The best quadrature formula based on Hermite information for the class <Emphasis Type="Italic">KW</Emphasis><Superscript>2</Superscript> [<Emphasis Type="Italic">a,b</Emphasis>]

The best quadrature formula has been found in the following sense: for a function whose norm of the second derivative is bounded by a given constant and the best quadrature formula for the approximate evaluation of integration of that function can minimize the worst possible error if the values of the function and its derivative at certain nodes are known. The best interpolation formula used to get the quadrature formula above is also found. Moreover, we compare the best quadrature formula with the open compound corrected trapezoidal formula by theoretical analysis and stochastic experiments.     

三种不同意义下的最佳求积公式之间的关系

详细讨论了函数类KWr[a,b]上Sard和N iko lsk ii意义下以及基于给定信息的最佳求积公式三者之间的关系,并且提供了一种由基于给定信息的最佳求积公式得到其它两种求积公式的方法.     

The best quadrature based on given Hermite information for the Sobolev class KW~r[a,b]

As usual, denote by KWr[a,b] the Sobolev class consisting of every function whose (r-1)th derivative is absolutely continuous on the interval [a,b] and rth derivative is bounded by K a.e. in [a, b]. For a function f∈KWr[a, b], its values and derivatives up to r -1 order at a set of nodes x are known. These values are said to be the given Hermite information. This work reports the results on the best quadrature based on the given Hermite information for the class KWr[a. b]. Existence and concrete construction issue of the best quadrature are settled down by a perfect spline interpolation. It turns out that the best quadrature depends on a system of algebraic equations satisfied by a set of free nodes of the interpolation perfect spline. From our another new result, it is shown that the system can be converted in a closed form to two single-variable polynomial equations, each being of degree approximately r/2. As a by-product, the best interpolation formula for the class KWr[a, b] is also obtained.     

The best quadrature based on given Hermite information for the Sobolev class KWr[a, b]

As usual, denote by KW ^r[a, b] the Sobolev class consisting of every function whose (r − 1)th derivative is absolutely continuous on the interval [a, b] and rth derivative is bounded by K a.e. in [a, b]. For a function f ∈ KW ^r [a, b], its values and derivatives up to r − 1 order at a set of nodes x are known. These values are said to be the given Hermite information. This work reports the results on the best quadrature based on the given Hermite information for the class KW ^r [a, b]. Existence and concrete construction issue of the best quadrature are settled down by a perfect spline interpolation. It turns out that the best quadrature depends on a system of algebraic equations satisfied by a set of free nodes of the interpolation perfect spline. From our another new result, it is shown that the system can be converted in a closed form to two single-variable polynomial equations, each being of degree approximately r/2. As a by-product, the best interpolation formula for the class KW ^r [a, b] is also obtained.     

Derivatives of a finite class of orthogonal polynomials related to inverse Gamma distribution

In this work, we consider derivatives of a finite class of orthogonal polynomials with respect to weight function which is related to the probability density function of the inverse gamma distribution over the positive real line. General properties for this derivative class such as orthogonality, Rodrigues’ formula, recurrence relation, generating function and various other related properties such as self-adjoint form and normal form are indicated. The corresponding Gaussian quadrature formulae are introduced with examples. These examples are provided to support the advantages of considering the derivatives class of the finite class of orthogonal polynomials related to inverse gamma distribution. The orthogonality property related to the Fourier transform of the derivative class under discussion is also given.     

A GENERALIZED GAUSS-TYPE QUADRATURE FORMULA AND ITS APPLICATIONS TO PSEUDOSPECTRAL METHOD

1　ＩｎｔｒｏｄｕｃｔｉｏｎＳｐｅｃｔｒａｌｍｅｔｈｏｄｅｍｐｌｏｙｓｇｌｏｂａｌｐｏｌｙｎｏｍｉａｌｓａｓｔｒｉａｌｆｕｎｃｔｉｏｎｓｉｎｓｐａｔｉａｌｄｉｓｃｒｅｔｉｚａｔｉｏｎｏｆｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓ.Ｉｔｏｆｔｅｎｐｒｏｖｉｄｅｓａｃｃｕｒａｔｅｎｕｍｅｒｉｃａｌｒｅｓｕｌｔｓｗｉｔｈｒｅｌａｔｉｖｅｌｙｌｅｓｓｄｅｇｒｅｅｏｆｆｒｅｅｄｏｍｓ,ａｎｄｔｈｕｓｈａｓｂｅｅｎｓｕｃｃｅｓｓｆｕｌｌｙｕｓｅｄｆｏｒｃｏｍｐｕｔａｔｉｏｎｓｉｎｓｃｉｅｎｃｅａｎｄｅｎｇｉｎ…     

Spherical Bessel functions and explicit quadrature formula

An evaluation of the derivative of spherical Bessel functions of order at its zeros is obtained. Consequently, an explicit quadrature formula for entire functions of exponential type is given.

     

经典SIMPSON求积公式的新证明

对于定积分近似计算中常使用的经典SIMPSON求积公式介绍一种新的简洁的证明方法并给出误差的最佳估计.     

有界变差函数的加权Ostrowski型不等式的改进及其应用(英文)

本文改进了有界变差函数的加权Ostrowski型不等式,并给出了其在求积公式中的应用.     

