Can anyone please explain the steps of Laguerre's method? I searched for it but I couldn't really understand them. I am a high school student and things in Wikipedia didn't really help me understand.
For example I don't know why this is needed (from Wikipedia):
where the square root of a complex number is chosen to produce larger absolute value of the denominator, or equivalently, to satisfy: $${Re}\,(\overline{G} \sqrt{(n-1)(nH - G^2)}\,)>0$$
I understand the notation but I don't know how it is related to the conjugate of G and why this is required.
I've also seen things like $$\frac{n}{max[G\pm\sqrt{(n-1)(nH-G^2)}]}$$I don't know why it should be max. Could anyone please clearly explain the whole method?
Apart from the mystical origins, what you want, as in any iterative root approximation method, is an improvement of the current root approximation $a$. This is with high likelihood to be found close to $a$, especially if the polynomial value at $a$ is already rather small. So you have to chose between two corrections corresponding to the two signs of the root, to the smaller correction belongs the larger denominator.
Now you have to select the larger one of $|G\pm R|$. Squaring that expression gives $$ |G\pm R|^2=|G|^2+|R|^2\pm2Re(\bar G R) $$ So for the larger value you want $\pm Re(\bar G R)\ge 0$, so you chose the sign for which this inequality is true.
Computing the Fatou components of the Julia-fractal of this method for the polynomial $x^{10}+8x^3-x+1$ gives the picture below.
Points of the same color converge to the same root, the shading indicates the numbers of steps to reach the white region around the root. As one can see, there is a region around the root where the smaller update indeed leads closer to the root, and patches outside this central basin where that is not necessarily the case. However, for most polynomials the method converges in a small number of steps to some root.