Editing Laguerre's method (section)

==Definition==
The algorithm of the Laguerre method to find one root of a polynomial {{math|''p''(''x'')}} of degree {{mvar|n}} is:
* Choose an initial guess {{math|''x''<sub>0</sub>}}
* For {{math|''k'' {{=}} 0, 1, 2, ...}}
** If <math>p(x_k)</math> is very small, exit the loop
** Calculate <math>G = \frac{p'(x_k)}{p(x_k)}</math>
** Calculate <math>H = G^2 - \frac{p''(x_k)}{p(x_k)}</math>
** Calculate <math>a = \frac{n}{G \plusmn \sqrt{(n-1)(nH - G^2)}}</math>, where the sign is chosen to give the denominator with the larger absolute value, to avoid [[catastrophic cancellation]].
** Set <math>x_{k+1} = x_k - a</math>
* Repeat until ''a'' is small enough or if the maximum number of iterations has been reached.

If a root has been found, the corresponding linear factor can be removed from ''p''. This deflation step reduces the degree of the polynomial by one, so that eventually, approximations for all roots of ''p'' can be found. Note however that deflation can lead to approximate factors that differ significantly from the corresponding exact factors. This error is least if the roots are found in the order of increasing magnitude.