插值多项式

First, we introduce and discuss the various methods of multivariate polynomial interpolation in the literature. Based on this study, we state multivariate Lagrange interpolation over again from algebraic geometry viewpoint:Given different interpolation nodes A1,A2 .....,An in the affine n-dimensional space Kn, and accordingly function values fi(i = 1,..., m), the question is how to find a polynomial p K[x1, x2,...,xn] satisfying the interpolation conditions:where X=(x1,X2,....,xn). Similarly with univariate problem, we have provedTheorem If the monomial ordering is given, a minimal ordering polynomial satisfying conditions (1) is uniquely exsisted.Such a polynomial can be computed by the Lagrange-Hermite interpolation algorithm introduced in chapter 2. Another statement for Lagrange interpolation problem is:Given monomials 1 ,2 ,.....,m from low degree to high one with respect to the ordering, some arbitrary values fi(i= 1,..., m), find a polynomial p, such thatIf there uniquely exists such an interpolation polynomial p{X, the interpolation problem is called properly posed.

文中首先对现有的多元多项式插值方法作了一个介绍和评述，在此基础上我们从代数几何观点重新讨论了多元Lagrange插值问题：给定n维仿射空间K~n中两两互异的点A_1，A_2，…，A_m，在结点A_i处给定函数值f_i(i=1，…，m)，构造多项式p∈K[X_1，X_2，…，X_n]，满足Lagrange插值条件：p=f_i，i=1，…，m (1)其中X=(X_1，X_2，…，X_n)，与一元情形相似地，本文证明了定理满足插值条件(1)的多项式存在，并且按"序"最低的多项式是唯一的，上述多项式可利用第二章介绍的Lagrange-Hermite插值算法求出，Lagrange插值另一种描述是：按序从低到高给定单项式ω_1，ω_2，…，ω_m，对任意给定的f_1，f_2，…，f_m，构造多项式p，满足插值条件：p=sum from i=1 to m=Ai=f_i，i=1，…，m (2)如果插值多项式p存在且唯一，则称插值问题适定。

The main purpose is to construct cubature formulae on an arbitrary spherical triangle. Firstly a method for accurately computing the definite integral of spherical polynomial functions on a spherical triangle is proposed.

本文主要研究定义在球面三角形上函数的数值积分，通过积分的插值多项式函数构造具有多项式精度的插值型求积公式，以及给出精确计算球面三角形上多项式函数方法。

Polynomial smooth techniques are applied to SVM model and replace x+ by a very accurate smooth approximation that is Hermite Interpolation polynomial,thus the undifferential model is converted into a differential model.The deduction procedure of Hermite Interpolation polynomial smoothing x+ is extended.

三次Hermite插值多项式光滑的支持向量机模型采用的是一种多项式光滑技术，用三次Hermite插值多项式代替单变量函数x+，将原来不可微的模型变为可微的模型，并且给出了三次Hermite插值多项式光滑化单变量函数x+的推导过程。

The second chapter is the main part of this paper, in which the formulation of the Riemann boundary value problem of non-normal type on the real axis, the solution method of homogeneous problem, the relation between the two kinds of different derivatives and the inhomogeneous problem will be thoroughly given. In this paper, the solution and the solvability of the Riemann boundary value problem of non-normal type on the real axis will be given. Furthermore, it is shown that the twokinds of derivatives of the function Ψ are existing and equivalent in the case ofthe solution about the original problem, therefore, we get uniformly Hermite interpolatory polynomial. The relation between the two kinds of different derivativesof the function Ψ are similar for smooth closed contours by means of the same proof.

第二章是本文的主要部分，分别给出了实轴上一类非正则型Riemann边值问题的提法、齐次问题的解法、两种导数的关系及非齐次问题的求解，本文运用杜金元教授[11]的方法获得了实轴上非正则型Riemann边值问题的封闭解及可解性条件，且在问题可解的情况下论证了函数Ψ的非切向极限导数和Peano导数存在且相等，从而获得了统一的Hermite插值多项式，同样关于封闭曲线上非正则型Riemann边值问题，采用本文论证方法证得了函数Ψ的非切向极限导数和Peano导数存在且相等，从而较好地统一了[10]、[11]中的Hermite插值多项式。

The problem of Lagrange interpolation of polynomial space in space Rs is studied,and the construction of Lagrange interpolation polynomial in space R1 and space R2 is proposed.

