Multiple choice

The square root of a number N is to be obtained by applying the Newton Raphson iterations to the equation x2 – N = 0. If i denotes the iteration index, the correct iterative scheme will be

  1. $X_{i+1} = \frac{1}{2}\Big(X_i + \frac{N}{X_i}\Big)$
  2. $X_{i+1} = \frac{1}{2}\Big(X_i^2 + \frac{N}{X_i^2}\Big)$
  3. $X_{i+1} = \frac{1}{2}\Big(X_i + \frac{N^2}{X_i}\Big)$
  4. $X_{i+1} = \frac{1}{2}\Big(X_i - \frac{N}{X_i}\Big)$
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A Correct answer
Explanation

Newton-Raphson iteration for f(x) = 0 is x_{i+1} = x_i - f(x_i)/f'(x_i). For f(x) = x^2 - N, f'(x) = 2x. Thus, x_{i+1} = x_i - (x_i^2 - N)/(2x_i) = (2x_i^2 - x_i^2 + N)/(2x_i) = (x_i^2 + N)/(2x_i) = 1/2 * (x_i + N/x_i).

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