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A fast structure-preserving method for computing the singular value decomposition of quaternion matrices

APPLIED MATHEMATICS AND COMPUTATION 2014年 235卷 157-167页

作者： Li, Ying Wei, Musheng Zhang, Fengxia Zhao, Jianli Liaocheng Univ Coll Math Sci Shandong 252000 Peoples R China Shanghai Normal Univ Coll Math & Sci Shanghai 200234 Peoples R China

In this paper we propose a fast structure-preserving algorithm for computing the singular value decomposition of quaternion matrices. The algorithm is based on the structure-preserving bidiagonalization of the real counterpart for quaternion matrices by applying orthogonal JRS-symplectic matrices. The algorithm is efficient and numerically stable. (C) 2014 Elsevier Inc. All rights reserved.

关键词： Quaternion matrix Quaternion singular value decomposition structure-preserving algorithm

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Local structure-preserving algorithms for the "good'' Boussinesq equation

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JOURNAL OF COMPUTATIONAL PHYSICS 2013年 239卷 72-89页

作者： Cai, Jiaxiang Wang, Yushun Huaiyin Normal Univ Sch Math Sci Huaian 223300 Jiangsu Peoples R China Nanjing Normal Univ Sch Math Sci Jiangsu Key Lab NSLSCS Nanjing 210046 Jiangsu Peoples R China

In this paper, we derive a series of local structure-preserving algorithms for the "good'' Boussinesq equation, including multisymplectic geometric structure-preserving algorithms, local energy-preserving algorithms and local momentum-preserving algorithms. The outstanding advantage of the proposed algorithms is that they conserve these local structures in any time-space region exactly. For example, the proposed local energy-preserving algorithms preserve the local energy conservation law in any local domain. Therefore, the local structure-preserving algorithms overcome the shortage of global structure-preserving algorithms on the boundary conditions. Especially, with suitable boundary conditions such as periodic or homogeneous boundary conditions, the local structure-preserving algorithms will be global structure-preserving algorithms. Numerical results verify the theoretical analysis. (C) 2013 Elsevier Inc. All rights reserved.

关键词： Boussinesq equation structure-preserving algorithm Multisymplectic Energy Momentum

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structure-preserving algorithms for autonomous Birkhoffian systems

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Journal of Beijing Institute of Technology 2012年第1期21卷 1-7页

作者：孔新雷吴惠彬梅凤翔 School of Science Beijing Institute of Technology School of Aerospace Engineering Beijing Institute of Technology

The Pfaff-Birkhoff variational principle is discretized, and based on the discrete variational principle the discrete Birkhoffian equations are obtained. Taking the discrete equations as an algorithm, the corresponding discrete flow is proved to be symplectic. That means the algorithm preserves the symplectic structure of Birkhofflan systems. Finally, simulation results of the given example indicate that structure-preserving algorithms have great advantage in stability and energy conserving.

关键词： Birkhoffian system discrete Birkhoffian equations structure-preserving algorithm

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Efficient energy-preserving integrators for oscillatory Hamiltonian systems

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JOURNAL OF COMPUTATIONAL PHYSICS 2013年 235卷 587-605页

作者： Wu, Xinyuan Wang, Bin Shi, Wei Nanjing Univ Dept Math Nanjing 210093 Jiangsu Peoples R China Nanjing Univ State Key Lab Novel Software Technol Nanjing 210093 Jiangsu Peoples R China

In this paper, we focus our attention on deriving and analyzing an efficient energy-preserving formula for the system of nonlinear oscillatory or highly oscillatory second-order differential equations q ''(t) + Mq(t) - f(q(t)), where M is a symmetric positive semi-definite matrix with parallel to M parallel to >> 1 and f(q) = -del U-q(q) is the negative gradient of a real-valued function U(q). This system is a Hamiltonian system with the Hamiltonian H(p, q) = 1/2p(T)p + 1/2q(T)Mq+U(q). The energy-preserving formula exactly preserves the Hamiltonian. We analyze in detail the properties of the energy-preserving formula and propose new efficient energy-preserving integrators in the sense of numerical implementation. The convergence analysis of the fixed-point iteration is presented for the implicit integrators proposed in this paper. It is shown that the convergence of implicit Average Vector Field methods is dependent on parallel to M parallel to, whereas the convergence of the new energy-preserving integrators is independent of parallel to M parallel to. The Fermi-Pasta-Ulam problem and the sine-Gordon equation are carried out numerically to show the competence and efficiency of the novel integrators in comparison with the well-known Average Vector Field methods in the scientific literature. (C) 2012 Elsevier Inc. All rights reserved.

