Problem 4 (20 points, 5 points each) Consider the quadratic convergence theorem of Newton's Method (Multi- variate case). Let F(x) : Rn - R" be a continuously differentiable function on an open convex set D C R". Assume that there exists a* E R" and r > 0 and B > 0 such that N(x*, r) C D, F(x*) = 0, VJF(x*)-1 exists and ||VJF(x*)-'|| 0 such that for all x ) E N(a*, E) the sequence generated by x(k+1) _ x(k) - VJF(x(k) )-1F(x(k) ) for k = 0, 1, 2, .... is well-defined and converges to x* with 1/ ac (4: +1) - 20* | |