Question

Consider the fixed-point iteration \[ x_{n+1} = \varphi(x_n), n \geq 0, \] with \[ \varphi(x) = 3 + (x - 3)^3, x \in (2.5, 3.5), \] and the initial approximation \( x_0 = 3.25 \). Then, the order of convergence of the fixed-point iteration method is

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

If the first derivative of \( \varphi(x) \) at the fixed point is zero, the fixed-point iteration method converges cubically (order 3).

Answer 1

The Correct Option is C

The fixed-point iteration method converges with an order of convergence given by the derivative of the function \( \varphi(x) \) at the fixed point. The fixed point of the iteration is \( x = 3 \), since: \[ \varphi(3) = 3 + (3 - 3)^3 = 3. \] To determine the order of convergence, we compute the derivative of \( \varphi(x) \): \[ \varphi'(x) = 3(x - 3)^2. \] At the fixed point \( x = 3 \): \[ \varphi'(3) = 3(3 - 3)^2 = 0. \] The convergence order is determined by the behavior of the derivative at the fixed point. Since the first derivative is zero, the convergence is of order 3, meaning the method converges cubically. Final Answer: \[ \boxed{3} \]