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An application of the induction method of V. Pták to the study of regula falsi
Applications of Mathematics, Tome 26 (1981) no. 2, pp. 111-120
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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.
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.
DOI : 10.21136/AM.1981.103902
Classification : 47H17, 49A51, 58C15, 65H10, 65J10
Keywords: induction method; regula falsi; $p$-dimensional rate of convergence; secant method; iterative procedure 
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Potra, Florian Alexandru. An application of the induction method of V. Pták to the study of regula falsi. Applications of Mathematics, Tome 26 (1981) no. 2, pp. 111-120. doi: 10.21136/AM.1981.103902

[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

[2] M. Fréchet: La notion de differentielle dans l'analyse générale. Ann. Ec. Norm. Sup, 42, (1925) 293-323. | DOI | MR

[3] T. Popoviciu: Introduction à Ia théorie des differences divisées. Bull. Math. Soc. Roum. Sci., 42 (1941), 65-78. | MR

[4] V. Pták: The rate of convergence of Newton's process. Numer. Math., 25 (1976), 279 - 285. | DOI | MR | Zbl

[5] V. Pták: Nondiscrete mathematical induction and iterative existence proofs. Linear algebra and its applications 13 (1976), 233 - 238. | MR

[6] V. Pták: What should be a rate of convergence?. R. A.I. R. O. , Analyse Numérique 11,3 (1977), 279-286. | DOI | MR

[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. | DOI | MR

[8] J. Schröder: Nichtlineare Majoranten beim Verfahren der schrittweissen Näherung. Arch. Math. (Basel) 7 (1956), 471-484. | DOI | MR

[9] А. С. Сергеев: О метоге хорд. Сибир. Матем. Ж. 2 (1961), 282-289. | MR | Zbl

[10] С. Улъм: Об обобщенных разделенных разностях. I, II И АН ЭССР, Физика, математика, 16 (1967) р. 13-26, 146-156. | MR | Zbl

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