On a quadratically convergent method using divided differences of order one under the gamma condition
The authors study the functional equation: \(F(x)=0\), where \(F\) is a twice Fréchet differentiable operator between a convex subset of a Banach space \(X\) and a Banach space \(Y\). They construct a sequence \(x_{n}\) approximating the unique solution \(x^*\) of the functional equation \[ x_{n+1}=x_{n}-[2x_{n}-x_{n-1},x_{n-1}]^{-1}F(x_{n}) ;\quad n>0 . \] Here \([x,y]\) denotes the divided difference of order 1 at the point \((x,y)\) and \[ [x,y](x-y)=F(x)-F(y). \] The first theorem of the paper gives conditions on the first and second Fréchet derivative of \(F\) under which the approximating sequence is well defined, remains in a certain neighborhood of \(x^*\) and converges to \(x^*\). The second theorem provides conditions under which semilocal convergence of the sequence takes place. Numerical examples illustrate the results.
scientific article; zbMATH DE number 5292959
scientific article