POLYNOMIAL ACCELERATION OF THE PICARD-LINDELOF ITERATION

On a finite time interval the convergence of the iteration is characte rized by the order and the type of the iteration operator. In this pap er we investigate the effect that polynomial acceleration has on the r ate of convergence of the Picard-Lindelof iteration. We show that if t he decomposition matrices commute, then the type of the polynomially a ccelerated iteration can be derived from the geometry of the spectra o f the decomposition matrices. In the best case the type can be dropped to one fourth of the original by polynomial: acceleration, whereas in the worst case it stays the same. The order is not affected by polyno mial acceleration. The non-commutative case is illustrated by an examp le.