Local optimization of the summation of divergent power series

A method of optimizing a sequence of economized rational approximants (ERAs ) to produce a sequence of approximants with enhanced convergence propertie s is described. It is shown that such a technique improves upon the error o f the Pade approximants at a chosen value of the independent variable, and in some cases leads to dramatic improvement, even in cases where Pade appro ximants behave erratically. The procedure is tested on six known functions, with improved convergence and accuracy in each case. The procedure is then applied to the problem of evaluating a perturbation series of an atomic sy stem, diamagnetic hydrogen, with significant improvement in both convergenc e and accuracy as well. (C) 2000 American Institute of Physics. [S0022-2488 (00)02008-9].