Question

Iterative function for calculating $(N{)}^{1/2N}$ by Newton-Raphson method is

• A. $\left(\frac{2N-1}{N}\right)x_o-x_k^{1-2N}$
• B. $(2N-1)x_k-x_k^{2N-1}$
• C. $\frac{1}{2N}(x_k^{2N}-N)$
• D. $\left(\frac{2N-1}{2N}\right)x_k+\frac{x_k^{1-2N}}{2}$

Answer 1

The correct option is D $\left(\frac{2N-1}{2N}\right)x_k+\frac{x_k^{1-2N}}{2}$

$x=(N{)}^{1/2N}$
$x^{2N}=N$
$x^{2N}-N=0$
$f\left(x\right)=x^{2N}-N$
$f^{\prime }\left(x\right)=2N.x^{2N-1}$

Newton Raphson iteration
$x_{k+1}=x_k-\frac{f\left(k\right)}{f^{\prime }\left(k\right)}=x_k-\frac{x_k^{2N}-N}{2N{x}_k^{2N-1}}$

$=\frac{2N{x}_k^{2N}-x_k^{2N}+N}{2N{x}_k^{2N-1}}$

$x_{k+1}=\left(\frac{2N-1}{2N}\right)x_k+\frac{x_k^{1-2N}}{2}$