Runge现象的研究

Research on Runge Phenomenon

摘要: 首先，本文解释了高次多项式插值时产生的Runge现象，并通过计算系统误差证明了插值多项式发散，从而导致了该现象。其次，以Runge函数、反三角函数和分式函数为例，运用等间距牛顿插值求出函数的插值多项式，进而求出其插值余项函数表达式，然后计算每相邻两个节点的中点处的误差，判断上述三个函数产生了Runge现象。第三，介绍并验证了采用切比雪夫节点、分段线性插值和三次样条插值三个常用的算法，能够避免上述函数产生Runge现象。最后，创新性地提出逼近性能指标，并基于最优多项式构造系数与阶次双确定法，该算法在避免Runge现象的同时具有优异的函数逼近效果，且运行速度有极大提升。

Abstract: Firstly, this paper explains the Runge phenomenon generated by high-order polynomial inter- polation, and proves that the interpolation polynomial divergence is obtained by calculating the systematic error. Secondly, taking the Runge function, inverse trigonometric function and fractional function as examples, the interpolation polynomial of the function is obtained by using the equally spaced Newton interpolation, and then the interpolation residual function expression is obtained, and then the midpoint of adjacent two nodes is calculated. The error at the location determines that the above three functions have generated the Runge phenomenon. Thirdly, the three algorithms of Chebyshev node, piecewise linear interpolation and cubic spline interpolation are introduced and verified, which can avoid the Runge phenomenon. Finally, the approximation performance index is proposed, and based on the optimal polynomial construction coefficient and order double determination method, the algorithm has excellent function approximation effect while avoiding the Runge phenomenon.