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A Simple Regularization of the Polynomial Interpolation for the Resolution of the Runge Phenomenon

机译：多项式插值的简单正则化以解决Runge现象

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A polynomial interpolation based on a uniform grid yields the well-known Runge phenomenon, where maximum error is unbounded for functions with complex roots in the Runge zone. In this paper, we investigate the Runge phenomenon with the finite precision operation. We first show that the maximum error is bounded because of round-off errors inherent to the finite precision operation. Then we propose a simple truncation method based on the truncated singular value decomposition. The method consists of two stages: In the first stage, a new interpolating matrix is constructed using the assumption that the function is analytic. The new interpolating matrix is preconditioned using the statistical filter method. In the second stage, a truncation procedure is applied such that singular values of the new interpolating matrix are truncated if they are equal to or lower than a certain tolerance level. We generalize the method, by analyzing the singular vectors of both the original and new interpolation matrices based on the assumption in the first stage. We show that the structure of the singular vectors makes the first stage essential for an accurate reconstruction of the original function. Numerical examples show that exponential decay of the error can be achieved if an appropriate truncation is chosen.
机译：基于均匀网格的多项式插值会产生众所周知的Runge现象，对于Runge区域中具有复杂根的函数，最大误差是无限的。在本文中，我们以有限精度操作研究了Runge现象。我们首先证明最大误差是有限的，这是因为有限精度运算固有的舍入误差。然后我们提出了一种基于截断奇异值分解的简单截断方法。该方法包括两个阶段：在第一阶段，使用函数为解析的假设构造新的插值矩阵。使用统计滤波器方法对新的插值矩阵进行预处理。在第二阶段，应用截断过程，以便如果新插值矩阵的奇异值等于或小于某个公差级别，则将其截断。我们基于第一阶段的假设，通过分析原始和新插值矩阵的奇异矢量来推广该方法。我们表明奇异矢量的结构使第一阶段对于原始功能的准确重建至关重要。数值示例表明，如果选择适当的截断，可以实现误差的指数衰减。

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