Question: numerical analysis suppose that x gx has unique solution...

(Numerical Analysis)

Suppose that x = g(x) has unique solution ξ with g′(ξ) = 0. Also assume that g is twice differentiable on some closed interval I containing ξ and x0, and that g′′(x) is continuous on I.

(a) Show that if g′(ξ) = 0, then the simple iteration given by xk+1 = g(xk), with x0 given above, converges quadratically to ξ. (hint: Perform a Taylor expansion of g(xk) at x = ξ)

(b) Show that with g(x) = x − f(x) , x = g(x ) yields Newton’s method.f′(x) k+1 k