Question

Given \( x_0 \neq 0 \), the iteration \[ x_{n+1} = \frac{1}{2} \left( -\frac{9}{x_n} + x_n \right), n \geq 0 { is a} \]

• A. Newton's method for \( f(x) = -9 + x^2 \)
• B. Fixed point iteration for \( f(x) = \frac{9 + x^2}{2x} \)
• C. Newton's method for \( f(x) = 9 + x^2 \)
• D. Fixed point iteration for \( f(x) = 9 + x^2 \)

Hint:

When solving for roots using Newton's method, the iteration formula is typically in the form \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \). Recognizing this pattern helps identify the method used.

Answer 1

The Correct Option is C

The given iteration is: \[ x_{n+1} = \frac{1}{2} \left( -\frac{9}{x_n} + x_n \right) \] This is a form of Newton's method. In general, Newton's method is used to find successively better approximations to the roots of a real-valued function. The standard form of Newton's method is: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] In our case, the iteration formula matches the form of Newton's method, with \( f(x) = 9 + x^2 \). 
Therefore, the correct answer is option (3), which corresponds to Newton's method for \( f(x) = 9 + x^2 \).