In this paper the following results are got: 1. We prove that the order definition which is given by J.F. Trawb on the iterative function is wrong. 2. A existent theorem on the repulsive point is given. 3. We give some theorems to judge a fixed point of iterative function. They all improve the Ostrowski's theorems.

By applying the canonical correlation decomposition (CCD) of matrix pairs, we obtain a general expression of the least-squares solutions of the matrix equation ATXA = D under the restriction that the solution matrix ∈ Rn×n is bisymmetric, where A ∈Rn×m and D ∈Rm×m are given matrices.

The boundary measure method is applied to transfer the form of the integral equation in order to use the collocation method or Galerkin method. A simple way to computer the coefficients of the wavelet series is also introduced. The way presented in this paper can be used to solve PDE problem in the two dimension region with any form of boundary.

This paper presents an augmented Lagrangian algorithm for nonlinear opti- mization of equality and bounded constraints. The method includes internal it- erations and outer iterations, which uses a trust region interior-point method in internal iteration. Under some conditions, the paper proves finite termination of internal iteration and analyses the local convergence of accelerating internal mini- mizer iterations. It also proves the global convergence of main algorithm when the approximate solution of internal minimizer is satisfied some conditions.

No error estimate of the spectral Galerkin approximation for the steady-state Navier-Stokes equations was given without assuming that the data of the external force field and the boundary conditions are small enough. In this paper, under the condition that the solutions of the Navier-Stokes equations are nonsingular, we proved the existence and convergence of the spectral Galerkin approkimation solutions and gave the error estimate. At last, this approximation method was applied to simulate the spherical Couette flow.

In thc paper, the nonperidic initial value problem for a class of semilinear parabolic equations is considered. We construct the full discrete Chebyshev pseu- dospectral scheme and analyze the error of approximate solution for it. We obtain the error estimation on large time using the local continuation method and the existence of approximate global attractor. This method can be applied to other nonlinear problems too.

A new kind of bivariate vector-valued rational interpolants is recursively estab- lished by means of Samelson inverse over rectangular grids, with scalar numerator and vector-valued denominator. In this respect, it is essentially different from that of the previous work. Sufficient conditions for existence, characterization and uniqueness in some sense are proved respectively. And the resluts in the paper are illustrated with some numerical examples.

The L∞ estimation and local superconvergence estimation are obtained for piecewise linear Galerkin approximation of Drichlet probolem

In this paper, we develop a rapidly convergent algorithm for mixed constrained signomial geometric programming. The algorithm makes use of the characteristics of signomial geometric programming, and establishes a new active-set strategy on the basis of trust region method. The global convergence is proved, and some numerical tests are given to illustrate the effectiveness.

A new NCP-function for the box constrained variational inequality VI([a, b], F) is proposed and its properties are investigated. Using this NCP-function the box constrained variational inequality is reformulated as a system of semismooth equa- tions whose merit function is differentiable every where. For the P0-function F, any stationary point of the merit function solves the VI([a, b], F). The related Newton-type method is proposed. For continuously differentiable and monotone function F, the generalized Newton equation involved in the method is always a uniquely solvable system of linear equations and affords a direction of sufficient decrease for the merit function. Under the condition of BD-regular solution, the algorithm is globally convergent and has a superlinear or possibly quadratic rate of convergence. The numerical results suggest that the algorithm is robust and efficient.

In this paper, an extragradient method for solving variational inequalities was proposed, which extends the method in [1] and converges under the condition that the underlying mapping is pesudomotone. The numerical analysis was also given in this paper.

