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Using Newton Raphson, show that root of the equation $N=AB$ is $$\sqrt {N}= \frac S4 + \frac NS, \text{ where } S=A+B.$$

amWhy
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ketankk
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    Please start using Mathjax. Also, defining $N$ in the title and $S$ in the body is confusing. The question should be complete without the title. Finally, you seem to be confusing the algorithm with the value it is designed to approximate. The square root of $3\times5$ is not $\frac{3+5}{4}+\frac{3\cdot5}{3+5}$. – almagest Jan 29 '20 at 07:56

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HINTS:

As almagest pointed out, it is not true that $\sqrt N = S/4+N/S$. But it is true that $\sqrt N \approx S/4+N/S$, and we can see that by using the Newton method. It will be the first approximation in the algorithm, if you choose the initial guess $x_0$ right. So there are two things to do:

  1. Find the function $f$ you want to find the root (i.e. the zero) of. That is $x=\sqrt N \implies f(x)=0$ should hold.
  2. Make a first guess at the square root. It should be something that is a good guess for $A\approx B$.
Milten
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