| Title:
|
An application of the induction method of V. Pták to the study of regula falsi (English) |
| Author:
|
Potra, Florian Alexandru |
| Language:
|
English |
| Journal:
|
Aplikace matematiky |
| ISSN:
|
0373-6725 |
| Volume:
|
26 |
| Issue:
|
2 |
| Year:
|
1981 |
| Pages:
|
111-120 |
| Summary lang:
|
English |
| Summary lang:
|
Czech |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we introduce the notion of "$p$-dimensional rate of convergence" which generalizes the notion of rate of convergence introduced by V. Pták. Using this notion we give a generalization of the Induction Theorem of V. Pták, which may constitute a basis for the study of the iterative procedures of the form $X_{n+1}=F(x_{n-p+1},X_{n-p+2},\ldots, x_n)$, $n=0,1,2,\ldots$. As an illustration we apply these results to the study of the convergence of the secant method, obtaining sharp estimates for the errors at each step of the iterative procedure. (English) |
| Keyword:
|
induction method |
| Keyword:
|
regula falsi |
| Keyword:
|
$p$-dimensional rate of convergence |
| Keyword:
|
secant method |
| Keyword:
|
iterative procedure |
| MSC:
|
47H17 |
| MSC:
|
49A51 |
| MSC:
|
58C15 |
| MSC:
|
65H10 |
| MSC:
|
65J10 |
| idZBL:
|
Zbl 0486.65038 |
| idMR:
|
MR0612668 |
| DOI:
|
10.21136/AM.1981.103902 |
| . |
| Date available:
|
2008-05-20T18:16:31Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/103902 |
| . |
| Reference:
|
[1] M. Balazs G. Goldner: On existence of divided differences in linear spaces.Revue d'analyse numérique et de la théorie de l'approximation, 2 (1973), 5-9. MR 0378398 |
| Reference:
|
[2] M. Fréchet: La notion de differentielle dans l'analyse générale.Ann. Ec. Norm. Sup, 42, (1925) 293-323. MR 1509268, 10.24033/asens.766 |
| Reference:
|
[3] T. Popoviciu: Introduction à Ia théorie des differences divisées.Bull. Math. Soc. Roum. Sci., 42 (1941), 65-78. MR 0013171 |
| Reference:
|
[4] V. Pták: The rate of convergence of Newton's process.Numer. Math., 25 (1976), 279 - 285. Zbl 0304.65037, MR 0478587, 10.1007/BF01399416 |
| Reference:
|
[5] V. Pták: Nondiscrete mathematical induction and iterative existence proofs.Linear algebra and its applications 13 (1976), 233 - 238. MR 0394119 |
| Reference:
|
[6] V. Pták: What should be a rate of convergence?.R. A.I. R. O. , Analyse Numérique 11,3 (1977), 279-286. MR 0474799, 10.1051/m2an/1977110302791 |
| Reference:
|
[7] J. Schmidt: Eine Übertragung der Regula Falsi auf Gleichungen in Banachraum.I, II, Z. Angew. Math. Mech., 43 (1963), p. 1-8, 97-11.0. MR 0147930, 10.1002/zamm.19630430102 |
| Reference:
|
[8] J. Schröder: Nichtlineare Majoranten beim Verfahren der schrittweissen Näherung.Arch. Math. (Basel) 7 (1956), 471-484. MR 0088047, 10.1007/BF01899031 |
| Reference:
|
[9] А. С. Сергеев: О метоге хорд.Сибир. Матем. Ж. 2 (1961), 282-289. Zbl 1160.68305, MR 0130517 |
| Reference:
|
[10] С. Улъм: Об обобщенных разделенных разностях.I, II И АН ЭССР, Физика, математика, 16 (1967) р. 13-26, 146-156. Zbl 1103.35360, MR 0215489 |
| . |
