A globally convergent ball Stirling method (Q1886274)

A Stirling iterative method for solving an operator equation \(P(x)=0\), or equivalently, a fixed point equation \(F(x)=x\) can be viewed as a combination of fixed point iteration and Newton iteration. The iterative scheme \(x_{n+1}=x_n-[I-F'(y_n)]^{-1}[x_n-F(x_n)]\) gives a general class of schemes. For \(y_n=x_n\), one has the standard Newton method, and for \(y_n=F(x_n)\), one has the (point) Stirling method. Analogously, ball methods generate a sequence of balls of shrinking radius, which contain a solution. A local convergence result is obtained for a ball-Stirling method. A numerical example is given in which a nonlinear Poisson problem is treated.