Question

To solve an algebraic equation \( f(x) = 0 \), an iterative scheme of the type \( x_{n+1} = g(x_n) \) is proposed, where \( g(x) = x - \frac{f(x)}{f'(x)} \). At the solution \( x = s \), \( g'(s) = 0 \) and \( g''(s) \neq 0 \). The order of convergence for this iterative scheme near the solution is \(\underline{\hspace{1cm}}\).
 

Hint:

For Newton-Raphson methods, when \( g'(s) = 0 \), the convergence order is quadratic (\(p = 2\)).

Answer 1

Correct Answer: 2

Given the iterative scheme \( x_{n+1} = g(x_n) \), the convergence order can be determined by the following:
The convergence order \( p \) of an iterative method is given by the condition:
\[ |x_{n+1} - s| \sim C |x_n - s|^p \]
For Newton-Raphson method (as suggested by the given \( g(x) \)), the order of convergence is typically quadratic when \( g'(s) = 0 \) and \( g''(s) \neq 0 \).
Thus, the order of convergence is:
\[ \boxed{2} \]