The strength of nonstationary iteration
The paper deals with the iterative solution of a nonlinear operator equation \(F=0\) where F is an analytical multivariate or abstract function having only simple zeros. A very general definition of information and iteration without memory is introduced. The definiton of globally convergent iterations is discussed and the constant of global convergence is defined. It is proved that, for the class \({\mathcal F}_ 1\) of all analytic operators having simple zeros, the constant of global convergence is not larger than 1/2 for any iteration. Furthermore it is proved that any globally convergent iteration is a ''one-point'' iteration. A globally convergent nonstationary interpolatory iteration \(I_ 0\) for operators \(F\in {\mathcal F}_ 1\) is constructed where the constant of global convergence is not less than 1/3. The i-th step of this iteration requires the computation of \(F(x_ 0),F'(x_ 0),...,F^{(i-1)}(x_ o)\) and the solution of a polynomial equation of degree i-1. The generalization of \(I_ 0\) to iterations with memory is derived with the constant of global convergence not less than 1/4.
scientific article; zbMATH DE number 3864390