【作者】 盛敏；

【导师】 唐烁；

【作者基本信息】 合肥工业大学 ， 计算数学， 2006， 硕士

【摘要】 本文针对基于连分式的多元混合切触有理插值问题展开了深入的讨论,其内容主要包括基于连分式的一阶混合切触有理插值、基于连分式的高阶混合切触有理插值以及基于连分式的高阶复合型混合切触有理插值。 给定型值点及相应的一阶偏导数值,本文将扩展的Newton多项式插值和Salzer型切触连分式插值相融合,构造了一阶混合型二元切触连分式插值公式,即Salzer-Newton型混合切触有理插值。进一步地,采用递归和循环格式给出了该插值格式的递推算法,该算法易于编程实现。数值例子显示本文构造的二元一阶混合切触有理插值具有较好的逼近效果。 高阶混合切触有理插值是一阶混合切触有理插值的推广,但需要充分考虑多元函数在x与y方向的偏导数以及二阶混合偏导数,其算法复杂度也必然大大增加。在一阶混合切触有理插值构造的基础上,本文构造了基于连分式的高阶多元切触有理插值公式,即Newton-Salzer型和Salzer-Newton型混合切触有理插值,并给出相应的递推算法。同时通过特征定理定性地给出了Newton-Salzer型有理分式函数中分子、分母的次数估计;并通过固定一个变量,将另一个变量按重节点个数离散展开而得到了误差估计式。 考虑到有理插值的适定性问题,本文采用将正方形网格分成两个三角网格方法,同时构造两个调配函数,使其对称作用于这两个三角网格。在此基础上,将原函数分别与这两个调配函数作适当组合以消除数据的病态特征同时又满足切触插值条件,构造了复合形式的混合切触有理插值公式并给出了误差估计。更多还原

【Abstract】 The summaries of this paper are the researches on the multivariate blending osculatory rational interpolants based on continued fractions, which include bivariate blending osculatory rational interpolants of one order, bivariate blending osculatory rational interpolants of two orders and composite scheme of multivariate blending osculatory rational interpolants of two orders.By the blending of expansive Newton polynomial and Salzer osculatory continued fraction, we construct the blending osculatory rational interpaltion polynomial of one order (SNBORIs). Then, a recursive algorithm is presented by the recursive and circulating format. Numerical examples illustrate that the blending osculatory rational interpolation of one order have good approximate effects which interpolate a series of given data.We get blending osculatory rational interpaltion of two orders by extending the blending osculatory rational interpaltion of one order. We need to consider about partial derivative with respect to x, y and blending partial derivative in (x_i,y_j) which lead to complex algorithm. In this paper the multivariate blending osculatory rational interpolation formulas (NSMORIs and SNMORIs) of two orders are constructed and recursive algorithms are deduced. Furthermore, characterization theorem and error estimate are presented.Considering about the existence of the rational interpolation, a square grid can be divided into two triangular subgrids and two allocated functions are constructed which lead to composite scheme of multivariate blending osculatory rational interpolation of two orders.更多还原