2003年, 第21卷, 第2期　刊出日期：2003-03-15

  

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  EXPLICIT BOUNDS OF EIGENVALUES FOR STIFFNESS MATRICES BY QUADRATIC HIERARCHICAL BASIS METHOD)

  Sang Dong KIM+

  Journal of Computational Mathematics. 2003, 21(2): 113-124.

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  The bounds for the eigenvalues of the stiffness matrices in the finite element discretization corresponding to Lu := - u" with zero boundary conditions by quadratic hierarchical basis are shown explicitly. The condition number of the resulting system begaves like $O(\frac{1}{h})$ where h is the mesh size. We also analyze a main diagonal preconditioner of the stiffness matrix which reduces the condition number of the preconditioned system to O(1).
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  AD GALERKIN ANALYSIS FOR NONLINEAR PSEUDO-HYPERBOLIC EQUATIONS

  Xia CUI

  Journal of Computational Mathematics. 2003, 21(2): 125-134.

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  AD (Alternating direction) Galerkin schemes for d-dimensional nonlinear pseudo-hyperbolic equations are studied. By using patch approximation technique, AD procedure is realized, and calculation work is simplified. By using Galerkin approach, highly computational accuracy is kept. By using various priori estimate techniques for differential equations, difficulty coming from non-linearity is treated, and optimal H1 and L2 convergence properties are demonstrated. Moreover, although all the existed AD Galerkin schemes using patch approximation are limited to have only one order accuracy in time increment, yet the schemes formulated in this paper have second order accuracy in it. This implies an essential avancement in AD Galerkin analysis.
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  VARIATIONAL INTEGRATORS FOR HIGHER ORDER DIFFERENTIAL EQUATIONS

  Ya Juan SUN(1),Meng Zhao Qin(2)

  Journal of Computational Mathematics. 2003, 21(2): 135-144.

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  We analyze three one parameter families of approximations and show that they are symplectic in Lagrangian sence and can be related to symplectic schemes in Hamiltonian sense by different symplectic mappings. We also give a direct generalization of Veselov variational principle for construction of scheme of higher order differential equations. At last, we present numerical experiments.
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  UNIFORM CONVERGENCE OF HERMITE INTERPOLATION OF HIGHER ORDER

  Yong Ping FENG(1),Wen Ming HE(2)

  Journal of Computational Mathematics. 2003, 21(2): 145-156.

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  In this paper the uniform convergence of Hermite-Fejer interpolation and Griinwald type theorem of higher order on an arbitrary system of nodes are presented.
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  ON THE GENERALIZED INVERSE NEVILLE-TYPE MATRIX-VALUED RATIONAL INTERPOLANTS

  Zhi Bing CHEN

  Journal of Computational Mathematics. 2003, 21(2): 157-166.

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  A new kind of matrix-valued rational interpolants is recursively established by means of generalized Samelson inverse for matrices, with scalar numerator and matrix-valued denominator. In this respect, it is essentially different from that of the previous works [7,9], where the matrix-valued rational interpolants is in Thiele-type continued fraction form with matrix-valued numerator and scalar denominator. For both univariate and bivariate cases, sufficient conditions for existence, characterisation and uniqueness in some sense are proved respectively, and an error formula for the univariate interpolating function is also given. The results obtained in this paper are illustrated with some numerical examples.
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  LEAST-SQUARES SOLUTIONS OF X~TAX = B OVER POSITIVE SEMIDEFINITE MATRIXES A

  Dong Xiu XIE(1),Lei ZHANG(2)

  Journal of Computational Mathematics. 2003, 21(2): 167-174.

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  This paper is mainly concerned with solving the following two problems: \par Problem Ⅰ. Given$X\in R^{n\times m},B\in R^{m\times m}$.Find $A\in P_n$such that $$\|X^TAX-B\|_F=\min,$$ where $P_n=\{A\in R^{n\times n}| x^TAx\geq 0, \forall\,x\in R^n\}$.\par Problem Ⅱ. Given $\widetilde{A}\in R^{n\times n}.$ Find $\widetilde{A}\in S_E$such that $$\|\widetilde{A}-\hat{A}\|_F=\min_{A\in S_E}\|\widetilde{A}-A\|_F,$$ where $\|\cdot\|_F$ is Frobenius norm, and $S_E$ denotes the solution set of Problem I.\par The general solution of problem Ihas been given. It is proved that there esists a unique solution for Problem II. The espression of this solution for corresponding Problem II for fome special case will be derived.
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  LEAST-SQUARES SOLUTION OF AXB = DOVER SYMMETRIC POSITIVE SEMIDEFINITE MATRICES X

  An Ping LIAO(1),Zhong Zhi BAI(2)

  Journal of Computational Mathematics. 2003, 21(2): 175-182.

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  Least-squares solution of AXB = D with respect to symmetric positive semidefinite matrix X is considered. By making use of the generalized singular value decomposition, we derive general analytic formulas, and present necessary and sufficient conditions for guaranteeing the existence of the solution. By applying MATLAB 5.2, we give some numerical examples to show the feasibility and accuracy of this construction technique in the finite precision arithmetic.
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  GENERALIZED NEKRASOV MATRICES AND APPLICATIONS

  Ming Xian PANG,Zhu Xiang LI

  Journal of Computational Mathematics. 2003, 21(2): 183-188.