研究空间Rs 中多项式空间中的Lagrange插值问题。给出了R1和R2上Lagrange插值多项式的构造，同时，给出了R2上插值问题的几个例子。另外，给出了矩形网点上的Lagrange插值多项式和三角形网点上的Lagrange插值多项式。讨论了Rs空间中的Lagrange插值多项式及其余项

Romberg first use of the method is integral for integration, Then the results obtained by using the interpolation method were obtained Lagrange polynomial interpolation polynomial interpolation and Newton, re-use of least squares fitting of thinking obtained polynomial, the last of these different types of polynomial, identify their respective strengths and weaknesses.

首先运用Romberg积分方法对给出定积分进行积分，然后对得到的结果用插值方法，分别求出Lagrange插值多项式和Newton插值多项式，再运用最小二乘法的思想求出拟合多项式，最后对这些不同类型多项式进行比较，找出它们各自的优劣。

It first uses the integrating method of Romberg , which is an improved trapezoidal integration, to solve the given definite integral,then we create Lagrange's interpolation polynomial and Newton's interpolation polynomial.

首先运用Romberg积分方法对给出定积分进行积分，然后对得到的结果用插值方法，分别求出Lagrange插值多项式和Newton插值多项式，再运用最小二乘法的思想求出拟合多项式，最后对这些不同类型多项式进行比较，找出它们各自的优劣。

For the interpolation problems in which the nodes are on one or more algebraic manifolds, we pointed out the interpolation bases of the lowest degree w.r.t.a graded term order and gave a new algorithm. We also gave a estimation for the degree of the interpolation polynomial.

我们针对插值节点位于一个或几个代数流形上的插值问题，从理论上指出了插值问题在某一分次序下最低次的插值基并给出了新的算法，同时也对插值多项式的次数作了估计。

Jacobian Davis iterative method of solving equations AX = B, least squares polynomial fitting, portfolio Simpson formula for integration, with a triangular decomposition method of solving equations AX = B.

拉格朗日插值多项式拟合，牛顿插值多项式，欧拉方程解偏微分方程，使用极限微分求解导数，微分方程组的N=4龙格库塔解法，雅可比爹迭代法解方程AX=B，最小二乘多项式拟合，组合辛普生公式求解积分，用三角分解法解方程

First, we introduce and discuss the various methods of multivariate polynomial interpolation in the literature. Based on this study, we state multivariate Lagrange interpolation over again from algebraic geometry viewpoint:Given different interpolation nodes A1,A2 .....,An in the affine n-dimensional space Kn, and accordingly function values fi(i = 1,..., m), the question is how to find a polynomial p K[x1, x2,...,xn] satisfying the interpolation conditions:where X=(x1,X2,....,xn). Similarly with univariate problem, we have provedTheorem If the monomial ordering is given, a minimal ordering polynomial satisfying conditions (1) is uniquely exsisted.Such a polynomial can be computed by the Lagrange-Hermite interpolation algorithm introduced in chapter 2. Another statement for Lagrange interpolation problem is:Given monomials 1 ,2 ,.....,m from low degree to high one with respect to the ordering, some arbitrary values fi(i= 1,..., m), find a polynomial p, such thatIf there uniquely exists such an interpolation polynomial p{X, the interpolation problem is called properly posed.

文中首先对现有的多元多项式插值方法作了一个介绍和评述，在此基础上我们从代数几何观点重新讨论了多元Lagrange插值问题：给定n维仿射空间K~n中两两互异的点A_1，A_2，…，A_m，在结点A_i处给定函数值f_i(i=1，…，m)，构造多项式p∈K[X_1，X_2，…，X_n]，满足Lagrange插值条件：p=f_i，i=1，…，m (1)其中X=(X_1，X_2，…，X_n)，与一元情形相似地，本文证明了定理满足插值条件(1)的多项式存在，并且按&序&最低的多项式是唯一的，上述多项式可利用第二章介绍的Lagrange-Hermite插值算法求出，Lagrange插值另一种描述是：按序从低到高给定单项式ω_1，ω_2，…，ω_m，对任意给定的f_1，f_2，…，f_m，构造多项式p，满足插值条件：p=sum from i=1 to m=Ai=f_i，i=1，…，m (2)如果插值多项式p存在且唯一，则称插值问题适定。

On the other hand, the more important thing is because the urban housing is a kind of heterogeneity products.

另一方面，更重要的是由于城市住房是一种异质性产品。

Climate histogram is the fall that collects place measure calm value, cent serves as cross axle for a few equal interval, the area that the frequency that the value appears according to place is accumulated and becomes will be determined inside each interval, discharge the graph that rise with post, also be called histogram.

气候直方图是将所收集的降水量测定值，分为几个相等的区间作为横轴，并将各区间内所测定值依所出现的次数累积而成的面积，用柱子排起来的图形，也叫做柱状图。