Search

• Conservative discontinuous Galerkin schemes for the nonlinear Serre equations. • The well-balanced property of the scheme for the equations with non-flat bottoms. • Hamiltonian conservative scheme based on the equations in non-conservative form. • Arbitrary high order accuracy and capability of the schemes are verified numerically. In this paper, we develop three conservative discontinuous Galerkin (DG) schemes for the one-dimensional nonlinear dispersive Serre equations, including two conserved schemes for the equations in conservative form and a Hamiltonian conserved scheme for the equations in non-conservative form. One of the schemes owns the well-balanced property via constructing a high order approximation to the source term for the Serre equations with a non-flat bottom topography. By virtue of the Hamiltonian structure of the Serre equations, we introduce an Hamiltonian invariant and then develop a DG scheme which can preserve the discrete version of such an invariant. Numerical experiments in different cases are performed to verify the accuracy and capability of these DG schemes for solving the Serre equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the quaternion matrix equation X − A X ˆ B = C , and study its minimal norm least squares solution, j-self-conjugate least-squares solution and anti-j-self-conjugate least-squares solution. By the real representation matrices of quaternion matrices, their particular structure and the properties of Frobenius norm, we convert above least-squares problems into corresponding problems of real matrix equations. The final results of the expressions only involve real matrices, and thus, the corresponding algorithms only involve real operations. Compared with the existing results, they are more convenient and efficient, which are also illustrated by the last two numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we consider the split quaternion matrix equation X − A f (X) B = C , f (X) ∈ { X , X H , X i H , X j H X k H } . The H representation method has the characteristics of transforming a matrix with a special structure into a column vector with independent elements. By using the real representation of split quaternion matrices, H representation method, the Kronecker product of matrices and the Moore-Penrose generalized inverse, we convert the split quaternion matrix equation into the real matrix equation, and derive the sufficient and necessary conditions and the general solution expressions for the (skew) bisymmetric solution of the original equation. Moreover, we provide numerical algorithms and illustrate the efficiency of our method by two numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, by using the special structure of the real representation of quaternion matrices, the properties of the Moore–Penrose inverse of quaternion matrices and the Kronecker product of matrices, we study least-squares bihermitian and skew bihermitian solutions of the quaternion matrix equation AXB = C, respectively. First we study the special structures of quaternion bihermitian and skew bihermitian matrices. Then the problem of solving the least-squares bihermitian and skew bihermitian solutions of the quaternion matrix equation AXB = C can be transformed into particular real linear system using these special structures. We deduce the form of all least-squares bihermitian and skew bihermitian solutions of the quaternion matrix equation AXB = C and propose real structure-preserving algorithms to compute the minimal norm least-squares bihermitian and skew bihermitian solution. All the computation process only involve real arithmetic. Numerical examples are provided to verify the effectiveness of our algorithms. Particularly, we also propose the conditions and computation method of bihermitian and skew bihermitian solutions of quaternion matrix equation AXB = C. [ABSTRACT FROM AUTHOR]

Let η H Q k × k and η A Q k × k represent the sets of all k × k η-Hermitian quaternion matrices and η-anti-Hermitian quaternion matrices, respectively. On the basis of the real representation matrix of a quaternion matrix and its particular structure, we convert the least squares problem of the quaternion matrix equation AXB + CY D = E over X ∈ η H Q n × n , Y ∈ η A Q k × k into the corresponding problem of the real matrix equation over free variables, and then we establish its unique minimal norm least squares solution. Our resulting expressions are expressed only by real matrices, and the algorithm only includes real operations. Consequently, they are very simple and convenient. Compared with the existing method [S.F. Yuan, Q.W. Wang, and X. Zhang, Least-squares problem for the quaternion matrix equation AXB + CYD = E over different constrained matrices, Int. J. Comput. Math. 90 (2013), pp. 565–576], the final two examples show that our method is more efficient and superior. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose an efficient method for special least squares solution of the complex matrix equation (A X B , C X D) = (E , F). By using the real representation matrices of complex matrices, the particular structure of the real representation matrices, the Moore–Penrose generalized inverse and the Kronecker product, we obtain the explicit expression of the minimal norm least squares Hermitian solution of the complex matrix equation (A X B , C X D) = (E , F) , which was studied by a product of matrices and vectors in Wang et al. (2016). Our resulting formulas only involve real matrices, and the corresponding algorithm only performs real arithmetic. Therefore our proposed method is more effective and portable. Finally, we give three numerical examples to illustrate the effectiveness of our proposed method. [ABSTRACT FROM AUTHOR]

In this paper, by using the real representations of quaternion matrices, the particular structure of the real representations of quaternion matrices, the Kronecker product of matrices and the Moore–Penrose generalized inverse, we obtain the expressions of the minimal norm least squares solution, the pure imaginary least squares solution, and the real least squares solution for the quaternion matrix equation A X B + C X D = E , respectively. Our resulting formulas only involve real matrices, and therefore are simpler than those reported in Yuan (2014). The corresponding algorithms only perform real arithmetic which also consider the particular structure of the real representations of quaternion matrices, and therefore are very efficient and portable. Numerical examples are provided to illustrate the efficiency of our algorithms. [ABSTRACT FROM AUTHOR]