查询词典 interpolation quadrature formula

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。

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。

Lagrange interpolation focus on the application of the formula .The comparison between the method of Transform and that of Lagrange's interpolation formula in square matrix form is made. Also the form of Lagrange's interpolation formula is generalized to include the double root case.The application of the generalized lagrange's interpolation formula in square matrix to solving state equations is found to be effective.

重点介绍了Lagrange插值公式在状态方程求解中的应用，用对角矩阵及相似矩阵的性质证明了，当状态转移矩阵中方阵的特征根相异时，拉格朗日插值公式与Laplace变换法对表达具有一致性，并用极限方法对方阵含二重根时拉格朗日插值公式的应用进行了推导。

The objective of this study is aimed at the implementation of parallelized numerical quadrature routines. The algorithm we choose for numerical quadrature routines is the so called "Adaptive Romberg Tree integration scheme"(abbr."ARTint") proposed by Lin et. al.[1997]. This numerical quadrature scheme can arrange the function evaluation points adaptively in accordance with characteristic of the trend of variation in integrand function values. When the accuracy of the numerical quadrature value is requested to be increased, a newer and larger set of function evaluation points must be incurred. The main advantage of ARTint method is the inclusion of all function evaluation points used in previous stages. It is this feature of ARTint method that makes it a quadratue scheme with prominent computation efficiency.

本研究在探讨数值积分程式之平行化，所使用的演算法为林聪悟与林佳慧[1997]一书所提出的「Romberg树自动布点数值积分」架构，此一数值积分方法能以适应被积函数值变化情形方式自动安排积分点来进行数值求积计算，在提高阶数以获得更高精度积分值的过程中，此一方法只须再计算新增加积分点的函数值，搭配先前已计算过的函数值的重覆运用即可获得更精确的数值积分估计值，如此将能减少升阶过程中函数求值计算的次数，有效率地完成单重积分的数值计算。

According to a given function, namely, composite trapezoidal formula using composite Simpson formula Romberg quadrature formula and the variable step-size integration quadrature formula results, and compare the accuracy of points.

详细说明：根据给定的函数，分别用复合梯形公式、复合辛普森公式、龙贝格求积公式和变步长的求积公式计算积分结果，并且比较积分的精确度。

Based on the relationship between the numerical aperture of the objective lens and scale pitch, this study chooses a suitable Plano-convex lenses available on the market, so as to approximate the diameter of the focusing spots to scale pitch. Also, by using the optical path of an optical encoder similar to the nanoscale, the changes in the surface profile of the scale can be measured directly, so as to obtain the two quadrature sinusoidal signals. By counting the zero crossing of these two quadrature sinusoidal signals and calculating the interpolation value of these two quadrature sinusoidal signals, the moving direction and moving displacement value of the scale can be obtained.

研究中再以搭配物镜的数值孔径与尺条间距的关系，选择ㄧ适当的市售平凸透镜，使聚焦光点的直径与尺条的间距相近，并使用相同於奈米级长行程双雷射聚焦读头装置的光路设计，如此将可藉由直接量测尺条的表面轮廓变化，以获得一组正交弦波讯号，继而利用讯号四分割与细分割处理，计算出尺条之移动方向与移动量，研制出一实用性长行程双雷射聚焦读头光学尺。

By adjusting the positions and angles of the two DVD optical pick-up heads, the relative distance of the two laser spots can be equivalent to one quarter of the grating pitch. When the grating moves, the surface profile of the grating can be measure directly, in order to obtain the two quadrature sinusoidal signals. Then by counting the zero crossing of these two quadrature sinusoidal signals and calculating the interpolation value of these two quadrature sinusoidal signals, the moving direction and moving displacement value of the grating can be obtained.

采用两个光学读取头，且两个光学读取头共用同一分光镜与同一聚焦物镜，於搭配光路的设计，可使两个光学读取头之光束打入同一聚焦物镜，藉由调整两个DVD光学读取头的位置与角度，使聚焦物镜聚焦的两相近聚焦光点的距离相等於四分之ㄧ光栅尺间距，当光栅尺移动时，可直接量测光栅尺的表面轮廓变化，以获得一组正交的弦波讯号，继而利用讯号四分割与细分割处理，计算出光栅尺之移动方向与移动量。

At the same time , we discuss the determinism and nondeterminism of formula-clock automata and the closure propertied under Boolean operation of the language which is accepted by formula-clock automata. We also prove the expressive equivalence between the deterministic formula-clock automata and the nondeterministic formula-clock automata. This means that every nondeterministic formula-clock automata can be transformed to a deterministic formula-clock automata which exactly accept the timed language identified by the fomer.We also extend timed words to infiniteness and define the formula-clock Buchi automata and the formula-clock Muller automata. At last, we show its application in formal verification and modeling of real-time system.

然后讨论了公式时钟自动机的确定性和非确定性，公式时钟自动机识别的语言类在并、交、补运算下的封闭性；并证明了确定的公式时钟自动机和非确定的公式时钟自动机表达能力的等价性，这意味这每一个非确定的公式时钟自动机都能转换为一个与之识别相同时间语言的确定的公式时钟自动机；我们将时间字扩展为无穷的，从而定义了公式时钟Buchi自动机与公式时钟Muller自动机；最后给出了用它进行实时系统的形式化验证方面及对实时系统建模的应用。

Furthermore,we present the method for designing quadratic Bezier developable surface:give four corner points of developable surface and two free designing parameters,the other two control vertexes are on the line connecting the linear interpolation point between the first two control vertexes and the linear interpolation point between the last two control vertexes,and they are the linear interpolation points between these two linear interpolation points respectively,namely,these four linear interpolation points are colinear.

提出了二次Bézier可展曲面的设计方法：给定可展曲面的4个角点a0、b0、a2、b2和两个自由设计参数？姿、？滋，则待求的2个控制顶点a1、b1是在前2个控制顶点a0、b0的线性插值点a*与后2个控制顶点a2、b2的线性插值点b*的连线上，并且也是a*、b*这2个线性插值点的线性插值，即这4点a*、a1、b1、b*共线。

On the basis of determining interpolation neighborhood, this paper factures many geochemistry plots by using more spatial interpolation of the deep penetrating geochemical data in the study area. The methods include inverse distance weighted interpolation, global polynomial interpolation, local polynomial interpolation, radial basis function, simple Kriging and universal Kriging, etc.

文中在确定插值邻域的基础上，应用多种空间插值方法对研究区的深穿透地球化学数据制作了多个地球化学图，如反距离加权插值法、全局多项式插值法、局部多项式插值法、径向基函数法、简单克里金法、泛克里金法等。

"Yes, now you can give yourself airs," she said, you have got what you wanted.

"对了，您现在高兴了，"她说道，这是您所期待的。

Then the LORD said to me: Rebel Israel is inwardly more just than traitorous Judah.

上主于是对我说：＂失节的以色列比失信的犹大，更显得正义。