Robust formula for N-point Padé approximant calculation based on Wynn identity

• Published in: Applied numerical mathematics 2020-11, Vol.157, p.291-306
• Main Authors: Mishonov, Todor M., Varonov, Albert M.
• Format: Article
• Language: English
• Subjects: Cauchy-Jacobi problem; Error criterion; Multipoint Padé approximant; N-point Padé approximant; Numerical extrapolation; Padé approximants; Wynn identity
• ISSN: 0168-9274; 1873-5460
• DOI: 10.1016/j.apnum.2020.06.007

The performed numerical analysis reveals that Wynn's identity for the compass 1/(N−C)+1/(S−C)=1/(W−C)+1/(E−C)=1/η (here C stands for centre, the other letters correspond to the four directions of the compass) gives the long sought criterion, the minimal |η| or ▪-criterion, for the choice of the optimal Padé approximant. The work of this method is illustrated by calculation of multipoint Padé approximation by a new formula for calculation of this best rational approximation. The calculation of the optimal Padé approximant by this criterion is demonstrated in calculation of series summation – frequently encountered problem in theoretical physics. This study originates from a magneto-hydrodynamic problem of heating of solar corona by Alfvén waves, where the present method is used for a predictor in solution of differential equations. In such a way, an efficient and valuable control mechanism for N-point Padé approximant calculation is proposed. We believe that the suggested method and criterion can be useful for many applied problems in numerous areas not only in physics but in any scientific application where differential equations are solved. The obtained new solution of the Cauchy-Jacobi problem is illustrated by a Fortran program. The algorithm is generalized for the case of the first K-derivatives at N-nodal points. The numerical analysis of probability distribution function of the errors will review, which formula for extrapolation of functions deserves to be implemented in the applied mathematics software like Mathematica and Maple.