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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
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.
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