关键词： Hamiltonian system Energy-preserving formula structure-preserving algorithm Oscillatory differential equation Average Vector Field formula Fermi-Pasta-Ulam problem Sine-Gordon equation

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A new structure-preserving method for quaternion Hermitian eigenvalue problems

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2013年第1期239卷 12-24页

作者： Jia, Zhigang Wei, Musheng Ling, Sitao Shanghai Normal Univ Coll Math & Sci Shanghai 200234 Peoples R China Jiangsu Normal Univ Sch Math Sci Jiangsu 221116 Peoples R China China Univ Min & Technol Coll Sci Jiangsu 221116 Peoples R China

In this paper we propose a novel structure-preserving algorithm for solving the right eigenvalue problem of quaternion Hermitian matrices. The algorithm is based on the structure-preserving tridiagonalization of the real counterpart for quaternion Hermitian matrices by applying orthogonal JRS-symplectic matrices. The algorithm is numerically stable because we use orthogonal transformations;the algorithm is very efficient, it costs about a quarter arithmetical operations, and a quarter to one-eighth CPU times, comparing with standard general-purpose algorithms. Numerical experiments are provided to demonstrate the efficiency of the structure-preserving algorithm. (C) 2012 Elsevier B.V. All rights reserved.

关键词： Quaternion Hermitian operator Quaternionic right eigenvalue problem structure-preserving algorithm

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A structure-preserving method for the quaternion LU decomposition in quaternionic quantum theory

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COMPUTER PHYSICS COMMUNICATIONS 2013年第9期184卷 2182-2186页

作者： Wang, Minghui Ma, Wenhao Qingdao Univ Sci & Technol Dept Math Qingdao 266061 Peoples R China

In this paper, for the first time, the structure-preserving Gauss transformation is defined. Then by means of its real representation matrix, we present a novel structure-preserving algorithm for the LU decomposition of a quaternion matrix. Numerical experiments show that the structure-preserving algorithm is better than that in the newest quaternion toolbox for matlab (QTFM). (C) 2013 Elsevier B.V. All rights reserved.

关键词： Quaternion matrix LU decomposition structure-preserving algorithm

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Conformal conservation laws and geometric integration for damped Hamiltonian PDEs

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JOURNAL OF COMPUTATIONAL PHYSICS 2013年第1期232卷 214-233页

作者： Moore, Brian E. Norena, Laura Schober, Constance M. Univ Cent Florida Dept Math Orlando FL 32816 USA

Conformal conservation laws are defined and derived for a class of multi-symplectic equations with added dissipation. In particular, the conservation laws of energy and momentum are considered, along with those that arise from linear symmetries. Numerical methods that preserve these conformal conservation laws are presented in detail, providing a framework for proving a numerical method exactly preserves the dissipative properties considered. The conformal methods are compared analytically and numerically to standard conservative methods, which includes a thorough inspection of numerical solution behavior for linear equations. Damped Klein-Gordon and sine-Gordon equations, and a damped nonlinear Schrodinger equation, are used as examples to demonstrate the results. (C) 2012 Elsevier Inc. All rights reserved.

关键词： Multi-symplectic PDE Linear dissipation structure-preserving algorithm Preissman box scheme Discrete gradient methods

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ON A NONLINEAR MATRIX EQUATION ARISING IN NANO RESEARCH

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2012年第1期33卷 235-262页

作者： Guo, Chun-Hua Kuo, Yueh-Cheng Lin, Wen-Wei Univ Regina Dept Math & Stat Regina SK S4S 0A2 Canada Natl Univ Kaohsiung Dept Appl Math Kaohsiung 811 Taiwan Natl Taiwan Univ Dept Math Taipei 106 Taiwan Natl Chiao Tung Univ Ctr Math Modelling & Sci Comp Hsinchu 300 Taiwan

