Exercise 2.17 Analyze the convergence of the fixed point iterations p(k+1) _ x(*)[(x(k))2 + 3a] 3(x(k))2...
2. (25 pts) Consider the fixed point problem with g(x) 3 Use the fixed point theorem to show fixed point iterations using g(r) converge to fixed point p E (0, 1] for all initial guesses po E [0, 1]. a Remember, the fixed point theorem: If g(x) is continuously differentiable in [a, b] and g [a, b g(x) converge to a fixed point p E [a, b] for all initial guesses po E [a, b. a, band g (x k...
2a², where [Fixed Point Iterations, 15 pts). Let g(2) = -22 + 3x + a a is a parameter. (a) Show that a is a fixed point of g(x). (b) For what values of a does the iteration scheme On+1 = g(n) converge linearly to the fixed point a (provided zo is chosen sufficiently close to a)? (c) Is there a value of a for which convergence is quadratic?
Obtain a rough estimate of all real roots of the function f(x) = ex-x-2 by incremental searching in [-2,2]. Use Ax- 1. b) Obtain two iterating functions for finding each of these roots by fixed-point iteration by solving for each x which appears in the equation. c) Without doing any iterations, determine if each iterating function will converge to each root and ether the convergence or divergence will be monotonic or oscillatory [25] a) 1. d) From the iterati ng...
2. (a) Suppose we have to find the root xof x); that is, we have to solve )0. Fixed-point methods do this by re-writing the equation in the form x·= g(x*) , and then using the iteration scheme : g(x) Show this converges (x-→x. as n→o) provided that K < 1 , for all x in some interval x"-a < x < x*+a ( a > 0 ) about the rootx 6 points] (b) Newton's method has the form of...
*3. Consider a function, f(x,y) = x3 + 3(y-1)2 . Starting from an initial point, X0 = [1 1] T , perform 2 iterations of conjugate gradient method (also known as Fletcher-Reeves method) to minimize the above function. Also, please check for convergence after each iteration.
q = 4 Q2 Consider the equation x -3x'te0 (a) Write this equation as x =g(x) in three different forms. Apply convergence test to each of these forms. Which g(r) is more suitable for the fixed point iteration. (b) Compute first 4 iterations by taking x 1 and graph each value of x and g(x) to show convergence or divergence of the scheme. Find the fixed point of g(x) correct to 5 decimal digits using the following fixed point iteration...
gol The fixed-point iteration Pn+1 = g(P) converges to a fixed point p = 0 of g(x) = x for all 0 < po < 1. The order of convergence of the sequence {n} is a > 0 if there exists > O such that lim Pn+1-pl =X. -00 P -plº Use the definition (6) to find the order of convergence of the sequence in (5).
7. Problem: (Fixed point iterations) Let f [0,3] [0,3] be a contraction with contraction constant q = 1 /3. Hence, f has a unique fixed point х* є [0,3]. n+1 f (xn), start: xo-0. Determine the smallest valid lower bound on the number of iterations n that are required to guarantee that x -n000 7. Problem: (Fixed point iterations) Let f [0,3] [0,3] be a contraction with contraction constant q = 1 /3. Hence, f has a unique fixed point...
1. tain a rough estimate of all real roots of the function f(x) searching in [-2,2]. Use Ax1 ex-2 by incremental b) Obtain two iterating functions for finding each of these roots by fixed-point iteration by solving for each x which appears in the equation c) Without doing any iterations, determine if each iterating function will converge to each root and state whether the convergence or divergence will be monotonic or oscillatory d) From the iterating functions obtained in part...
1 10 onvelge a636lutely, converges conditionally, or diverges. Justify your answer, including naming the convergence test you use. (1n(b) n7/3-4 (2k)! n-2 k-0 (-1)k 2k 4. (a) (10) Let* Find a power series for h(), and find the radius of convergence Ri for h'(x). Find the smallest reasonable positive integer n so that - (b) (10) Let A- differs from A by less than 0.01. Give reasons. 5. (a) (10) Let g(x) sin z. Write down the Taylor series for...