(a), (b), (c)
(d)
3. Newton's method Let f:R → R be given by f(x):= { x - a3, where...
detailed answer and thumbs up guaranteed Newton's method Let f: R + R be given by f(x) := }\x – al, where a € R is a constant. The minimizer is obviously 2* = a. Suppose that we apply Newton's method to the following problem: minimize f(x):= با این | 2 – al: from an initial point x° ER \ {a}. (a) (3 points) Write down f'(x) and f'(). You need to consider two cases: < > a and x...
5. Let f(x) = ax2 +bx+c, where a > 0. Prove that the secant method for minimization will terminate in exactly one iteration for any initial points Xo, X1, provided that x1 + xo: 6. Consider the sequence {x(k)} given by i. Write down the value of the limit of {x(k)}. ii. Find the order of convergence of {x(k)}. 7. Consider the function f(x) = x4 – 14x3 + 60x2 – 70x in the interval (0, 2). Use the bisection...
(4) Let f R -R be a strictly conve:r C2 function and let 0 a) Write the Euler-Lagrange equation for the minimizer u.(x) of the following problem: minimize u subject to: u E A, where A- 0,REC1[0 , 1and u (0 a u(1)b) b) Assuming the minimizer u(a) is a C2 function, prove t is strictly convex (4) Let f R -R be a strictly conve:r C2 function and let 0 a) Write the Euler-Lagrange equation for the minimizer u.(x)...
12. Let f: x> (x-1)2-1. (a) Apply fixed-point iteration to f with ro-1. What is the next iterate? (b) Apply Newton's method to f with ro- 1. What is the next iterate? (c) Apply the secant method to f with 20 1 andェ,-2. What is the next iterate? CD 12. Let f: x> (x-1)2-1. (a) Apply fixed-point iteration to f with ro-1. What is the next iterate? (b) Apply Newton's method to f with ro- 1. What is the next...
(6) Let fel ), where is Lebesgue measure on R. Define F:R → R by F(x) = f' f(t) dx. (a) Prove that F is a continuous function. (b) Prove that F is uniformly continuous on R. (Note that R is not compact.)
Problem 3. (30 pts.) Let f(x) 32-1 (a) Calculate the derivative (the gradient) (r) and the second derivative (the Hessian) "() (4pts) (b) Using ro = 10, iterate the gradient descent method (you choose your ok) until s(k10-6 (11 pts) (c) Using zo = 10, iterate Newton's method (you choose your 0k ) until Irk-rk-1 < 10-6. (15 pts) Problem 4. (30 pts.) Let D ), (1,2), (3,2), (4,3),(4,4)] be a collection of data points. Your task is to find...
Can you help me with parts A to D please? Thanks 3 Newton and Secant Method [30 pts]. We want to solve the equation f(x) 0, where f(x) = (x-1 )4. a) Write down Newton's iteration for solving f(x) 0. b) For the starting value xo 2, compute x c) What is the root ξ of f, i.e., f(5) = 0? Do you expect linear or quadratic order of convergence to 5 and why? d) Name one advantage of Newton's...
3. Let f(a) 990 (a) Use the differentials to estimate 990 (b) Apply Newton's method to the equation f(z) = 0, derive the recurrence relation of r, and 2,-,, (c) Use Newton's method with initial approximation 퍼 10 to find 8p the third approximation to the root of the equation f(a)0. 3. Let f(a) 990 (a) Use the differentials to estimate 990 (b) Apply Newton's method to the equation f(z) = 0, derive the recurrence relation of r, and 2,-,,...
Problem 3: Let f: X -> R, XC R2, be given by f(x, y)n(x 2y 1), V(r,y) e X Find the maximal domain X and write the second-order Taylor polynomial for f around the point (2,1) E X. (6 points) Problem 3: Let f: X -> R, XC R2, be given by f(x, y)n(x 2y 1), V(r,y) e X Find the maximal domain X and write the second-order Taylor polynomial for f around the point (2,1) E X. (6 points)
*14. Let A be an n x n matrix. Define f:R" R by f(x) = Ax.x = x'AX. (a) Show that f is differentiable and Df (a)h = Aah + Ah a. (b) Deduce that when A is symmetric, Df(a)h = 2Aa . h. 15. Let a € R", 8 >0, and suppose f: B(a, 8) - R is differentiable at a. Suppose f(a) f(x)