本项目拟建立差商形式的Faa di Bruno公式及其推广和扩展形式，并应用于数值逼近问题，主要围绕下列四个具体目标展开研究：（1）通过差商的链式法则，以Jablonski矩阵恒等式的形式建立求复合函数高阶差商的Faa di Bruno公式的一般形式，并导出整数次迭代函数的高阶差商的显式公式以及各类推广和扩展；（2）建立差商意义下的反函数和隐函数的高阶差商的显式公式；（3）建立多元复合函数的差商形式Faa di Bruno公式；（4）利用差商形式Faa di Bruno公式以及它的拓广形式为解决数值计算中饱和逼近下曲线、曲面插值和数值微分的误差估计问题提供新的有效的方法。本项目的研究把代数组合与数值计算两个不同的方向进行交叉，为数值计算的发展和创新提供理论基础和新的方法。

This project aims to establish a divided difference version of Faa di Bruno's formula, its generalizations and extended forms, then to apply them to numerical approximation problems. Our research is divided into the following four parts: (1) We will employ the chain rule of divided differences to establish a general form of Faa di Bruno's formula for divided differences of a composite function in the form of Jablonski's matrix identity, which leads to the explicit expression for the divided differences of integer iteration functions and various generalizations and extensions; (2) We will establish an explicit formula for the divided differences of inverse functions and implicit functions; (3) We will establish a divided difference version of Faa di Bruno's formula for multivariate functions; (4) Making use of the divided difference version of Faa di Bruno's formula and its extended forms we will provide an effective method to solve the problem of the error estimations of curve and surface interpolation and their numerical differentiations under saturated approximation. This project is an interdisciplinary research of algebraic combinatorics and numerical analysis, which will provide a theoretical basis and new methods for the development and innovation of numerical analysis.