For large sparse system of linear equations with the coefficient matrix with a dominant indefinite symmetric part, we present a class of splitting minimal resid- ual method, briefly called as SMINRES-method, by making use of the inner/outer iteration technique. The SMINRES-method is established by first transforming the linear system into an equivalent fixed-point problem based on the symmetric/skew- symmetric splitting of the coefficient matrix, and then utilizing the minimal resid- ual (MINRES) method as the inner iterate process to get a new approximation to the original system of linear equations at each of the outer iteration step. The MINRES can be replaced by a preconditioned MINRES (PMINRES) at the inner iterate of the SMINRES method, which resulting in the so-called preconditioned splitting minimal residual (PSMINRES) method. Under suitable conditions, we prove the convergence and derive the residual estimates of the new SMINRES and PSMINRES methods. Computations show that numerical behaviours of the SMIN- RES as well as its symmetric Gauss-Seidel (SGS) iteration preconditioned variant, SGS-SMINRES, are superior to those of some standard Krylov subspace meth- ods such as CGS, CMRES and their unsymmetric Gauss-Seidel (UGS) iteration preconditioned variants UGS-CGS and UGS-GMRES.

In this paper, the convergency of spline interpolation operators is obtained, these spline operators are determined by linear differential operators and con straint functionals. The errors of the interpolating spline with EHB fanctionals are estimated. The best approximation of linear functionals on W2m spaces are investigated, which let to a useful computational method for the approximation so- lution of higher order linear differential equations with multipoint boundary value conditions.

We present a generalized numerical embedding algorithm for solvillg nonsmooth equations based on the results in [1]. Convergence of the algorithm is proved care- fully and implementation is discussed. Application of the algorithm to the com- plementarity problem, variational inequalities and nonlinear optimization problem is discussed.

In this paper, a difference scheme in a special mesh is presented for solving the singluarly perturbed two-point boundary value problems. It is shown that the convergence order is O(N-2) uniformly with respect to the perturbation parameter, ∈. The numerical examples are given underline the theoretical results.

In this paper, the inertia of a symmetric Z-matrix is studied, and bounds of the number of its positive eigenvaues are obtained. Also the interlacing theorem for Schur complement of a symmetric Z-matrix is established, which can be considered as a generalization Cauchy interlacing theorem in some extent.

In this paper, the following problems are considered Problem I. Given A ∈Rm×n, D ∈ R n×n. a) Let S1 = {X: X ∈ Rm×n, ||ATX - XTA - D|| = min} find X ∈ S1 such that ||X|| = min; b) Let S2 = {X: X ∈ Rm×n, ATX - XTA = D} find X ∈ S2 such that ||X|| = min. Problem II. Given A ∈ Rm×n,B B∈Rn×p, D ∈Rm×p. Let L1 = {X: X ∈ SRn×n, AXB = D} find X ∈ L1 such that ||X|| = min. Problem III. Given A ∈ Rm×n,B ∈ Rp×q, C ∈ Rm×q, G ∈ Rl×n, H ∈ Rp×t,D ∈ Rl × t. Let L2 = {X: X ∈ Rn×p, AXB = C, GXH = D} find X ∈ L2 such that ||X|| = min. Using singularvalue and canonical correlation decompositions, the necessary and sufficiellt conditions, under which S2, L1 and L2 are nonempty, are studied. The expressions for the solutions of Problems I, II and III are given.

In this paper, a general construction of biorthonormal multiresolution analyses with length 5 is studied. Both the existence of unique symmetric biorthonormal MRAs and the inexistence of antisymmetric ones are also proved. The regularity of the scale functions is analyzed and some examples are given at last.

This paper, as a natural sequel to [1], gives the further consideration of problem I posed by Liao Anping and Guo Zhong in [2]: given X, Z ∈ Rn×m, Y, W ∈ Rn×l, find A ∈ R0n×n such that AX = Z, yTA = WT, where R0n×n = {A ∈ Rn×n| X ∈ Rn×l,, XTAX ≥ 0}. In [1], we gave a necessary and sufficiellt condition for the solvability and the expression of the general solution of Problem I. In this papar,we will show a better expression of the general solution of Problem I.

