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Multiquadric and its shape parameter: A numerical investigation of error estimate, condition number, and round-off error by arbitrary precision computation

机译：多二次方及其形状参数：通过任意精度计算对误差估计，条件数和舍入误差进行数值研究

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摘要

Hardy's multiquadric and its related interpolators have been found to be highly efficient for interpolating continuous, multivariate functions, as well as for the solution of partial differential equations. Particularly, the interpolation error can be dramatically reduced by varying the shape parameter to make the interpolator optimally flat. This improvement of accuracy is accomplished without reducing the fill distance of collocation points, that is, without the increase of computational cost. There exist a number of mathematical theories investigating the multiquadric family of radial basis functions. These theories are often not fully tested due to the computation difficulty associated with the ill-conditioning of the interpolation matrix. This paper overcomes this difficulty by utilizing arbitrary precision arithmetic in the computation. The issues investigated include conditional positive definiteness, error estimate, optimal shape parameter, traditional and effective condition numbers, round-off error, derivatives of interpolator, and the edge effect of interpolation.
机译：已经发现，Hardy的多二次方及其相关的内插器对于内插连续的多元函数以及偏微分方程的求解非常有效。特别地，通过改变形状参数以使内插器最佳地平坦，可以显着减小内插误差。在不减小并置点的填充距离的情况下，即在不增加计算成本的情况下，实现了精度的提高。存在许多研究径向基函数的多二元族的数学理论。由于与插值矩阵的不良条件相关的计算难度，这些理论通常未得到充分测试。本文通过在计算中使用任意精度算法克服了这一困难。研究的问题包括条件正定性，误差估计，最佳形状参数，传统有效条件数，舍入误差，插值器的导数以及插值的边缘效应。

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