Parallel laguerre iterations: The complex case

Summary

Consider a polynomialP(z) of degreeN whose zeros are known to lie insideN closed disks, each disk containing one and only one root. In this paper we show that if the given disks are “sufficiently well separated”, then the first derivative ofP(z) never vanishes inside the initial inclusion regions. The formulation of the square-root iteration in terms of circular regions is then possible and leads to an iterative scheme with degree four convergence. The corresponding algorithm makes use of circular arithmetic and in particular of the definition of square root of a disk. A criterion for the selection of the appropriate square-root set is also given. The procedure can be used to simultaneously refine all (complex or real) roots ofP(z) together with their error bounds.