The interpolation errors of the higher order bivariate Lagrange polynomial interpolation based on the rectangular, right and equilateral triangular interpolations are measured by using the maximum and root-mean-square (RMS) errors. The error distributions of above three kinds of interpolations are analyzed to find the regions having the smallest interpolation error. Both analytical and numerical results show that the right triangular interpolation is the most efficient interpolation method. Although both the maximum and RMS errors of the right triangular interpolation are larger than that of the rectangular interpolation, the number of data points used in the triangular interpolation is up to 50% less than that used in the rectangular one. On the other hand, the equilateral triangular interpolation using the regions inside a big triangle as interpolation area is proved to be the most accurate interpolation method. The interpolation errors of the equilateral triangular are almost half of that of the rectangular interpolation. In addition, the right triangular, equilateral triangular and rectangular interpolations for the third and fourth orders are applied to accelerate the calculation of doubly periodic Green’s function (PGF).

中文翻译：

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电磁学中高阶二元拉格朗日和三角插值的误差分析

通过使用最大和均方根（RMS）误差来测量基于矩形，右和等边三角形插值的高阶二元Lagrange多项式插值的插值误差。分析以上三种内插的误差分布，以找到具有最小内插误差的区域。分析和数值结果均表明，直角三角形插值是最有效的插值方法。尽管直角三角形插值的最大误差和均方根误差均大于矩形插值，但是在三角形插值中使用的数据点数比矩形插值中的数据点数最多少50％。另一方面，用大三角形内部的区域作为插值区域的等边三角形插值被证明是最准确的插值方法。等边三角形的插值误差几乎是矩形插值的一半。此外，三阶和四阶的直角三角形，等边三角形和矩形插值用于加速双周期格林函数（PGF）的计算。