非矩形网格上的多元连分式逼近及其应用研究

The study on multivariate continued fraction interpolation over non-rectangular grids and its application in numerical integration is an important topic of basic research in numerical approximation and computational geometry. Recently, the subject on multivariate continued fractions is mostly restricted to the rectangular grids, while few work is done on the topic of multivariate continued fraction interpolation with its application. Based on the research work in recent years, the objective of the topic is firstly to investigate the interpolating polynomial over general triples, to work out the explicit coefficients in term of B-net for the construction of numerical integration formula over triangular domain, to construct bivariate continued fraction interpolating functions over general triples, and to study bivariate osculatory interpolation formula with one-order partial derivatives at all corner points of each triple. Secondly, the objective of the topic is to show which of polynomial interpolation or continued fraction interpolation plays a better role in approximation by numerical examples, and to discuss the best distribution of the general triples. Thirdly, the objective of the topic is to establish trivaraite branched continued fraction interpolation over non-rectangular grids and to work out the error estimation. Finally, the objective of the topic is to investigate the application of the multivariate continued fraction interpolation in numerical integration, and to compare it with bivariate splines. Hence, the purpose is to construct bivariate polynomial and continued fraction interpolating functions over general triples, to complete trivariate branched continued fraction interpolation over non-rectangular grids, to establish the corresponding numerical integration formula, and to disclose the inherent relation between multivariate continued fractions and bivariate splines.

非矩形网格上多元连分式插值及其在数值积分中的应用研究是数值逼近与计算几何中的一个重要基础研究课题. 目前，多元连分式的研究主要局限于矩形网格，而非矩形网格上的多元连分式插值及其应用的研究成果极少. 基于申请者近年来的研究工作，本课题首先研究一般三点组上插值多项式，计算B网系数，由此建立三角域上的数值积分公式，构造一般三点组上的二元连分式插值函数，研究其于每个三点组角点处的切触插值性质. 接着，通过数值算例比较三点组上的多项式插值与连分式插值逼近效果，进一步分析三点组在某种意义下的最优分布. 然后，构造三元分叉连分式插值函数，算出插值余项. 最后，研究多元连分式插值函数在数值积分中的应用，并比较多元连分式与二元样条函数的逼近效果. 本课题的目标是构造一般三点组上的二元多项式与连分式插值函数，非矩形网格上的三元分叉连分式插值函数，建立相应的数值积分公式，揭示多元连分式与二元样条之间的内在关系.

本项目对非矩形网格上二元连分式与二元多项式插值展开研究，为散乱点插值内容的发展提供理论支持与有效算法. 研究内容包括如下，一方面，首先利用两种新的非张量积型二元偏逆差商算法，分别构造基于二元连分式具有奇数与偶数个插值节点的散乱点插值格式，接着建立被插函数与二元连分式之间的恒等式，然后利用连分式的三项递推关系式得到特征定理，以此揭示插值连分式渐近式分子分母次数，最后给出若干算例，研究表明所提出算法可行有效，且所构造的二元插值连分式渐近式分子分母次数小于张量积型二元Thiele型连分式渐近式分子分母次数，这是由于节省了冗余插值节点信息. 另一方面，首先基于非张量积型二元差商递推算法，构造非矩形网格上的二元多项式插值格式，这可以转化为散乱点插值，研究表明插值公式随插值节点个数为奇数与偶数而不同，然后算出插值余项，并分别建立基于奇数与偶数个插值节点的高阶非张量积型差商与高阶偏导数之间关系式，最后通过若干数值算例说明算法的有效性，且插值多项式随节点顺序的改变而改变，虽然插值节点集合不变，算例还表明，由于减少使用冗余插值节点，所构造的插值多项式次数低于二元张量积型Lagrange插值多项式次数. 该项目的研究将不断丰富和完善数值逼近与计算几何理论与应用内容，为多元数值逼近的发展注入新的活力.