
| MSC:
|
65G99 |
| MSC:
|
65H10 |
| MSC:
|
65J15 |
| MSC:
|
65N30 |
| idZBL:
|
Zbl 1081.65043 |
| idMR:
|
MR2121665 |
| . |
| Date available:
|
2009-09-24T11:22:00Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127968 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[6] J. M. Gutiérrez: A new semilocal convergence theorem for Newton’s method.J. Comput. Appl. Math. 79 (1997), 131–145. MR 1437974, 10.1016/S0377-0427(97)81611-1 |
| Reference:
|
[7] J. M. Gutiérrez, M. A. Hernandez and M. A. Salanova: Accessibility of solutions by Newton’s method.Intern. J. Comput. Math. 57 (1995), 239–247. 10.1080/00207169508804427 |
| Reference:
|
[8] M. A. Hernandez, M. J. Rubio and J. A. Ezquerro: 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:
|
[9] L. V. Kantorovich and G. R. Akilov: Functional Analysis in Normed Spaces.Pergamon Press, Oxford Press, , 1982. MR 0664597 |
| Reference:
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[10] J. M. Ortega and W. C. Rheinboldt: Iterative Solution of Nonlinear Equations in Several Variables.Academic Press, New York, 1970. MR 0273810 |
| Reference:
|
[11] F. A. Potra: Sharp error bounds for a class of Newton-like methods.Libertas Mathematica 5 (1985), 71–84. Zbl 0581.47050, MR 0816258 |
| Reference:
|
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| . |
