Question

Give the answer in one word:
Write the formula of square root of a quantity $y$ by Newton-Raphson method.

Answer 1

Let $x=\sqrt{y}$

$⟹x^2-y=0$

Newton-Raphson method $x_{n+1}=x_n-\frac{f\left(x_n\right)}{f^{\prime }\left(x_n\right)}$

$f\left(x_n\right)=x_n^2-y$

$f^{\prime }\left(x_n\right)=2x_n$

$x_{n+1}=x_n-\frac{x_n^2-y}{2x_n}$

$=$ $x_n-\frac{x_n}{2}+\frac{y}{2x_n}=\frac{x_n}{2}+\frac{y}{2x_n}$

$x_{n+1}=\frac{1}{2}[x_n+\frac{y}{x_n}]$ $\Rightarrow$ Formula