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
(Tiberiu Popoviciu Institute of Numerical Analysis)

Ioan Şerb
(Tiberiu Popoviciu Institute of Numerical Analysis)

Original title (in French)

Sur des méthodes itératives optimales

English translation of the title

On optimal iterative methods

multistep successive approximations; optimal iterative methods

Scanned paper.

PDF-Latex version of the paper.

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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