An application of the induction method of V. Pták to the study of regula falsi

Florian-Alexandru Potra

Aplikace matematiky (1981)

  • Volume: 26, Issue: 2, page 111-120
  • ISSN: 0862-7940

Abstract

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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 , ... , x n ) , n = 0 , 1 , 2 , ... . 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.

How to cite

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Potra, Florian-Alexandru. "An application of the induction method of V. Pták to the study of regula falsi." Aplikace matematiky 26.2 (1981): 111-120. <http://eudml.org/doc/15187>.

@article{Potra1981,
abstract = {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.},
author = {Potra, Florian-Alexandru},
journal = {Aplikace matematiky},
keywords = {induction method; regula falsi; $p$-dimensional rate of convergence; secant method; iterative procedure; induction method; regula falsi; p-dimensional rate of convergence; secant method; iterative procedure},
language = {eng},
number = {2},
pages = {111-120},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {An application of the induction method of V. Pták to the study of regula falsi},
url = {http://eudml.org/doc/15187},
volume = {26},
year = {1981},
}

TY - JOUR
AU - Potra, Florian-Alexandru
TI - An application of the induction method of V. Pták to the study of regula falsi
JO - Aplikace matematiky
PY - 1981
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 26
IS - 2
SP - 111
EP - 120
AB - 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.
LA - eng
KW - induction method; regula falsi; $p$-dimensional rate of convergence; secant method; iterative procedure; induction method; regula falsi; p-dimensional rate of convergence; secant method; iterative procedure
UR - http://eudml.org/doc/15187
ER -

References

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  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. (1973) MR0378398
  2. M. Fréchet, La notion de differentielle dans l'analyse générale, Ann. Ec. Norm. Sup, 42, (1925) 293-323. (1925) MR1509268
  3. T. Popoviciu, Introduction à Ia théorie des differences divisées, Bull. Math. Soc. Roum. Sci., 42 (1941), 65-78. (1941) MR0013171
  4. V. Pták, 10.1007/BF01399416, Numer. Math., 25 (1976), 279 - 285. (1976) Zbl0304.65037MR0478587DOI10.1007/BF01399416
  5. V. Pták, Nondiscrete mathematical induction and iterative existence proofs, Linear algebra and its applications 13 (1976), 233 - 238. (1976) MR0394119
  6. V. Pták, What should be a rate of convergence?, R. A.I. R. O. , Analyse Numérique 11,3 (1977), 279-286. (1977) MR0474799
  7. J. Schmidt, 10.1002/zamm.19630430102, I, II, Z. Angew. Math. Mech., 43 (1963), p. 1-8, 97-11.0. (1963) MR0147930DOI10.1002/zamm.19630430102
  8. J. Schröder, 10.1007/BF01899031, Arch. Math. (Basel) 7 (1956), 471-484. (1956) MR0088047DOI10.1007/BF01899031
  9. А. С. Сергеев, О метоге хорд, Сибир. Матем. Ж. 2 (1961), 282-289. (1961) Zbl1160.68305MR0130517
  10. С. Улъм, Об обобщенных разделенных разностях, I, II И АН ЭССР, Физика, математика, 16 (1967) р. 13-26, 146-156. (1967) Zbl1103.35360MR0215489

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