中文摘要：

数据插值方法的精度和效率是人们在插值方法的研究中关注的主要问题。在理论分析的基础上给出了自仿射分形插值函数的表达形式和垂直比例因子的显式表达形式。分别利用分形插值方法和拉格朗日插值方法对给定数据进行了插值拟合处理,结果表明分形插值方法对数据的拟合精度整体上高于拉格朗日插值方法,而且不会出现数据拟合中常见的“龙格现象”。通过对拟合曲线的分析,发现由于垂直比例因子采用了局部显式表达形式,从而将局部信息与全局信息有机地结合了起来,既突出了局部信息,避免了“龙格现象”,又保持了数据总体的变化趋势。拉格朗日插值在插值区间的中部精度很高,而靠近区间两端则会出现严重的“龙格现象”。

英文摘要：

The main issues paid attention to in the research into interpolation method are the precision and the efficiency. The expressions of the affine fractal interpolation function and the locally explicit presentation of the vertical scaling factors are presented on the basis of fractal geometry. Data - oriented interpolation fitting treatment has been made by using fractal interpolation and Lagrange interpolation method, respectively. The numerical results show that the fitting accuracy by fractal interpolation method is higher than that obtained by Lagrange method, and can not appear ＂Runge- phenomenon＂ which often appears in data -fitting by high -order polynomial interpolation. By analyzing the fitting curves, and by using the locally explicit vertical scaling factors, local information and global information is integrated appropriately, at the same time, this method not only makes the local information obvious and avoids ＂Runge - phenomenon＂ but preserves the overall characteristics of the data investigated. Lagrange interpolation method can yields result with high accuracy in the middle of the interval while the result obtained on the two ends of the interval will be deteriorated for the reason of ＂Runge - phenomenon＂.