Examples from a general theory

For 'any' equation F(x) = 0 with an isolated solution x* which we want to compute, it will be shown that quantities p. can be defined and computed so that XPn+1 := Xp(n) + P-n . F(xp(n)) will 'converge' with 'any' speed needed, i.e. starting with an xp(0) in a neigbourhood of x*, xp(N) will for for some small value of N give x* within the precision you need. And it will do so even if the 'standard' iteration Xn+1 := X-n+ F(x(n)) is divergent.