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2. (25 pts) Consider the fixed point problem with g(x) 3 Use the fixed point theorem...
4. Let F be a continuously differentiable function, and let s be a fixed point of...
4. Let F be a continuously differentiable function, and let s be a fixed point of F (a) Prove if F,(s)| < 1, then there exists α > 0 such that fixed point iterations will o E [s - a, s+a]. converge tO s whenever x (b) Prove if IF'(s)| > 1, then given fixed point iterations xn satisfying rnメs for all n, xn will not converge to s.
3. (25 pts) Suppose f(x) is twice continuously differentiable for all r, and f"(x) > 0...
3. (25 pts) Suppose f(x) is twice continuously differentiable for all r, and f"(x) > 0 for all , and f(x) has a root at p satisfying f'(p) < 0. Let p, be Newton's method's sequence of approximations for initial guess po < p. Prove pi > po and pı < p Remember, Newton's method is Pn+1 = pn - f(pn)/f'(P/) and 1 f"(En P+1 P2 f(pP-p)2. between pn and p for some
3. (25 pts) Suppose f(x) is twice...
2a², where [Fixed Point Iterations, 15 pts). Let g(2) = -22 + 3x + a a...
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?
1. (a) State and prove the Mean-Value Theorem. You may use Rolle's Theorem provided you state...
1. (a) State and prove the Mean-Value Theorem. You may use Rolle's Theorem provided you state it clearly (b) A fired point of a function g: (a, bR is a point cE (a, b) such that g(c)-c Suppose g (a, b is differentiable and g'(x)< 1 for all x E (a, b Prove that g cannot have more than one fixed point. <「 for (c) Prove, for all 0 < x < 2π, that sin(x) < x.
2. Rolle's theorem states that if F : [a, b] → R is a continuous function,...
2. Rolle's theorem states that if F : [a, b] → R is a continuous function, differentiable on Ja, bl, and F(a) = F(b) then there exists a cela, b[ such that F"(c) = 0. (a) Suppose g : [a, b] → R is a continuous function, differentiable on ja, bl, with the property that (c) +0 for all cela, b[. Using Rolle's theorem, show that g(a) + g(b). [6 Marks] (b) Now, with g still as in part (a),...
Suppose you want to find a fixed point of a smooth function g(x) on the interval...
Suppose you want to find a fixed point of a smooth function g(x)
on the interval [a,b]
a. Give conditions which would be sufficient to show that fixed
point iteration on g(x), starting with some
[a,b], will converge to the fixed point p.
b. When is this convergence only linear?
c. When is this convergence only quadratic?
d. Suppose a smooth function f(x) has a root p with f '(p) != 0.
Assuming you choose the initial guess close enough...
2. [10 pts ] Use fixed-point iteration to determine a solution accurate to within 10-3 for f(x) x...
2. [10 pts ] Use fixed-point iteration to determine a solution accurate to within 10-3 for f(x) x - cos(x)/2, for x in [ 0,1]. Use your calculator to calculate values, but be sure to show what values are being calculated. (a) show the function g(o) that you use: (b) show the initial value po that you use: (c) show the computations for the successive values of the pi until convergence:
2. [10 pts ] Use fixed-point iteration to determine...
gol The fixed-point iteration Pn+1 = g(P) converges to a fixed point p = 0 of...
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).
II. Using Newton’s method, write a MATLAB program to find the fixed point of the following...
II. Using Newton’s method, write a MATLAB program to find the fixed point of the following function: ?(?) = √? + ?? accurate to at least 8 decimal places. (HINT: finding the fixed point of f(x) is the same as finding the zero of g(x) = f(x) − x. ) The output of this program should display in a single table (i) the solution for the fixed point, (ii) the initial guess, (iii) the number of iterations it took to...
6.5 Employ the Newton-Raphson method to determine a real root for 4x20.5 using initial guesses of...
6.5 Employ the Newton-Raphson method to determine a real root for 4x20.5 using initial guesses of (a) 4.52 f(x) 15.5x Pick the best numerical technique, justify your choice and then use that technique to determine the root. Note that it is known that for positive initial guesses, all techniques except fixed-point iteration will eventually converge. Perform iterations until the approximate relative error falls below 2 %. If you use a bracket- ing method, use initial guesses of x 0 and...
4. Let F be a continuously differentiable function, and let s be a fixed point of F (a) Prove if F,(s)| < 1, then there exists α > 0 such that fixed point iterations will o E [s - a, s+a]. converge tO s whenever x (b) Prove if IF'(s)| > 1, then given fixed point iterations xn satisfying rnメs for all n, xn will not converge to s.
3. (25 pts) Suppose f(x) is twice continuously differentiable for all r, and f"(x) > 0 for all , and f(x) has a root at p satisfying f'(p) < 0. Let p, be Newton's method's sequence of approximations for initial guess po < p. Prove pi > po and pı < p Remember, Newton's method is Pn+1 = pn - f(pn)/f'(P/) and 1 f"(En P+1 P2 f(pP-p)2. between pn and p for some
3. (25 pts) Suppose f(x) is twice...
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?
1. (a) State and prove the Mean-Value Theorem. You may use Rolle's Theorem provided you state it clearly (b) A fired point of a function g: (a, bR is a point cE (a, b) such that g(c)-c Suppose g (a, b is differentiable and g'(x)< 1 for all x E (a, b Prove that g cannot have more than one fixed point. <「 for (c) Prove, for all 0 < x < 2π, that sin(x) < x.
2. Rolle's theorem states that if F : [a, b] → R is a continuous function, differentiable on Ja, bl, and F(a) = F(b) then there exists a cela, b[ such that F"(c) = 0. (a) Suppose g : [a, b] → R is a continuous function, differentiable on ja, bl, with the property that (c) +0 for all cela, b[. Using Rolle's theorem, show that g(a) + g(b). [6 Marks] (b) Now, with g still as in part (a),...
Suppose you want to find a fixed point of a smooth function g(x)
on the interval [a,b]
a. Give conditions which would be sufficient to show that fixed
point iteration on g(x), starting with some
[a,b], will converge to the fixed point p.
b. When is this convergence only linear?
c. When is this convergence only quadratic?
d. Suppose a smooth function f(x) has a root p with f '(p) != 0.
Assuming you choose the initial guess close enough...
2. [10 pts ] Use fixed-point iteration to determine a solution accurate to within 10-3 for f(x) x - cos(x)/2, for x in [ 0,1]. Use your calculator to calculate values, but be sure to show what values are being calculated. (a) show the function g(o) that you use: (b) show the initial value po that you use: (c) show the computations for the successive values of the pi until convergence:
2. [10 pts ] Use fixed-point iteration to determine...
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).
6.5 Employ the Newton-Raphson method to determine a real root for 4x20.5 using initial guesses of (a) 4.52 f(x) 15.5x Pick the best numerical technique, justify your choice and then use that technique to determine the root. Note that it is known that for positive initial guesses, all techniques except fixed-point iteration will eventually converge. Perform iterations until the approximate relative error falls below 2 %. If you use a bracket- ing method, use initial guesses of x 0 and...
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