The incompressible Navier-Stokes equations, upon spatial discretization, be- come a system of differential algebraic equations, formally of index 2. But due to the special forms of the discrete gradient and discrete divergence, its index can be regarded as 1. Thus, in this paper, a systematic approach following the ODE theory and methods is presented for the construction of high-order time-accurate implicit schemes for the incompressible Navier-Stokes equations, with projection methods for efficiency of numerical solution. The 3rd order 3-step BDF with component- consistent pressure-correction projection method is a first attempt in this direction; the related iterative solution of the auxiliary velocity the boundary conditions and the stability of the algorithm are discussed. Results of numerical tests on the incom- pressible Navier-Stokes equations with an exact solution are presented, confirming the accuracy stability and component- consistency of the proposed method.

In this paper some common used numerical schemes for solving discrete ordinate equations are considered and the error estimates are studied for the combined spatial and angular approximations. The conclusions show that the error order of scalar flux in all of these schemes can not be second order even if the source term f is smooth enough. In addition, when we introduce a kind of graded grids, the simple step character scheme has same accuracy as "high order" ones.

We consider the rank-constrained subset selection problem (RCSS): Given a matrix A and an integer p ≤ rank(A), find the largest submatrix A0 consisting of some columns of A with rank(A0) = p. The RCSS problem is generally NP- hard. This paper focuses on a divide-and-conquer (DC) algorithm for solving the RCSS problem: partition the matrix A into several small column blocks: A = [Al,……) Ak] with a certain column permutation II and decompose p to p1 + p2 + ……+ pk such that solutions of the RCSS problems for smaller couples form a solution of the original RCSS problem. We show that the optimal solution of the RCSS problem can be found by DC algorithm for each p ≤ rank(A), if and only if A is column-partitionable, i. e., rank(A) = Σ rank(Ai). Based upon QR decomposition, a fast algorithm for determining the column partition is offered. Our divide-and-conquer algorithm is also quite efficient even A is approkimately column-partitionable.

In this paper, we discusse the locking phenomenon of the finite element method for the pure displacement boundary value problem in the planar elasticity as Lame constant λ- ∞. The locking-free scheme of Crouziex-Raviart element was pro- posed and anaIyzed by Brenner et al.[2] and [3]. We firstly present the derivation of Brenner's scheme, then propose and analyse a locking-free scheme of noncon- forming rectangle finite element.

1.引言 基算法是我们近几年来研究的一种线性规划新算法,它具有操作方便,高效,数值稳定性好等特点.它可以看作单纯形法的发展. 本文是文[1,2]的续篇,有关基线算法的基本概念和基本理论请参看文[1,2].这里简单介绍如下. Rn表示n维欧几里得空间.如果不特别指明,x∈Rn视情况可以是行向量或列向量.考虑标准型线性规划问题（LP）

0.引言 分形集的分形维数估计及在一定维数下的Hausdorff测度估计是分形理论中的基本问题,对一般的分形集,这两个问题往往十分困难,特别后一个,即使对一些具有严格自相似性的经典分形集也是如此.Cantor尘[1]是经典分形集之一,它具有严格的自相似性,结构十分良好,虽然它的Hausdorff维数α=log3 4不难得到,但求它的α维Hausdorff测度准确值却是一个十分困难的问题.本文构造了Cantor尘上的一个质量分布分形插值函数,将求Cantor尘的Hausdorff测度转化为求分形插值函数的极值问题,并利用质量分布原理,给出了Canor尘的Hausdorff测度的表达式和近似值.

1.引 言 自从Levenberg（1944）和Marguart（1963）关于无约束优化信赖域算法的工作开始,这类算法得到深入研究和迅速发展[1-7],正如Fletcher[1]及Powell[7]等人的工作所表明的,漂亮的收敛性和有效的计算性确定了信赖域算法是一类重要和实用的方法,因此研究约束优化问题的信赖域算法具有重要的意义,但目前这方面的研究多半集中于等式约束最优化[8-11]或线性约束最优化[12],且采用罚函数技术[8-11],而有关非线性不等式约束最优化的信赖域算法的研究成果尚不多见,尤其是可行单调的信赖算法研究更少.本文吸取信赖域法、

