A QR DECOMPOSITION BASED SOLVER FOR THE LEAST SQUARES PROBLEMS FROM THE MINIMAL RESIDUAL METHOD FOR THE SYLVESTER EQUATION

Based on the generalized minimal residual （GMRES） principle, Hu and Reichel proposed a minimal residual algorithm for the Sylvester equation. The algorithm requires the solution of a structured least squares problem. They form the normal equations of the least squares problem and then solve it by a direct solver, so it is susceptible to instability. In this paper, by exploiting the special structure of the least squares problem and working on the problem directly, a numerically stable QR decomposition based algorithm is presented for the problem. The new algorithm is more stable than the normal equations algorithm of Hu and Reichel. Numerical experiments are reported to confirm the superior stability of the new algorithm.     

Reducibility of hyperplane arrangements

Certain problems on reducibility of central hyperplane arrangements are settled. Firstly, a necessary and sufficient condition on reducibility is obtained. More precisely, it is proved that the number of irreducible components of a central hyperplane arrangement equals the dimension of the space consisting of the logarithmic derivations of the arrangement with degree zero or one. Secondly, it is proved that the decomposition of an arrangement into a direct sum of its irreducible components is unique up to an isomorphism of the ambient space. Thirdly, an effective algorithm for determining the number of irreducible components and decomposing an arrangement into a direct sum of its irreducible components is offered. This algorithm can decide whether an arrangement is reducible, and if it is the case, what the defining equations of irreducible components are.     

ON DIRECT METHOD OF SOLUTION FOR A CLASS OF SINGULAR INTEGRAL EQUATIONS

In this article, by introducing characteristic singular integral operator and associate singular integral equations (SIEs), the authors discuss the direct method of solution for a class of singular integral equations with certain analytic inputs. They obtain both the conditions of solvability and the solutions in closed form. It is noteworthy that the method is different from the classical one that is due to Lu.     

The Mechanical Quadrature Methods and Their Extrapolation for Solving BIE of Steklov Eigenvalue Problems

By means of the potential theory Steklov eigenvalue problems are transformed into general eigenvalue problems of boundary integral equations (BIE) with the logarithmic singularity.Using the quadrature rules, the paper presents quadrature methods for BIE of Steklov eigenvalue problem, which possess high accuracies O(h^3) and low computing complexities. Moreover, an asymptotic expansion of the errors with odd powers is shown. Using h^3-Richardson extrapolation, we can not only improve the accuracy order of approximations, but also derive a posterior estimate as adaptive algorithms. The efficiency of the algorithm is illustrated by some examples.     

Adaptive Image Zo oming Algorithm Combining Total Variation Minimization and a Second-order Functional

An image zooming algorithm by using partial differential equations(PDEs) is proposed here. It combines the second-order PDE with a fourth-order PDE. The combined algorithm is able to preserve edges and at the same time avoid the blurry effect in smooth regions. An adaptive function is used to combine the two PDEs. Numerical experiments illustrate advantages of the proposed model.     

A DIRECT METHOD OF OPERATIONAL CALCULUS(Ⅰ)——Research for the solution of second-order linear ordinary differential equations with variable coefficients

According to Mikusinski's fundamental view, we reconstruct a direct method of operation acalbulus which may be adapted to search for the solution of differential equations with variablecoefficients.The present paper is devoted to solutions of two types——indirect (undetermined) method and direct (series of integral) method-for solving second-order linear differential equations: The latter displays forthright feature while the former can solve second-order linear differential equation and interrelate first-order Riccati's non-linear differential equation at the same time.     

An extended fast algorithm for constructing the Dixon resultant matrix

In recent years,the Dixon resultant matrix has been used widely in the re-sultant elimination to solve nonlinear polynomial equations and many researchers havestudied its efficient algorithms.The recursive algorithm is a very efficient algorithm,butwhich deals with the case of three polynomial equations with two variables at most.Inthis paper,we extend the algorithm to the general case of n 1 polynomial equations in nvariables.The algorithm has been implemented in Maple 9.By testing the random polyno-mial equations,the results demonstrate that the efficiency of our program is much betterthan the previous methods,and it is exciting that the necessary condition for the existenceof common intersection points on four general surfaces in which the degree with respectto every variable is not greater than 2 is given out in 48×48 Dixon matrix firstly by ourprogram.     

A COLLOCATION METHOD FOR THE CONDUCTIVITY PROBLEM WITH DISCONTINUOUS COEFFICIENT

In this paper, a new collocation BEM for the Robin boundary value problem of the conductivity equation ▽(γ▽u) = 0 is discussed, where the 7 is a piecewise constant function. By the integral representation formula of the solution of the conductivity equation on the boundary and interface, the boundary integral equations are obtained. We discuss the properties of these integral equations and propose a collocation method for solving these boundary integral equations. Both the theoretical analysis and the error analysis are presented and a numerical example is given.     

