Let $a\gt0$. Start from a convenient equation and use Newton's method to deduce a method to approximate $\frac{1}{\sqrt{a}}$ without divisions. How is the starting value chosen? What is the termination criteria? Deduce from here a method to approximate $\sqrt{a}$ without divisions.
Now I see that in the solution, they start from the equation $f\left(x\right)=\frac{1}{x^2}-a=0$ for which they apply Newton's method. Why is this the equation? I don't understand the reasoning behind it.
Also, they say that any value $x_0\gt0$ can be chosen as a starting value. I don't really understand this either. What about the $\sqrt{a}$ part?
Any tips would be greatly appreciated!