1.引 言 许多物理应用问题归结为求微分方程数值解,而这可以通过离散化为求解稀疏线性方程组,所以稀疏线性方程组求解的有效性在很大程度上决定了原问题求解算法的有效性.直接

1.引 言 差分-流线扩散法（Finite Difference-Streamline Diffusion Method,简称FDSD方法）于1998年由文[1]提出并对线性对流占优扩散问题给出分析,随后文[2],[3]就非线性问题的FDSD格式及FDSD预测-校正格式,分别作出了分析,文[4]讨论了FDSD方法的后验估计及自适应技术,[5],[6]则分别讨论了FDSD方法的某些重要应用.与基于时-空有限元的传统流线扩散法相比,FDSD方法的计算工作量已有成数量级的减少,且较易于推广到非线性问题,然而,对于高维问题,在每一时间层,仍然需要求解一大型线性或非线性方程组,工作量仍然很大.参照J.Douglas与T.Dupont关于抛物问题交替方向

1.引 言 数值模拟流体自由界面运动一直是研究水波的主要方法.海浪攀爬海岸的研究是水动力学中的一个很经典且很具有挑战性的课题,因为在水面附近的方程是高度非线性的.对于二维情形下有一致倾斜度的海岸上的水波,Carrier＆Greeenspan[7]建立了基于浅水模型的非线性理论,Tuck＆Hwang引入变量代换,把最初的非线性方程转变成更容易分析的线性方程.这种直接对单一流体用浅水方程计算自由界面的办法仍然被广泛应用.Zhang,Wu,Hou[23]给出了这个问题的Euler-Langrange混合格式,Li ＆ Zhang[13]借助这个格式,并引入人工边界条件对海浪攀爬海岸问题进行了整体的数值模拟.

1.引 言 非线性方程组F（x）=0的数值求解,经典的算法是Newton迭代;xk+1=xk+sk,k=0,1,2,…,（1.1）其中的sk满足F’（xk）sk=-F（xk）;k=0,1,2,….（1.2）这里x0为迭代的初始点,{xk}称为Newton迭代序列.当变量个数比较多时,每一步Newton迭代中计算Jacobi矩阵F’（xk）和求解线性方程组（1.2）的代价非常高;特别当xk远离方程组的解x＊时,高精度地求解线性方程组（1.2）

1.引 言 由于采用高精度算法能大大提高有限元计算的精度,因此有许多专家对它进行了多方面研究,取得了一批卓有成效的成果[16]研究有限元高精度的方法主要有两种: （1）美国H.A.Schatz.B.wahlbin[4,5]等发现的直接考察u-uh或　（u-uh）在局部对称点所具有的超收敛性的方法. （2）中国林群,朱起定[1,2],陈传淼[3]等所发现的通过研究uI-uh或　（uI-uh）所具有的整体超收敛与外推性质来得到u-uh或　（u-uh）在剖分点与其他某些特殊点的超收敛与外推性质.

1.引 言 广义差分法作为处理偏微分方程的离散技术,能够保持质量,动量,能量等物理量的守恒.广义差分法（有些文献称为box method[3];finite volume element method[4],[5],[6]）利用在对偶剖分体积单元积分原始方程,并将近似解限制于某一有限元空间而得到离散方程.因此,它在局部区域保持了原始方程的物理守恒性和其他重要特性.从而被广泛地应用于数值求解数学物理方程,特别是计算流体力学和热传导问题[11]. 对广义差分法的研究已有许多文献,专著[10]有详细的介绍.早期的工作主要考虑标准的重心对偶剖分.近年来Cai et,al[4],[5],[6],在某些假定下对较一般的对偶剖分给出了能量模误差估计,Huang and Xi[9]去掉了文献[6]中的这些限制.Chou,Li[8]和Li,

1.引 言考虑观测线性系统AX=B,（1.1a）其中A∈Cm×n,B∈Cm×d（本文通篇假设m≥n+d）,分别是精确但不可观测的A0∈Cm×n,B0∈Cm×d的近似,即精确线性系统是A0X=B0.（1.1b）Golub和Van Loan于1980年提出的总体最小二乘问题（以下简称TLS问题）就是求解线性系统AX=B（1.2）

