一种新型高效的振荡函数数值积分方法

§1.引言 目前,求振荡函数的数值积分的方法已有多种,如Lobatto法则,Filon法则和Price法则等.为了得到较精确的计算值,迄今各种方法需要的计算量都相当大. 本文给出一种新型高效的求解振荡函数数值积分的方法.与其他方法相比,具有简便易行、计算量小而求积精度高等特点.     

二元直交多项式的不变因子与数值积分公式

A．H．Stroud给出了关于二元m^2点2m-1次求积公式存在性的充分条件,即两个m次直交多项式P1(x,y)和p2(x,y)存在m^2个不同的公共零点,并且都不是无穷远点。本文用不变因子的方法给出了当m=2时这种直交多项式对的一种选取方法．另外,本文最后给出了一些2m-1次积分公式．     

振荡函数的Hermite数值积分公式

本文讨论了振荡函数形如∫-1^1 f(x)sinwxdx,∫-1^1 f(x)coswxdx的Hermite积分公式，它基于f(x)的Hermite插值多项式的一些结论，导出了依赖于xnj的am1及不依赖于xn1的g(k,w)的权数因子的递推关系式，并给出误差分析。     

对振荡函数数值积分方法的进一步探讨

本文在 [1 ]等成果的基础上 ,对振荡函数数值积分的方法做了进一步的探讨 ,给出了一种代数精确度更高、具有函数振荡越剧烈求积结果越精确的特点的、优于 [1 ]的新的对振荡函数的 Gauss型积分 .     

数值积分校正公式

通过两个例子说明一旦具有数值积分公式的余项表达式,只需利用代数精度概念即可确定余项里的中间点的具体数值,从而获得更高代数精度的数值积分校正公式.本文的方法可用于各类数值积分公式.     

一个新的数值积分公式

本文主要构造了一个数值积分公式,进一步研究了数值积分公式中的中间点的渐进性,基于中间点的渐进性质,得到了该数值积分公式的校正公式,最后给出数值试验,数值试验表明该公式的精度非常高。     

数值积分公式截断误差的简单求法

不用复杂的积分计算给出了求积公式截断误差的简单求法     

振荡函数的Hermite数值积分公式

本文讨论了振荡函数形如∫１－１ｆ（ｘ）ｓｉｎωｘｄｘ，∫１－１ｆ（ｘ）ｃｏｓωｘｄｘ的Ｈｅｒｍｉｔｅ积分公式，它基于ｆ（ｘ）的Ｈｅｒｍｉｔｅ插值多项式的一些结论，导出了依赖于ｘｎｊ的ａｎｊ及不依赖于ｘｎｊ的ｇ（ｋ，ｗ）的权数因子的递推关系式，并给出误差分析．     

高精度数值积分公式的构造及其应用

通过对一个给定的数值积分公式进行加速、改进,得到了两类新的精度更高的数值积分公式.然后将其进行复合,得到复合公式,并将复合公式推广到计算二重积分.最后进行了数值实验,数值计算结果表明:两类新的数值积分公式都具有比给定的公式更高阶的精度和更快的收敛速度.     

一类巴西列维奇函数的Fekete—Szegoe问题

本我们引入了一类巴西列维奇函数,讨论了该类函数上的Ｆｅｋｅｔｅ－Ｓｚｅｇｏｅ问题,所获得的结论是准确的。一些已知的结论是本的特例。     

FAST DENSE MATRIX METHOD FOR THE SOLUTION OF INTEGRAL EQUATIONS OF THE SECOND KIND

We present a fast algorithm based on polynomial interpolation to approximate matrices arising from the discretization of second-kind integral equations where the kernel function is either smooth, non-oscillatory and possessing only a finite number of singularities or a product of such function with a highly oscillatory coefficient function. Contrast to wavelet-like approximations, ourapproximation matrix is not sparse. However, the approximation can be construced in O(n) operations and requires O(n) storage, where n is the number of quadrature points used in the discretization. Moreover, the matrix-vector multiplication cost is of order O(nlogn). Thus our scheme is well suitable for conjugate gradient type methods. Our numerical results indicate that the algorithm is very accurate and stable for high degree polynomial interpolation.     

