Question

Let \( f(x) = x^4 + 2x^3 - 11x^2 - 12x + 36 \) for \( x \in \mathbb{R}. \) The order of convergence of the Newton-Raphson method \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}, n \geq 0, \] with \( x_0 = 2.1 \), for finding the root \( \alpha = 2 \) of the equation \( f(x) = 0 \) is \(\underline{\hspace{1cm}}\) .

Hint:

The Newton-Raphson method generally converges quadratically, which is the highest possible order of convergence for methods that use local information.

Answer 1

Correct Answer: 1

For the Newton-Raphson method, the order of convergence is determined by the behavior of the error in successive iterations. The method converges quadratically, which means that the error decreases by a factor of roughly the square of the previous error. Thus, the order of convergence of the method for this equation is \( \boxed{1}. \)