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Robust formula for N-point Padé approximant calculation based on Wynn identity
Applied Numerical Mathematics ( IF 2.2 ) Pub Date : 2020-11-01 , DOI: 10.1016/j.apnum.2020.06.007
Todor M. Mishonov , Albert M. Varonov

The performed numerical analysis reveals that Wynn's identity for the compass $1/(N-C)+1/(S-C)=1/(W-C)+1/(E-C)=1/\eta$ (here C stands for center, the other letters correspond to the four directions of the compass) gives the long sought criterion, the minimal $|\eta|$, for the choice of the optimal Pad\'e approximant. The work of this method is illustrated by calculation of multipoint Pad\'e approximation by a new formula for calculation of this best rational approximation. The work of the criterion for the calculation of optimal Pad\'e approximant is illustrated by the frequently seen in the theoretical physics problems of calculation of series summation and multipoint Pad\'e approximation used as a predictor for solution of differential equations motivated by the magneto-hydrodynamic problem of heating of solar corona by Alv\'en waves. In such a way, an efficient and valuable control mechanism for $N$-point Pad\'e 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.

中文翻译:

基于 Wynn 恒等式的 N 点 Padé 近似计算的鲁棒公式

进行的数值分析表明,永利对罗盘的身份 $1/(NC)+1/(SC)=1/(WC)+1/(EC)=1/\eta$(这里 C 代表中心,其他字母对应于罗盘的四个方向)给出了长期寻求的标准,最小的 $|\eta|$,用于选择最佳的 Pad\'e 近似值。该方法的工作通过使用用于计算此最佳有理近似的新公式计算多点 Pad\'e 近似来说明。计算最优 Pad\'e 逼近的判据的工作通过在计算级数求和和多点 Pad\' 的理论物理问题中经常看到的例子来说明 e 近似值用作微分方程解的预测因子,该微分方程由 Alv\'en 波加热太阳日冕的磁流体动力学问题驱动。以此方式,提出了一种用于$N$-point Pad\'e 近似计算的有效且有价值的控制机制。我们相信,所建议的方法和标准对许多领域的许多应用问题都有用,不仅在物理学中,而且在求解微分方程的任何科学应用中。Cauchy-Jacobi 问题的解决方案由 Fortran 程序说明。该算法适用于在 $N$ 节点处的第一个 $K$ 导数的情况。我们相信,所建议的方法和标准对许多领域的许多应用问题都有用,不仅在物理学中,而且在求解微分方程的任何科学应用中。Cauchy-Jacobi 问题的解决方案由 Fortran 程序说明。该算法适用于在 $N$ 节点处的第一个 $K$ 导数的情况。我们相信,所建议的方法和标准对许多领域的许多应用问题都有用,不仅在物理学中,而且在求解微分方程的任何科学应用中。Cauchy-Jacobi 问题的解决方案由 Fortran 程序说明。该算法适用于在 $N$ 节点处的第一个 $K$ 导数的情况。
更新日期:2020-11-01
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