FRACTIONAL INTEGRAL AND FRACTAL FUNCTION

In this paper, the relationship between Riemann-Liouville fractional integral and the box-counting dimension of graphs of fractal functions is discussed.     

TWO-DIMENSIONAL SINGULAR INTEGRAL EQUATIONS OF THE HYPERCOMPLEX FUNCTIONS

ＴＷＯ－ＤＩＭＥＮＳＩＯＮＡＬＳＩＮＧＵＬＡＲＩＮＴＥＧＲＡＬＥＱＵＡＴＩＯＮＳＯＦＴＨＥＨＹＰＥＲＣＯＭＰＬＥＸＦＵＮＣＴＩＯＮＳ（刘春根）￥ＬｉｕＣｈａｎｇｅｎ（Ｄｅｐｔ．ｏｆＭａｔｈＮａｎｋａｉＵｎｉｖ．，Ｔｉａｎｊｉｎ５００７１，Ｃｈｉｎａ．...     

积图P_n×P_m的奇优美性和奇强协调性

给出了积图P_n×P_m的奇优美标号和奇强协调标号.     

HIGH ORDER NUMERICAL METHODS TO TWO DIMENSIONAL HEAVISIDE FUNCTION INTEGRALS

In this paper we design and analyze a class of high order numerical methods to two dimensional Heaviside function integrals. Inspired by our high order numerical methods to two dimensional delta function integrals [19], the methods comprise approximating the mesh cell restrictions of the Heaviside function integral. In each mesh cell the two dimensional Heaviside function integral can be rewritten as a one dimensional ordinary integral with the integrand being a one dimensional Heaviside function integral which is smooth on several subsets of the integral interval. Thus the two dimensional Heaviside function integral is approximated by applying standard one dimensional high order numerical quadratures and high order numerical methods to one dimensional Heaviside function integrals. We establish error estimates for the method which show that the method can achieve any desired accuracy by assigning the corresponding accuracy to the sub-algorithms. Numerical examples are presented showing that the second- to fourth-order methods implemented in this paper achieve or exceed the expected accuracy.     

ON THE DISTRIBUTION OF VALUES AND DEFICIENT FUNCTION OF A GENERAL RANDOM INTEGRAL FUNCTION

We consider the random power serie8aoIw(z) = ZX.(w)z", (1)n =Owhere {Xn} is a 8equence of complex random variables in a probability spare (Q, F, P).Lemma 1 Supose that {Xn) is independent ai1d that there exists a positive coustallt crsuch thatVn 2 l, o'a: = a'ElXn l' S E'lX.l < oo. (2)Then for w e n, there exists a natural nunther N(w) a.s. such that for n > N(w)IXn(w)l 5 nrrn. (3)If {X.* } is any subsequence Of {X.}, thenP (ltL(lXn*l 2;rru*) = l. (4)Proof When ny = 0,X. = 0 a.s. …     

圆周上带希尔伯特核的柯西奇异积分方程的数值解(英文)

本文研究了圆周上带希尔伯特核的柯西奇异积分的复合梯型公式.利用连续的分片线性函数逼近被积函数,得到积分公式的误差估计;然后用积分公式构造求解对应奇异积分方程的两种格式;最后给出数值实验验证理论分析结果.     

核密度中含有参数的Cauchy主值积分的求积公式

在本文中，我们以二元三次复插值样条函数作为逼近工具，给出了光滑封闭曲线上的核密度中含有参数的各种Ｃａｕｃｈｙ主值积分的求积公式，误差估计及逼近性定理。     

一类改进的非光滑规划的填充函数法

本文考虑优化问题     

A SPLINE METHOD FOR SOLVING TWO-DIMENSIONAL FREDHOLM INTEGRAL EQUATION OF SECOND KIND WITH THE HYPERSINGULAR KERNEL

1. IntroductionIn recent years, the boundary element methods became a reliable and powerful numerical methods for solving the boundary value problems, such as elastoplasticitys etc. In thesemethods, the original problem is reduced to a boundary integral equation. For the one dimensional boundarys a lot of methods have been put forward recently. But for the two dimensionalboundary situation, it is not so easy to be done because the partition can be very complicated.Since P. Zwart obtained an …