Question

The Newton-Raphson method formula for finding the square root of a real number N from the equation $x^2-N=0$ is

• A. $x_{i+1}=\frac{5x_i}{2}$
• B. $x_{i+1}=\frac{3x_i}{2}$
• C. $x_{i+1}=\frac{1}{2}(x_i+\frac{N}{x_i^2})$
• D. $x_{i+1}=\frac{1}{2}(x_i+\frac{N}{x_i})$

Answer 1

The correct option is D $x_{i+1}=\frac{1}{2}(x_i+\frac{N}{x_i})$

$x_{i+1}=x_i-\frac{f\left(x_i\right)}{f^{\prime }\left(x_i\right)}$
$f\left(x\right)=x^2-N$
$f^{\prime }\left(x\right)=2x$
$x_{i+1}=x_i-\frac{x_i^2-N}{2x_i}=x_i-\frac{x_i}{2}+\frac{N}{2x_i}$
$=\frac{x_i}{2}+\frac{N}{2x_i}=\frac{1}{2}(x_i+\frac{N}{x_i})$