Robust formula for $N$-point Padé approximant calculation based on Wynn identity
T. M. Mishonov, A. M. Varonov
备注：
12 pages, 6 figures, 2 appendix sections; Title and abstract precised, section Introduction restated, figure captions shortened and a new appendix section added, where the Cauchy-Jacobi problem is solved for the case of the first $K$ derivatives at $N$ nodal points
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 center, the other letters correspond to the four directions of the compass) gives the long sought criterion, the minimal $|η|$, 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 work of the criterion for the calculation of optimal Padé approximant is illustrated by the frequently seen in the theoretical physics problems of calculation of series summation and multipoint Padé approximation used as a predictor for solution of differential equations motivated by the magneto-hydrodynamic problem of heating of solar corona by Alvén waves. 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 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.
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