# On optimal iterative methods

## Abstract

Let $$\left( X,\rho \right)$$ be a complete matrix space, the nonlinear mapping $$\varphi:I\subset X\rightarrow X$$ and the equation $$x=\varphi \left(x\right)$$ with solution $$x^{\ast}$$. We consider another application, $$F:X^{k}\rightarrow X$$ for which we assume the diagonal coincides with $$\varphi$$:  $$F(x,…,x)=\varphi(x)$$. In order to solve the mentioned equation we consider the iterative method $x_{n+1}=F\left(x_{n},x_{n-1},\cdots,x_{n-k+1}\right),$ $$n=k-1,k,…$$ Let $$i_{0},i_{1},….,i_{k-1}$$ be a permutation of the numbers $$0,1,…,k-1$$ and therefore $$i_{0}-n-k-1,~i_{1}+n-k+1,…,i_{k-1}+n-k+1$$ a  permutation of the numbers $$n-k+1,\ n-k+2,…,n$$. Among the class of methods given by $x_{n+1}=F\left( x_{i_{0}+n-k+1},x_{i_{1}+n-k+1},…,x_{i_{k-1}+n-k+1}\right)$ we determine the method for which the difference $$\mathcal{\rho}\left( x^{\ast},x_{n+1}\right)$$ is the smallest.

## Authors

Ion Păvăloiu, Ioan Şerb

## Title

### Original title (in French)

Sur des méthodes itératives optimales

### English translation of the title

On optimal iterative methods

## Keywords

multistep successive approximations; optimal iterative methods

## References

[1] I. Pavaloiu, Introducere in teoria aproximarii solutiilor ecuatiilor, Ed. Dacia,  Cluj-Napoca, 1976.

[2] I. Pavaloiu, Rezolvarea ecuatiilor prin interpolare, Ed. Dacia, Cluj-Napoca, 1981.

[3] Weinischke, J. H., Uber eine Klasse von Iterationsverfahren, Num. Math., 6 (1964), 395–404.

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##### Cite this paper as:

I. Păvăloiu, I. Şerb, Sur des méthodes itératives optimales, Seminar on functional analysis and numerical methods, Preprint no. 1 (1983), pp. 175-182 (in French).

##### Journal
Seminar on Functional Analysis and Numerical Methods
Preprint
##### Publisher Name
“Babes-Bolyai” University
Faculty of Mathematics and Physics
Research Seminars
##### DOI

Not available yet.