Numerical Methods for Roots of Polynomials - Part II

15.9 Accelerated Refinement of Splitting. An Algorithm and the Error Bound

It is hard to control the precision of the computations in Algorithm 15.7.1 and their Boolean complexity for an arbitrary input of this algorithm (see Schönhage (1985) and Emiris et al. (1997)). To our advantage, however, we have a special input class, and we will achieve the desired control based on an alternative to Algorithm 15.7.1, where we will exploit the isolation properties (15.9)–(15.11) for the zero sets of the factors F and G.

In this and the next two sections we will recall (and will slightly simplify by removing some contour integration techniques) such alternative algorithm from Schönhage (1982a), §11, which evaluates the polynomials and satisfying (15.40) ...