浅谈微积分所蕴含的特征值计算思想方法

数学思想方法是数学的灵魂.主要阐述微积分中泰勒公式的应用,泰勒公式不但能用来近似计算数学常数π,而且能用来理论证明两个粗糙的近似值通过简单的组合加工技术(现代人称之为"外推法")产生更准确的近似值,也把外推法推广用于求解Steklov特征值问题,数值算例表明非协调Crouzeix-Raviart有限元外推法既能提高解的精度又能提供准确特征值的下界.     

Pointwise bounds for integral type operators

The purpose of this paper is to study pointwise upper bounds for integral operators whenever some bound for the input function is known. Applications to the cases of Hardy, Riemann–Liouville or Volterra type operators and the Abel transform, among others, are given. The underlying techniques are closely related to Yano's extrapolation theory.     

Approximation and eigenvalue extrapolation of biharmonic eigenvalue problem by nonconforming finite element methods

In this paper, we analyze the biharmonic eigenvalue problem by two nonconforming finite elements, Q and E Q. We obtain full order convergence rate of the eigenvalue approximations for the biharmonic eigenvalue problem based on asymptotic error expansions for these two nonconforming finite elements. Using the technique of eigenvalue error expansion, the technique of integral identities, and the extrapolation method, we can improve the accuracy of the eigenvalue approximations. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

一种新的求解线性方程组的外推加速方法

本文提出两种优化模型, 通过在子空间{x^(k),…, x^(k-m)}上寻找最优解, 建立了一种新的外推加速方法. 讨论了该方法的收敛性和收敛速度. 最后, 通过三个数值实例展示了算法是可行的和有效的.     

Mechanical quadrature methods and their splitting extrapolations for boundary integral equations of first kind on open arcs

This paper presents the mechanical quadrature methods (MQMs) for solving boundary integral equations (BIEs) of the first kind on open arcs. The spectral condition number of MQMs is only O(h⁻¹), where h is the maximal mesh width. The errors of MQMs have multivariate asymptotic expansions, accompanied with for all mesh widths h_i. Hence, once discrete equations with coarse meshes are solved in parallel, the accuracy order of numerical approximations can be greatly improved by splitting extrapolation algorithms (SEAs). Moreover, a posteriori asymptotic error estimates are derived, which can be used to formulate self-adaptive algorithms. Numerical examples are also provided to support our algorithms and analysis. Furthermore, compared with the existing algorithms, such as Galerkin and collocation methods, the accuracy order of the MQMs is higher, and the discrete matrix entries are explicit, to prove that the MQMs in this paper are more promising and beneficial to practical applications.     

一种多光谱高温计无源温区的标定方法研究

目前,高温测量中,多光谱高温计具有高分辨率与高信噪比的优点,但无源温区(大于3 000℃)标定方法的发展却远滞后于多光谱高温计制造水平,并已严重影响高温计的测量精度和应用范围。为突破高温测量领域中无源温区标定方法的瓶颈,文章提出了一种多光谱高温计无源温区的标定方法。此方法利用已知温度点的多光谱高温计的输出电压U建立了一种基于幂函数的温度-电压(T-U)模型。在此基础上应用导数最小二乘法对模型参数求解,实现对无源温区的标定。理论和实测数据都验证了此方法的有效性与精度。在多光谱高温测量应用的光谱范围(0.4～1.1μm)内,对精度优于3‰,1%和3%的无源温区标定范围做出理论上的划分。     

On one-electron basis set extrapolation of atomic and molecular correlation energies

It is well known that the convergence of correlation energies in atomic and molecular calculations is relatively slow and that calculations aimed at high accuracy must explicitly make corrections for this. In this work we consider 1e ^? basis set extrapolation as a means of obtaining high accuracy. The correlation consistent basis sets of Dunning et al. have provided a convenient platform for extrapolation, with the independent variable being X?=?D,?T,?Q,?5,?…?. There has been much debate in the literature about the functional form to use for the extrapolation, with contention between the ‘theoretically justified’ (X?+?1)^?3 form and empirical forms based on exponentials or variable powers. We will dissect the theoretical justification of the (X?+?1)^?3 form by considering MP2 calculations on He and Ne as a function of the maximum angular momentum (?) in the basis using basis sets having converged radial extent. Calculations with ? up to 14 were carried out for Ne. It is shown that while the asymptotic form of (??+?1)^?3 is clearly reached, higher order terms are very important in the range of ? normally used in molecular calculations. We also use similar analysis techniques for an open shell atom and a small molecule. The functional form for the dependence of molecular properties with ? is complex and it is safer to extrapolate fitting parameters than energies.     

一维搜索的一个Anderson-Bj(o)rck外推加速算法

对一维搜索的二点二次插值法,利用Anderson—Bjorck外推的思想提出了一个加速收敛算法,得出并证明了两条收敛性定理,且通过数值试验验证了该算法比二点二次插值法收敛速度快很多。     

Rubio de Francía's weighted extrapolation in mixed-norm spaces and applications

Weighted extrapolation for pairs of functions in mixed-norm Banach function spaces defined on the product of quasi-metric measure spaces $(X,d,\mu )$ and $(Y,\rho ,\nu )$ are derived. As special cases, we have appropriate results for mixed-norm Lebesgue, Lorentz, and Orlicz spaces. Some of the derived results are applied to get weighted extrapolation in mixed-norm grand Lebesgue spaces.