# Laguerre's method and zero division

I'm trying to understand Laguerre's method for root finding and I have hit one road block.

Suppose I have a polynomial $p(x) = x^4 + 1$ and an initial guess $x_0 = 0$. This results in division by zero in the above formula.

What to do in such cases? Should one backtrack and try again with different guess? Are there any guides on how to pick the next $x_k$?

"Numerical Recipes" contain a test for the zero division, in which case, the next step becomes polar(1+abx, iter), where abx is computed above and iter is the number of iteration. Does anyone know what does that polar(1+..., iter) mean? Why does it use polar coordinates?

• I guess you should start with $x_0=1$ – kingW3 Apr 24 '14 at 9:38

What is worse is that the polynomials $x^m\pm1$ show some strange behaviour under the Laguerre method, they converge only in an annulus around the unit circle. I have some pictures of this where you can observe this behavior starting with $m=5$.
I would imagine that abx stands for the absolute value of $x$. So what the polar formula is doing is selecting a circle that is to the outside of $x$ and selecting a random point on this circle. The iteration count iter modulo $2\pi$ should be random enough. This has no deeper meaning, it just serves to get away from $x$ without losing the scale of $x$.
• This is very interesting, thanks! What is your recommendation then to avoid the zero in denominator? Do you perhaps know, what's with that polar(1+..., ...) from NR? Btw., from the pictures, Laguerre looks rather scary. However, I also tried Halley and, could be that I did something not exactly right, but Laguerre performed better as the pictures might suggest. Just saying... – prepaul Apr 24 '14 at 16:05