Talk:Laguerre's method

Particular case where the algorithm fails.

If your polynomial is with coefficient of x^2 and x equals to 0, and if you start the iterations with x=0, then G and H are both equal to 0 and then the correction, "a", equals to infinity.— Preceding unsigned comment added by 74.57.114.58 (talk • contribs) 02:57, 29 May 2014

That is why in a real implementation the first step is to test that the leading coefficient is different from zero and to deflate away all trivial roots x=0.--LutzL (talk) 10:45, 29 May 2014 (UTC)

The algorithm fails even for an initial polynomial such as x^3 Q(x) -1, where Q(x) is some polynomial , that is, if the initial polynomial is without x^2 coefficient and without x coefficient (but with a non null coefficient for some higher power of x), if we start with x = 0 at step 0. An implementation should check to see if there is a non null coeff and for x and for x^2, and if it is NOT the case, should start the iteration with another value than x= 0. — Preceding unsigned comment added by 74.57.192.157 (talk) 12:07, 28 January 2015 (UTC)

Connection to Newton's method

If we make the even stronger assumption that $x$ is close enough to a root $x_{1}$ for the terms in $G$ involving the other roots to be neglected, then we get

$G={\frac {d}{dx}}\ln |p(x)|={\frac {1}{x-x_{1}}},$

and this gives

$x-x_{1}={\frac {1}{G}}={\frac {p(x)}{p'(x)}},$

and so

$x_{1}=x-{\frac {p(x)}{p'(x)}},$

without even having to consider $H$, which gives us Newton's method.

Is it appropriate to mention this connection in this article and/or in the Newton's method article, or is this too much like original research? PMLawrence (talk) 07:35, 29 June 2014 (UTC)

In the absence of feedback, I have now provided a brief mention that this connection exists and the assumptions it needs, but without providing the derivation as I can't find outside sources for that yet. PMLawrence (talk) 14:18, 30 June 2014 (UTC)