On the Practical Realization of ε-algorithm for Calculation of Limits of Numerical Sequences and N-point Padé Approximations

摘要

It is shown that the Wynn's missing identity of Frobenius gives a good practical numerical criterion for the choice of the optimal Padé approximant. The method is illustrated by calculation of divergent series, numerical interpolation and extrapolation, and can be used as predictor method for numerical solution of ordinary differential equations. The performed numerical experiments reveal that practical realization of the Padé approximants is not problem of the theory of complex functions but belongs to mathematical statistics. A significant correlation between the empirical error extracted by the cross rule by Wynn and the real error reveals that the long sought criterion has already been found. Padé approximants are fast convergent but the noise of rounding requires a compromise. The optimal approximant is determined by modulus minimization of empirical error in the Padé table (N - C)~(-1) + (S - C)~(-1) =ε~(-1)_(emp) = (W - C)~(-1) + (E - C)~(-1).