Theorem 5 (Linearity of Fixed Point Iteration)

Let [Maple Math] be a continuous function over [Maple Math] , with [Maple Math] in the interval [Maple Math] for all [Maple Math] in [Maple Math] . Assume that [Maple Math] is continuous on ( [Maple Math] ) with [Maple Math] with [Maple Math] for all [Maple Math] in ( [Maple Math] ). Then, if [Maple Math] . the sequence [Maple Math] ( [Maple Math] ), converges linearly to the unique fixed point [Maple Math] for any [Maple Math] in [Maple Math] .