The matrix equation X + A (vertical bar) X(-1)A - Q arises in Green's function calculations in nano research, where A is a real square matrix and Q is a real symmetric matrix dependent on a parameter and is usually indefinite. In practice one is mainly interested in those values of the parameter for which the matrix equation has no stabilizing solutions. The solution of interest in this case is a special weakly stabilizing complex symmetric solution X-*, which is the limit of the unique stabilizing solution X-eta of the perturbed equation X + A(inverted perpendicular) X(-1)A = Q + i eta I, as eta -> 0(+). It has been shown that a doubling algorithm can be used to compute X-eta efficiently even for very small values of eta, thus providing good approximations to X-*. It has been observed by nano scientists that a modified fixed-point method can sometimes be quite useful, particularly for computing X-eta for many different values of the parameter. We provide a rigorous analysis of this modified fixed-point method and its variant and of their generalizations. We also show that the imaginary part X-I of the matrix X-* is positive semidefinite and we determine the rank of X-I in terms of the number of unimodular eigenvalues of the quadratic pencil lambda(2)A(inverted perpendicular) - lambda Q + A. Finally we present a new structure-preserving algorithm that is applied directly on the equation X + A(inverted perpendicular) X(-1)A = Q. In doing so, we work with real arithmetic most of the time.

关键词： nonlinear matrix equation complex symmetric solution weakly stabilizing solution fixed-point iteration structure-preserving algorithm Green's function

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Numerical solution of nonlinear matrix equations arising from Green's function calculations in nano research

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2012年第17期236卷 4166-4180页

作者： Guo, Chun-Hua Kuo, Yueh-Cheng Lin, Wen-Wei Univ Regina Dept Math & Stat Regina SK S4S 0A2 Canada Natl Univ Kaohsiung Dept Appl Math Kaohsiung 811 Taiwan Natl Chiao Tung Univ Dept Appl Math Hsinchu 300 Taiwan

The Green's function approach for treating quantum transport in nano devices requires the solution of nonlinear matrix equations of the form X + (C* + i eta D*)X-1(C + i eta D) = R+i eta P. where R and P are Hermitian, P + lambda D* + lambda D-1 is positive definite for all lambda on the unit circle, and eta -> 0(+). For each fixed eta > 0, we show that the required solution is the unique stabilizing solution X-eta. Then X-center dot = lim(eta -> 0+) X-eta is a particular weakly stabilizing solution of the matrix equation X + C*X-1C = R. In nano applications, the matrices R and C are dependent on a parameter, which is the system energy E. In practice one is mainly interested in those values of g for which the equation X + C*X-1C = R has no stabilizing solutions or, equivalently, the quadratic matrix polynomial P(lambda) = lambda C-2* - lambda R + C has eigenvalues on the unit circle. We point out that a doubling algorithm can be used to compute X-eta efficiently even for very small values eta, thus providing good approximations to X-*. We also explain how the solution X-* can be computed directly using subspace methods such as the QZ algorithm by determining which unimodular eigenvalues of P(lambda) should be included in the computation. In some applications the matrices C, D, R, P have very special sparsity structures. We show how these special structures can be exploited to drastically reduce the complexity of the doubling algorithm for computing X-eta. (C) 2012 Elsevier B.V. All rights reserved.

关键词： Nonlinear matrix equation Weakly stabilizing solution structure-preserving algorithm Green's function

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A structure-preserving iteration method of model updating based on matrix approximation theory

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COMPUTATIONAL & APPLIED MATHEMATICS 2010年第2期29卷 173-194页

作者： Xie, Dongxiu Beijing Informat Sci & Technol Univ Sch Sci Beijing 100192 Peoples R China

Some theories and a method are discussed on updating a generalized centrosymmetric model. It gives a generalized centrosymmetric modified solution with partial prescribed least square spectra constraints. The emphasis is given on exploiting structure-preserving algorithm based on matrix approximation theory. A perturbation theory for the modified solution is established. The convergence of an iterative solution is investigated. Illustrative examples are provided.

关键词： structure-preserving algorithm generalized centrosymmetric matrix model updating perturbation analysis

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