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  In this paper, the concept of generalized Nekrasov matrices is introduced, some properties of these matrices are discussed, obtained equivalent representation of generalized diagonally dominant matrices.
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  FOURIER-CHEBYSHEV COEFFICIENTS AND GAUSS-TURAN QUADRATURE WITH CHEBYSHEV WEIGHT

  Shi Jun YANG,Xing Hua WANG

  Journal of Computational Mathematics. 2003, 21(2): 189-194.

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  The main purpose of this paper is to derive an explicit expression for Fourier-Chebyshev coefficient $A_{kn}(f)=\frac{\displaystyle 2}{\displaystyle\pi}\int_{-1}^1f(x)T_{kn}(x)\frac{\displaystyle dx}{\displaystyle \sqrt{1-x^2}},k,n\in N_0$,which is initiated by L.Gori andC.A.Micchelli.
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  On the United Theory of the Family of Euler-Halley Type Methods with Cubical Convergence in Banach Spaces

  Xing Hua WANG(1),Chong LI(2)

  Journal of Computational Mathematics. 2003, 21(2): 195-200.

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  The convergence problem of the family of Euler-Halley methods is considered under the Lipschitz condition with the L-average, and a united convergence theory with its applications is presented.
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  A NOTE ON THE BACKWARD ERRORS FOR INVERSE EIGENVALUE PROBLEMS

  Xin Guo LIU,Zheng Jian BAI

  Journal of Computational Mathematics. 2003, 21(2): 201-206.

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  In this note, we consider the backward errors for more general inverse eigenvalue problems by extending Sun's approach. The optimal backward errors are defined for diagonal-ization matrix inverse eigenvalue problem with respect to an approximate solution, and the upper and lower bounds are derived for the optimal backward errors. The results may be useful for testing the stability of practical algorithms.
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  A TRUST-REGION ALGORITHM FOR NONLINEAR INEQUALITY CONSTRAINED OPTIMIZATION

  Xiao Jiao TONG(1),Shu Zi ZHOU(2)

  Journal of Computational Mathematics. 2003, 21(2): 207-220.

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  This paper presents a new trust-region algorithm for n-dimension nonlinear optimization subject to m nonlinear inequality constraints. Equivalent KKT conditions are derived, which is the basis for constructing the new algorithm. Global convergence of the trial steps, local quadratic convergence theorem is proved for nondegenerate minimizer point.Numerical experiment is presinted to show the effectiveness of our qpproach.
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  A NEW FAMILY OF TRUST REGION ALGORITHMS FOR UNCONSTRAINED OPTIMIZATION

  Yu Hong DAI,Da Chuan XU

  Journal of Computational Mathematics. 2003, 21(2): 221-228.

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  Trust region (TR) algorithms are a class of recently developed algorithms for nonlinear optimization. A new family of TR algorithms for unconstrained optimization, which is the extension of the usual TR method, is presented in this paper. When the objective function is bounded below and continuously differentiable, and the norm of the Hese approximations increases at most linearly weth the iteration number, we prove the global convergence of the algorithms. Limited numerical results are reported, which indicate that our new TR algorithm is competitive.
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  A SELF-ADAPTIVE TRUST REGION ALGORITHM

  Long HEI

  Journal of Computational Mathematics. 2003, 21(2): 229-236.

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  In this paper we propose a self-adaptive trust region algorithm. The trust region radius is updated at a variable rate according to the ratio between the actual reduction and the predicted reduction of the objective function, rather than by simply enlarging or reducing the original trust region radius at a constant rate. Weshow that this new algorithm preserves the strong convergence property of traditional trust region methods. Numerical results are also presented.
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  A NONMONOTONE TRUST REGION ALGORITHM FOR NONLINEAR OPTIMIZATION SUBJECT TO GENERAL CONSTRAINTS

  Hong Chao ZHANG

  Journal of Computational Mathematics. 2003, 21(2): 237-246.

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  In this paper we present a nonmonotone trust region algorithm for general nonlinear constrained optimization problems. The main idea of this paper is to combine Yuan's technique[1] with a nonmonotone method similar to Ke and Han [2]. This new algorithm may not only keep the robust properties of the algorithm given by yuan, but also have some advantages led by the nonmonotone technique. Under very mild conditions, global convergence for the algorithm is given. Numerical experiments demonstrate tre efficiency of the algorithm.
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  A ROBUST SQP METHOD FOR OPTIMIZATION WITH INEQUALITY CONSTRAINTS

  Ju Liang ZHANG(1),Xiang Sun ZHANG(2)

  Journal of Computational Mathematics. 2003, 21(2): 247-256.

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  A new algorithm for inequality constrained optimization is presented, which solves a linear programming subproblem and a quadratic subproblem at each iteration. The algorithm can circumvent the difficulties associated with the possible inconsistency of QP point which satisfies a certain first-order necessary condition even if the original problem is itself infeasible. Under certain condition, some global convergence results are proved and local superlinear convergence results are also lbtained. Preliminary numerical results are reported.