Abstract
Let \(\left( X,\rho \right)\) be a complete metric space, \(f:X\rightarrow X\) a nonlinear mapping. In order to solve the equation \(x=f\left( x\right) \) we consider a multistep method \[x_{n+k+1}=G(x_{n},x_{n+1},…,x_{n+k}), \quad n=1,2,… \] generated by a mapping \(G:X^{k+1}\rightarrow X\), whose diagonal restriction coincides with \(f\): \(G(x,…,x)=f(x)\). Under Lipschitz assumption on \(G\) we determine the algebraic equation whose unique positive solution leads to the convergence order of the iterations. We also study the case when the operator \(G\) replaced by an approximation of it.
Authors
Ion Păvăloiu
Title
Original title (in French)
Sur l’approximation des racines des equations dans une espace métrique
English translation of the title
On approximating the solutions of equations in metric spaces
Keywords
multistep iterative methods; convergence; successive approximations
References
[1] I. Pavaloiu, I., Serb, Sur des methodes iteratives optimales, Research Seminars, Seminar on Functional Analysis and Numerical Methods, Preprint Nr.1 (1983), 175–182.
[2] I.A. Rus, An iterative method for the solution of the equation x = f (x, x, . . . , x), Anal. Num´er. Theor. Approx., 10 (1981), 95–100.
[3] Weinischke, J.H., Uber eine klasse von Iterationverfahren, Numerische Mathematik 6 (1964), 395–404.
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About this paper
Cite this paper as:
I. Păvăloiu, Sur l’approximation des racines des equations dans une espace métrique, Seminar on functional analysis and numerical methods, Preprint no. 1 (1989), pp. 95-104 (in French).
Journal
Seminar on functional analysis and numerical methods,
Preprint
Publisher Name
“Babes-Bolyai” University,
Faculty of Mathematics,
Research Seminars
DOI
Not available yet.