A root-finding algorithm is a numerical algorithm for approximating a root of an equation 
. An iterative method constructs
successive approximations using evaluations of 
and possibly its derivatives.
A stationary method uses the same update rule at every step. A method without memory
computes its next approximation from the current approximation and the evaluations
made during that step, without retaining earlier evaluation data (Kung and Traub
1974).
For a simple root 
, a local order bound 
means that the update 
satisfies
 |
for some 
and every sufficiently close 
. This local condition concerns all sufficiently close
starting points, rather than only the points on a particular iteration. The Kung-Traub
conjecture relates this rate of convergence
to the number of scalar evaluations per step.
The Ehrlich-Aberth method approximates all roots of a polynomial simultaneously (Ehrlich 1967, Aberth 1973).
