A new approach to acceleration of convergence of a sequence of vectors

The vector epsilon algorithm (VEA) has many advantages as a method for accelerating the convergence of a sequence of vectors. A vector Padé approximantP(z)/Q(z) of type [n/2k] can be associated with each entry of the vector epsilon table. In the scalar case, it reduces to the Padé approximantp(z)/q(z) of type [n–k/k]. It is thought that the disadvantages of VEA are (indirectly) attributable to the positivity property ofQ(x), x , recalling that in the scalar case,Q(z)q(z)². In this paper, a specification of a polynomial (z) of degreek is given, such that (z)²Q(z). The coefficients of (z) specify an accelerator for a sequence of vectors which should avoid many of the numerical difficulties of VEA.This work was supported in part by the EC-HCM project ROLLS under contract CHRX-CT93-0416.