LetBbe a compact interval in ℝ,M=B×Band φ:M→Ba map inC3(M). Suppose that ξ is a fixed point of φ. We study the behaviour of the iteratesxn+2=φ(xn+1,xn) (x0,x1∈B). Of particular interest is the situation where ϕx(ξ,ξ)=ϕy(ξ,ξ)=0. In case of the wellknown “Regula falsi” we also have ϕxx(ξ,ξ)=ϕyy(ξ,ξ)=0 and the order of convergence is$$tfrac{1}{2}(1 + sqrt 5 )$$. We consider the case where ϕyy(ξ,ξ)≠0. It turns out that there is a constant γ∈(1,2) such that successive iterates gain factors γ, 2/γ, γ, 2/γ, ... on the number of valid decimals. Depending on the initial iteratesx0,x1the number λ may range over all of (1, 2) such that in the extreme cases an additional iterative step may have virtually no effect on the number o
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