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?
Each iterative method that can be described by a fraction of some fashion, Newton, Halley, Laguerre, have points where the denominator is zero. There you can not start. Since these are only a finite number of points, it is rather improbable that you encounter these points during an iteration from a random initial point.
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
abxstands 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 countitermodulo $2\pi$ should be random enough. This has no deeper meaning, it just serves to get away from $x$ without losing the scale of $x$.