Question

The Newton-Raphson iteration $$ x_{n+1} = \frac{1}{3} \left( 2 x_n + \frac{N}{x_n^2} \right) $$ is used to compute

• A. \( \sqrt[3]{N} \)
• B. \( \sqrt{N} \)
• C. \( \frac{1}{N} \)
• D. \( \frac{1}{\sqrt{N}} \)

Hint:

Newton-Raphson formula for cube root uses iteration with derivative \( 3x^2 \).

Answer 1

The Correct Option is A

The iteration formula corresponds to the Newton-Raphson method for finding cube root of \( N \) by solving \[ x^3 - N = 0 \] Update formula: \[ x_{n+1} = x_n - \frac{x_n^3 - N}{3 x_n^2} = \frac{1}{3} \left( 2 x_n + \frac{N}{x_n^2} \right) \]