1.引 言在孤立子的研究中起着重要作用的典型方程-KdV方程已有不少作者[1-5]在数学分析上做了许多深入的研究,文[6]讨论了如下一类高阶广义KdV方程组

1.引 言 本文的目的是用求解偏微分方程（PDE）的方法来消除离散三角形曲面的噪声,所使用的方程是热传导方程到曲面的推广.热传导方程应用于图像处理已有二十余年的历史,有关参考文献相当丰富（见[1,11,12,19]）.众所周知,对于给定的初始图像ρ0,热传导方程　　在τ时刻的解与用Gauss滤波器Gσ（x）=　（当标准差σ=2τ,时）和ρ0作卷积的结果相同.容易看出Gρ和ρ0的卷积运算相当于对ρ0做加权平均,当标准离差σ变大时,该加权平均在一个较大的范围实现,这解释了热传导方程的滤波作用.近来热传导方程已推广到空间曲面[4,5]以及高维空间中的二维流形（见[3]）,对

1.引 言为求解微分方程的外边值问题常需要引进人工边界（见[1-4]）,对人工边界外部区域作自然边界归化得到的自然积分方程即Dirichlet-Neumann映射,正是人工边界上的准确的边界条件（见[2-6]）,这是一类非局部边界条件.自然积分算子即Dirichlet-Neumann算子,

考虑下面的抛物型积分微分方程初边值问题: 　（a） ut+A（t）u+∫0tB（t,s）u（s）ds=f, （x,t）∈Q=Ω×J,J=（0,T] （b） u=0,（x,t）∈　Ω×J,（1） （c）　u（x,0）=u0,x∈Ω,其中Ω为Rd（d≤4）中具有分片光滑边界　Ω的有界域,A（t）是一致正定的二阶椭圆微分算子

本文我们考虑一类典型的椭圆型算子的障碍问题的区域分解算法,分析算法的单调收敛性并给出相应的收敛速度估计.障碍问题有着重要的物理背景（参见[3,9]）.近些年来,有关障碍问题的区域分解法方面的研究已经有一些成果.关于线性算子情形,读者可参看[1,2,5,7,8,10,12,13,14,15,17]等文献,而对于非线性算子情形,读者可参看[4,6,16,18].在这些文献中,已经有部分涉及到算法的收敛速度估计.例如,文[15,16]给出了有限元区域分解算法的迭代误差的渐近最大模估计,文[13]给出了求解具M-阵的有限维互补问题

1.引 言 考虑下列凸数学规划（CNLP）问题 min f（x）,s.t.x ∈ Ω,（1.1）严格可行集合Ω0={x∈Rn:gi（x）<0,i=1,…,m}集合Ω表示Ω0的闭包,f（x）,gi（x）均为充分光滑函数.Ω的边界集合　Ω=Ω\Ω0,g=（g,…,gm）T,　x∈Ω,

现有文献中对于非线性延迟微分方程渐近稳定性及其数值方法的稳定性研究大都局限于常延迟的情形,例如可参见匡蛟勋[1-3],黄乘明[4],Torelli[5]等人的大量工作.1994年A.Iserles[6] 首次研究了比例延迟微分方程数值方法的线性稳定性,随后有相当多的文献对比例延迟微分方程的各种数值方法的线性稳定性进行了讨论.1997年Zennaro[7]首次研究了非线性刚性变延迟微分方程的渐近稳定性,但该文中对于延迟量的限制十分苛刻,同时该文也首次研究了非线性刚性变延迟微分方程Runge-Kutta方法的非线性稳定性. 本文目的是试图在上述基础上进一步研究非线性刚性变延迟微分方程的渐近稳定性及其数值方法的稳定性.首先在第二节我们给出了非线性刚性变延迟微分方程模型问题（2.1）渐