积分第一中值定理中的ξ在数值积分上的应用

根据积分第一中值定理的中间点ξ的渐近性质推导出一种单节点数值求积公式，证明余项的表达式，进行数值实验，此求积公式还适于瑕积分的数值计算。     

基于Thiele型连分式的数值积分

基于Thiele型连分式构造求积公式,这类求积公式能再生由Thiele型连分式前三项渐近式的线性组合所表示的任意有理函数,接着算出求积余项,并推导出分母在给定区间上无零点的充分条件.更进一步,通过等分给定区间,构造相应的复化求积公式,并算出求积余项.研究表明,在若干条件满足的前提下,复化求积公式序列能一致收敛于积分真值,一些数值算例说明了这一点.     

已知三面及其两两夹角的四面体的求积公式

已知三面及其两两夹角的四面体的求积公式孔令恩（山东枣庄市立新学校２７７１００）文［１］介绍了四面体中，已知同一顶点三棱ａ，ｂ，ｃ及其两两夹角θ１，θ２，θ３的求积公式Ｖ＝１６ａｂｃ·Ｔ（Ｒ）①其中Ｔ２（Ｒ）＝１ｃｏｓθ１ｃｏｓθ２ｃｏｓθ１１ｃｏｓθ...     

积分第一中值定理中的ξ在数值积分上的应用

根据积分第一中值定理的中间点 ξ的渐近性质推导出一种单节点数值求积公式 ;证明余项的表达式 ;进行数值实验 .此求积公式还适于瑕积分的数值计算 .     

一类Gauss-Jacobi数值求积公式的极限性质

讨论了形如∫aa+h(x-a)βf(x)dx的Gauss-Jacobi求积公式,当积分区间长度趋向于零时,确定了求积公式的余项中介点η的渐近性,并给出了校正公式,比原公式提高了两次代数精度.此外,本文的结论包含了文[3]的结果.     

一种确定求积公式误差最优估计的简单方法

利用求积公式代数精度的概念,给出一种确定Newton-Cotes和Hermite插值型求积公式截断误差最优估计的简单方法,并通过实例验证其有效性.     

关于Euler-Maclaurin公式的一个注记

利用梯形公式的余项,将被积函数的二阶导数做幂级数展开,证明了余项是关于求积区间长度的奇数次幂级数.推导出了复合梯形公式的一类渐近展开式,从另一方面印证了Euler-Maclaurin公式.     

积分第一中值定理中间点的一般渐近性质与求积公式

证明关于积分第一中值定理的中间点ξ的渐近性质的一般结果.而且,由此自然地推导出单节点数值积分公式.此求积公式具有高精度,还适于瑕积分的数值计算.     

Bernoulli多项式与Euler—Maclaurin公式的推广

本文提出了广义Bernoulli多项式与广义Bernoulli数,并借此得到了一类含两端点连续阶导数值求积公式的误差渐近式和推广的Euler-Maclaurin求和公式.借助于计算机代数系统进行了公式的机械推导,并列出了部分推导结果.     

刚性多滞量积分微分方程的Runge-Kutta 方法

本文研究了求解刚性多滞量积分微分方程的Runge-Kutta方法的非线性稳定性和计算有效性.经典Runge—Kutta方法连同复合求积公式和Pouzet求积公式被改造用于求解一类刚性多滞量Volterra型积分微分方程.其分析导出了:在适当条件下,扩展的Runge-Kutta方法是渐近稳定和整体稳定的.此外,数值试验表明所给出的方法是高度有效的.     

A note on generalized averaged Gaussian formulas

We have recently proposed a very simple numerical method for constructing the averaged Gaussian quadrature formulas. These formulas exist in many more cases than the real positive Gauss–Kronrod formulas. In this note we try to answer whether the averaged Gaussian formulas are an adequate alternative to the corresponding Gauss–Kronrod quadrature formulas, to estimate the remainder term of a Gaussian rule.     

On families of quadrature formulas based on Euler identities

A family consisting of quadrature formulas which are exact for all polynomials of order ?5 is studied. Changing the coefficients, a second family of quadrature formulas, with the degree of exactness higher than that of the formulas from the first family, is produced. These formulas contain values of the first derivative at the end points of the interval and are sometimes called “corrected”.     

