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A rank-one Fitting Method with Descent Direction for Solving Symmetric Nonlinear Equations

International Journal of Communications, Network and System Sciences 2009年第6期2卷 555-561页

作者： Gonglin YUAN Zhongxing WANG Zengxin WEI College of Mathematics and Information Science Guangxi University

In this paper, a rank-one updated method for solving symmetric nonlinear equations is proposed. This method possesses some features: 1) The updated matrix is positive definite whatever line search technique is used;2) The search direction is descent for the norm function;3) The global convergence of the given method is established under reasonable conditions. Numerical results show that the presented method is interesting.

关键词： rank-one Update Global Convergence Nonlinear Equations Descent Direction

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Sparse two-sided rank-one updates for nonlinear equations

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Science China Mathematics 2010年第11期53卷 2907-2915页

作者： CHENG MingHou & DAI YuHong State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

The two-sided rank-one (TR1) update method was introduced by Griewank and Walther (2002) for solving nonlinear equations. It generates dense approximations of the Jacobian and thus is not applicable to large-scale sparse problems. To overcome this difficulty, we propose sparse extensions of the TR1 update and give some convergence analysis. The numerical experiments show that some of our extensions are superior to the TR1 update method. Some convergence analysis is also presented.

关键词： TR1 update Broyden rank-one update sparsity

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Computing the Moore-Penrose Inverse of a Matrix Through Symmetric rank-one Updates

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American Journal of Computational Mathematics 2011年第3期1卷 147-151页

作者： Xuzhou Chen Jun Ji Department of Computer Science Fitchburg State University Department of Mathematics and Statistics Kennesaw State University

This paper presents a recursive procedure to compute the Moore-Penrose inverse of a matrix A. The method is based on the expression for the Moore-Penrose inverse of rank-one modified matrix. The computational complexity of the method is analyzed and a numerical example is included. A variant of the algorithm with lower computational complexity is also proposed. Both algorithms are tested on randomly generated matrices. Numerical performance confirms our theoretic results.

关键词： Finite Recursive Algorithm Moore-Penrose Inverse Symmetric rank-one Update

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THE INVERSE EIGENPROBLEM WITH rank-one UPDATING AND ITS STABLITY

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Numerical Mathematics A Journal of Chinese Universities(English Series) 1994年第1期3卷 87-95页

作者：苏仰锋蒋尔雄

Let ∑, Г be two n-by-n diagonal matrices with σi,γi as their diagonals. For the inverse eigenvalue problem: look for y∈Rn such that Г + yyT is similar to ∑, we prove thatu also the sufficient condition for the solvability of this inverse problem. Its solution (set) is given explicitly. In some case, the problem is unstable. But we prove that the sums of the square of some contigious components keep stable, i.e., small sum keeps small, large sum has a small relative perturbation, see Theorem 3.

关键词： Inverse eigenproblem rank-one updating stablity

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ASYMPTOTIC EIGENVALUE ESTIMATION FOR A CLASS OF STRUCTURED MATRICES

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Annals of Applied Mathematics 2019年第2期35卷 152-158页

作者： Juan Liang Jiangzhou Lai Qiang Niu School of Math.and Statistics Minnan Normal UniversityZhangzhou 363000FujianPR China School of Math.and Computer Science Fuzhou UniversityFuzhou 350108FujianPR China Dept.of Mathematical Sciences Xi'an Jiaotong-Liverpool UniversitySuzhou 215123JiangsuPR China

In this paper we consider eigenvalue asymptotic estimations for a class of structured matrices arising from statistical applications. The asymptotic upper bounds of the largest eigenvalue(λmax) and the sum of squares of eigenvalues(■)are derived. Both these bounds are useful in examining the stability of certain Markov process. Numerical examples are provided to illustrate tightness of the bounds.

关键词： Toeplitz matrix eigenvalue rank-one modification trace

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