Recall the basic result on the rate of convergence for the Newton-Raphson method:

E(k) t

f(xr) 2f(xr)

[E(k1) t ]2

which holds in the vicinity of the root xr. We would like to develop a "real time" check of the rate of convergence based on the values of the function f during the iterations, rather than the total error for dierent iterations. Start from the Taylor series expansion of f(x) around xr. Keeping only the linear terms from the Taylor series expansion, obtain approximate expressions for f(x) at x(k1) and x(k), in the vicinity of the root xr. Use these expressions and the equation from above to show the following relation: f(x(k)) f(xr) 2[f(xr)]2 [f(x(k1))]2