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 co...
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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.
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...
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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-preservingalgorithms and local momentum-preservingalgorithms. 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-preservingalgorithms 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.
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 correspondin...
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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.
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) ...
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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.
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 r...
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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.
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 ...
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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.
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 a...
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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.
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 usuall...
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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.
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 Hermi...
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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.
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...
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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.
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