A DIRECT METHOD OF OPERATIONAL CALCULUS (Ⅰ)——Research for the solution of second-order linear ordinary differentiaI equations with Variable coefficients

According to Mikusinski's fundamental view,we reconstruct a direct method of operation a calbulus which may be adapted to search for the solution of differential equations with variable coefficients.The present paper is devoted to solutions of two types——indirect (undetermined) method and direct (series of integral) method-for solving second-order linear differential equations, The latter displays forthright feature while the former can solve second-order linear differential equation and interrelate first-order Riccati's non-linear differential equation at the same time.     

A reduced-order extrapolation algorithm based on CNLSMFE formulation and POD technique for two-dimensional Sobolev equations

A reduced-order extrapolation algorithm based on Crank-Nicolson least-squares mixed finite element （CNLSMFE） formulation and proper orthogonal decomposition （POD） technique for two-dimensional （2D） Sobolev equations is established. The error estimates of the reduced-order CNLSMFE solutions and the implementation for the reduced-order extrapolation algorithm are provided. A numerical example is used to show that the results of numerical computations are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order extrapolation algorithm is feasible and efficient for seeking numerical solutions to 2D Sobolev equations.     

OPTIMIZATION OF ADAPTIVE DIRECT METHOD FOR APPROXIMATE SOLUTION OF INTEGRAL EQUATIONS OF SEVERAL VARIABLES

This paper determines the exact error order on optimization of adaptive directmethods of approximate solution of the class of Fredholm integral equations of the secondkind with kernel belonging to the anisotropic Sobolev classes, and also gives an optimalalgorithm.     

多元Fredholm积分方程自适应直接方法的优化

考虑了以各向异性光滑函数为核及自由项的多元Fredholm积分方程类的自适应直接方法的优化,得到误差阶的精确估计及相应的最优算法。     

Optimization of adaptive direct algorithms for approximate solution of integral equation

The concern of this paper is to study the optimization of adaptive direct algorithms for approximate solution of the second kind Fredholm integral equations. We derive the exact order of the problem with integral kernels belonging to the Besov classes. Furthermore, we also present an almost optimal adaptive direct algorithm for above equation class.     

Optimization of Approximate Solution of Integral Equations of Several Variables

In this paper, we consider the problem of optimization of adaptive direct methods of operator equations. Adaptivity of a direct method is understood in the sense that the subspace on the basis of which it is constructed is chosen depending on the operator of the concrete equation (otherwise, nonadaptive direct method is then concerned), which would essentially let us increase the precision. For some classes of the second kind of Fredhlom integral equations with anisotropic smooth kernels we determine the exact order of the error of adaptive direct methods, and we also give an optimal algorithm. Received April 13, 2001, Accepted September 13, 2001     

多元积分方程自适应解法的优化

考虑核属于各向异性的Sobolev类的积分方程自适应直接方法的优化,得到误差阶的精确估计及相应的最优方法.     

多元积分方程自适应解法的优化

给出了核属于Besov空间的第二类Fredholm积分方程自适应的最优算法,并得到相关的误差阶的精确估计。     

Solving a large dense linear system by adaptive cross approximation

An efficient algorithm for the direct solution of a linear system associated with the discretization of boundary integral equations (in two dimensions) is described without having to compute the complete matrix of the linear system. This algorithm is based on the unitary-weight representation, for which a new construction based on adaptive cross approximation is proposed. This low rank approximation uses only a small part of the entries to construct the adaptive cross representation, and therefore the linear system can be solved efficiently.     

动力学方程的积分型直接积分法

提出了一个求解动力学问题的新方法(DIM-IM).将动力学方程化成积分方程的形式,借助于该方程构造出了具有显式预测-校正的单步、自起动和四阶精度的积分型直接积分算法.理论分析和算例指出,这一方法较中心差分法、Houbolt法、Newmark法和Wilson-θ法都有较高的精度.本方法适用于强非线性,非保守系统.     

核属于H函数类的多维积分方程近似解直接方法的优化

本文我们确定了核属于 H函数类的多维第二类 Fredholm积分方程类在自适直接方法意义下的最优近似解的精确阶估计 ,并给出了最优算法 .     

非光滑第二类Fredholm方程Collocation解的迭代──校正方法

本文对于一类具非光滑核第二类Ｆｒｅｄｈｏｌｍ方程的Ｃｏｌｌｏｃａｔｉｏｎ解提出一种迭代─校正方法，使得在计算量增加很少的前提下，成倍提高逼近解精度，并将此方法用于平面多角域上边界积分方程，从而给出其相应微分方程逼近解的高精度算法。此方法还是一种自适应方法。