Newton's Method Note: throughout this question do not use any existing implementations of any of the algorithms discussed unless explicitly asked to in the question. Using existing implementations can result in a grade of zero for the entire question. In homework I we studied gradient descent (GD), which is usually referred to as a first order method. Here, we study an alternative algorithm known as Newton's algorithm, which is generally referred to as a second order method. Roughly speaking, a second order method makes use of both first and second derivatives. Generally, second order methods are much more accurate than first order ones. Given a twice differentiable function g : R. - R, Newton's method generates a sequence {x(k )) iteratively according to the following update rule: k =0, 1, 2, .... (1) For example, consider the function g(x) = (x' - sin(x) with initial guess x(0) = 0 . Then g (x) = X - Com(X), and (x) = 1 + zin( x), and so we have the following iterations: I'l = xi . x - coax" ) =0 - 0 - coz(0) = 1 1 + gin( x(0) ) 1 + gin(0) x ]) = x(0) - > x() - com(x ) 1=1 - 1 - COB() = 0.750363867840244 1 + sin( x(1) ) 1 + gin(1) x(3) = 0.739112890911362 and this continues until we terminate the algorithm (as a quick exercise for your own benefit, code this up, plot the function and each of the iterates). We note here that in practice, we often use a different update called the dampened Newton method, defined by: x(k#1) = x(k) . q=(x) g(X)' k=0 , 1, 2, .... (2) Here, as in the case of GD, the step size "a has the effect of 'dampening' the update. (2) Consider the twice differentiable function f : R." - R. The Newton steps in this cage are now: ((4+0) = x(0) . (H(x))) 17f(x()), k =0, 1, 2, . ..; (3) where H(x) = V'f (x) is the Hessian of f. Explain heuristically (in a couple of sentences) how the above formula is a generalization of equation ( 1) to functions with vector inputs. what to submit: Some commentary (b) Consider the function f : R.' - R. defined by f (x, y) = 100( y - x]]' + (1 - x)'. Create a 3D plot of the function using mplot3d (see lab0 for example). Further, compute the gradient and Hessian of f. what to submit: A single plot, the code used to generate the plot, the gradient and Hessian calculated along with all working.