A High-Order Piecewise Quartic Spline Rule for Hadamard Integral and Its Application of the Cavity Scattering

We develop a fourth-order piecewise quartic spline rule for Hadamard integral. The quadrature formula of Hadamard integral is obtained by replacing the integrand function with the piecewise quartic spline interpolation function. We establish corresponding error estimates and analyze the numerical stability. The rule can achieve fourth-order convergence at any point in the interval, even when the singular point coincides with the grid point. Since the derivative information of the integrand is not required, the rule can be easily applied to solve many practical problems. Finally, the quadrature formula is applied to solve the electromagnetic scattering from cavities with different wave numbers, which improves the whole accuracy of the solution. Numerical experiments are presented to show the efficiency and accuracy of the theoretical analysis.     

基于切比雪夫节点的一类带显式系数的广义高斯求积公式(英文)

The goal here is to give a simple approach to a quadrature formula based on the divided diffierences of the integrand at the zeros of the nth Chebyshev polynomial of the first kind,and those of the（n-1）st Chebyshev polynomial of the second kind.Explicit expressions for the corresponding coefficients of the quadrature rule are also found after expansions of the divided diffierences,which was proposed in[14].     

The best quadrature based on given Hermite information for the Sobolev class KW r[a, b]

As usual, denote by KW ^r[a, b] the Sobolev class consisting of every function whose (r ? 1)th derivative is absolutely continuous on the interval [a, b] and rth derivative is bounded by K a.e. in [a, b]. For a function f ∈ KW ^r [a, b], its values and derivatives up to r ? 1 order at a set of nodes x are known. These values are said to be the given Hermite information. This work reports the results on the best quadrature based on the given Hermite information for the class KW ^r [a, b]. Existence and concrete construction issue of the best quadrature are settled down by a perfect spline interpolation. It turns out that the best quadrature depends on a system of algebraic equations satisfied by a set of free nodes of the interpolation perfect spline. From our another new result, it is shown that the system can be converted in a closed form to two single-variable polynomial equations, each being of degree approximately r/2. As a by-product, the best interpolation formula for the class KW ^r [a, b] is also obtained.     

On a quadrature formula of Micchelli and Rivlin

Micchelli and Rivlin (1972) obtained a quadrature formula of highest algebraic degree of precision for the Fourier-Chebyshev coefficients A_n(f), which is based on the divided differences of f′ at the zeros of the Chebyshev polynomial T_n(x). We give here a simple approach to questions of this type, which applies to the coefficients in arbitrary orthogonal expansion of f. As an auxiliary result we obtain a new interpolation formula and a new representation of the Turán quadrature formula.     

A generalized Birkhoff-Young-Chebyshev quadrature formula for analytic functions

A generalized N-point Birkhoff-Young quadrature of interpolatory type, with the Chebyshev weight, for numerical integration of analytic functions is considered. The nodes of such a quadrature are characterized by an orthogonality relation. Some special cases of this quadrature formula are derived.     

柯特斯校正公式及其误差估计

提出了一类计算定积分的高精度柯特斯校正公式,通过两种方法进行了推导,给出了它的复化公式及其加速公式,并得到了它们的误差估计和收敛阶.数值实验验证了复化柯特斯校正公式及其加速公式的高效性.     

具调节因子Hermite拟谱逼近的误差估计

本文研究了具调节因子的Hermite函数的拟谱方法在赋权Sobolev空间中函数的逼近.通过具调节因子的Hermite多项式的性质和相应的Gauss类型的求积公式,得到了在具调节因子的Hermite多项式的零点上的插值算子的稳定性以及误差界.并具有通常的高阶收敛性.     

Quadrature formulas for Bessel polynomials

A quadrature formula is a formula computing a definite integration by evaluation at finite points. The existence of certain quadrature formulas for orthogonal polynomials is related to interesting problems such as Waring’s problem in number theory and spherical designs in algebraic combinatorics. Sawa and Uchida proved the existence and the non-existence of certain rational quadrature formulas for the weight functions of certain classical orthogonal polynomials. Classical orthogonal polynomials belong to the Askey-scheme, which is a hierarchy of hypergeometric orthogonal polynomials. Thus, it is natural to extend the work of Sawa and Uchida to other polynomials in the Askey-scheme. In this article, we extend the work of Sawa and Uchida to the weight function of the Bessel polynomials. In the proofs, we use the Riesz–Shohat theorem and Newton polygons. It is also of number theoretic interest that proofs of some results are reduced to determining the sets of rational points on elliptic curves.     

Fourier and barycentric formulae for equidistant Hermite trigonometric interpolation

We consider the Hermite trigonometric interpolation problem of order 1 for equidistant nodes, i.e., the problem of finding a trigonometric polynomial t that interpolates the values of a function and of its derivative at equidistant points. We give a formula for the Fourier coefficients of t in terms of those of the two classical trigonometric polynomials interpolating the values and those of the derivative separately. This formula yields the coefficients with a single FFT. It also gives an aliasing formula for the error in the coefficients which, on its turn, yields error bounds and convergence results for differentiable as well as analytic functions. We then consider the Lagrangian formula and eliminate the unstable factor by switching to the barycentric formula. We also give simplified formulae for even and odd functions, as well as consequent formulae for Hermite interpolation between Chebyshev points.     

A Carleman Function and the Cauchy Problem for the Laplace Equation

We suggest an explicit formula for reconstruction of a harmonic function in a domain from its values and the values of its normal derivative on part of the boundary; i.e., we give an explicit continuation formula and a regularization procedure for a solution to the Cauchy problem for the Laplace equation.     

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.