Optimal quadrature formulas for Fourier coefficients in $W_2^{(m,m-1)}$ space

This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the $W_2^{(m,m-1)}[0,1]$ space for calculating Fourier coefficients. Using S.~L.\ Sobolev''s method we obtain new optimal quadrature formulas of such type for $N 1\geq m$, where $N 1$ is the number of the nodes. Moreover, explicit formulas for the optimal coefficients are obtained. We investigate the order of convergence of the optimal formula for $m=1$. The obtained optimal quadrature formula in the $W_2^{(m,m-1)}[0,1]$ space is exact for $\exp(-x)$ and $P_{m-2}(x)$, where $P_{m-2}(x)$ is a polynomial of degree $m-2$. Furthermore, we present some numerical results, which confirm the obtained theoretical results.     

Convergence of Gaussian quadrature formulas on infinite intervals

More general and stronger estimations of bounds for the fundamental functions of Hermite interpolation of high order on an arbitrary system of nodes on infinite intervals are given. Based on this result, convergence of Gaussian quadrature formulas for Riemann–Stieltjes integrable functions on an arbitrary system of nodes on infinite intervals is discussed.     

Efficiency of numerical integration algorithms related to divisor theory in cyclotomic fields

For function classes with dominant mixed derivative and bounded mixed difference in the metric ofL ^q (1<q≤2), quadrature formulas are constructed so that the following properties are achieved simultaneously: the grid is simple, the algorithm is efficient and close to the optimal algorithm for constructing the grid, and the order of the error on the power scale cannot be further improved. The caseq=2 was studied earlier. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 297–301, February, 1997. Translated by N. K. Kulman     

On computing rational Gauss-Chebyshev quadrature formulas

We provide an algorithm to compute the nodes and weights for Gauss-Chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary real poles outside . Contrary to existing rational quadrature formulas, the computational effort is very low, even for extremely high degrees, and under certain conditions on the poles it can be shown that the complexity is of order . This method is based on the derivation of explicit expressions for Chebyshev orthogonal rational functions, which are (thus far) the only examples of explicitly known orthogonal rational functions on with arbitrary real poles outside this interval.

     

三步五阶迭代方法解非线性方程组

本文根据求积公式, 给出了三种求解非线性方程组的迭代方法, 并证明了所提出的三步迭代方法具有五阶收敛性. 最后给出了四个数值实例, 将本文的实验结果与现有的几种迭代方法的实验结果作了比较分析, 表明本文所提出的方法具有明显的优越性.     

AN EXTREMAL APPROACH TO BIRKHOFF QUADRATURE FORMULAS

1. Introduction and Main ResultsIn tfor paPer we shaJl use the ddstions and notations of [3l. Let E = (e'k)7t' kt. be anincidence matrir with entries consisting of zeros and ones and satisfying lEl:= Z.,* ei* = n + 1(here we allow a zero row ). Furthermore, in wha follOws we assume that(A) E satisfies the P6lya condition(B) all sequences of E in the interior rows, 0 < i < m + 1, are even.Let Sm denote the set of poiats X = (xo, z1 l "') xm, x.+1) fOr whichand Sm its clOusure. If some O…     

Quadrature formulas for classes of functions with bounded mixed derivative or difference

Quadrature formulas are considered for classes of smooth functions W _p ^r , B _p ^r , θ with bounded mixed derivative or difference. For the classes of functions indicated above, the result that quadrature formulas constructed with the help of number-theoretic methods are optimal (in the sense of order) is proved, and the optimal order of the error estimates is obtained. Project supported by the National Natural Science Foundation of China and the Doctoral Program Foundation of the State Education Commission of China.     

A sequentially optimal algorithm for numerical integration

For the class of functions of one variable, satisfying the Lipschitz condition with a fixed constant, an optimal passive algorithm for numerical integration (an optimal quadrature formula) has been found by Nikol'skii. In this paper, a sequentially optimal algorithm is constructed; i.e., the algorithm on each step makes use in an optimal way of all relevant information which was accumulated on previous steps. Using the algorithm, it is necessary to solve an integer program at each step. An effective algorithm for solving these problems is given.The author is indebted to Professor S. E. Dreyfus, Department of Industrial Engineering and Operations Research, University of California, Berkeley, California, for his helpful attention to this paper.