On optimal iterative methods

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.

Ion Păvăloiu, Ioan Şerb

Original title (in French)

Sur des méthodes itératives optimales

English translation of the title

On optimal iterative methods

multistep successive approximations; optimal iterative methods

[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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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).

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