A necessary and sufficient condition is obtained for the generalized eigenfunction systems of 2 ×2 operator matrices to be a block Schauder basis of some Hilbert space, which offers a mathematical foundation of s...A necessary and sufficient condition is obtained for the generalized eigenfunction systems of 2 ×2 operator matrices to be a block Schauder basis of some Hilbert space, which offers a mathematical foundation of solving symplectic elasticity problems by using the method of separation of variables. Moreover, the theoretical result is applied to two plane elasticity problems via the separable Hamiltonian systems.展开更多
A block representation of the BLU factorization for block tridiagonal matrices is presented. Some properties on the factors obtained in the course of the factorization are studied. Simpler expressions for errors incur...A block representation of the BLU factorization for block tridiagonal matrices is presented. Some properties on the factors obtained in the course of the factorization are studied. Simpler expressions for errors incurred at the process of the factorization for block tridiagonal matrices are considered.展开更多
In this paper, we present a unified approach to decomposing a special class of block tridiagonal matrices <i>K</i> (<i>α</i> ,<i>β</i> ) into block diagonal matrices using similar...In this paper, we present a unified approach to decomposing a special class of block tridiagonal matrices <i>K</i> (<i>α</i> ,<i>β</i> ) into block diagonal matrices using similarity transformations. The matrices <i>K</i> (<i>α</i> ,<i>β</i> )∈ <i>R</i><sup><i>pq</i>× <i>pq</i></sup> are of the form <i>K</i> (<i>α</i> ,<i>β</i> = block-tridiag[<i>β B</i>,<i>A</i>,<i>α B</i>] for three special pairs of (<i>α</i> ,<i>β</i> ): <i>K</i> (1,1), <i>K</i> (1,2) and <i>K</i> (2,2) , where the matrices <i>A</i> and <i>B</i>, <i>A</i>, <i>B</i>∈ <i>R</i><sup><i>p</i>× <i>q</i></sup> , are general square matrices. The decomposed block diagonal matrices <img src="Edit_00717830-3b3b-4856-8ecd-a9db983fef19.png" width="15" height="15" alt="" />(<i>α</i> ,<i>β</i> ) for the three cases are all of the form: <img src="Edit_71ffcd27-6acc-4922-b5e2-f4be15b9b8dc.png" width="15" height="15" alt="" />(<i>α</i> ,<i>β</i> ) = <i>D</i><sub>1</sub> (<i>α</i> ,<i>β</i> ) ⊕ <i>D</i><sub>2</sub> (<i>α</i> ,<i>β</i> ) ⊕---⊕ <i>D</i><sub>q</sub> (<i>α</i> ,<i>β</i> ) , where <i>D<sub>k</sub></i> (<i>α</i> ,<i>β</i> ) = <i>A</i>+ 2cos ( <i>θ<sub>k</sub></i> (<i>α</i> ,<i>β</i> )) <i>B</i>, in which <i>θ<sub>k</sub></i> (<i>α</i> ,<i>β</i> ) , k = 1,2, --- q , depend on the values of <i>α</i> and <i>β</i>. Our decomposition method is closely related to the classical fast Poisson solver using Fourier analysis. Unlike the fast Poisson solver, our approach decomposes <i>K</i> (<i>α</i> ,<i>β</i> ) into <i>q</i> diagonal blocks, instead of <i>p</i> blocks. Furthermore, our proposed approach does not require matrices <i>A</i> and <i>B</i> to be symmetric and commute, and employs only the eigenvectors of the tridiagonal matrix <i>T</i> (<i>α</i> ,<i>β</i> ) = tridiag[<i>β b</i>, <i>a</i>,<i>αb</i>] in a block form, where <i>a</i> and <i>b</i> are scalars. The transformation matrices, their inverses, and the explicit form of the decomposed block diagonal matrices are derived in this paper. Numerical examples and experiments are also presented to demonstrate the validity and usefulness of the approach. Due to the decoupled nature of the decomposed matrices, this approach lends itself to parallel and distributed computations for solving both linear systems and eigenvalue problems using multiprocessors.展开更多
A fast algorithm FBTQ is presented which computes the QR factorization a block-Toeplitz matrix A (A∈R) in O(mns3) multiplications. We prove that the QR decomposition of A and the inverse Cholesky decomposition can be...A fast algorithm FBTQ is presented which computes the QR factorization a block-Toeplitz matrix A (A∈R) in O(mns3) multiplications. We prove that the QR decomposition of A and the inverse Cholesky decomposition can be computed in parallel using the sametransformation.We also prove that some kind of Toeplltz-block matrices can he transformed into the corresponding block-Toeplitz matrices.展开更多
In this article, the computation of μ-values known as Structured Singular Values SSV for the companion matrices is presented. The comparison of lower bounds with the well-known MATLAB routine mussv is investigated. T...In this article, the computation of μ-values known as Structured Singular Values SSV for the companion matrices is presented. The comparison of lower bounds with the well-known MATLAB routine mussv is investigated. The Structured Singular Values provides important tools to analyze the stability and instability analysis of closed loop time invariant systems in the linear control theory as well as in structured eigenvalue perturbation theory.展开更多
Block matrices associated with discrete Trigonometric transforms （DTT＇s） arise in the mathematical modelling of several applications of wave propagation theory including discretizations of scatterers and radiators ...Block matrices associated with discrete Trigonometric transforms （DTT＇s） arise in the mathematical modelling of several applications of wave propagation theory including discretizations of scatterers and radiators with the Method of Moments, the Boundary Element Method, and the Method of Auxiliary Sources. The DTT＇s are represented by the Fourier, Hartley, Cosine, and Sine matrices, which are unitary and offer simultaneous diagonalizations of specific matrix algebras. The main tool for the investigation of the aforementioned wave applications is the efficient inversion of such types of block matrices. To this direction, in this paper we develop an efficient algorithm for the inversion of matrices with U-diagonalizable blocks （U a fixed unitary matrix） by utilizing the U- diagonalization of each block and subsequently a similarity transformation procedure. We determine the developed method＇s computational complexity and point out its high efficiency compared to standard inversion techniques. An implementation of the algorithm in Matlab is given. Several numerical results are presented demonstrating the CPU-time efficiency and accuracy for ill-conditioned matrices of the method. The investigated matrices stem from real-world wave propagation applications.展开更多
基金supported by the National Natural Science Foundation of China (Grant Nos. 11361034 and 11371185)the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20111501110001)the Natural Science Foundation of Inner Mongolia, China (Grant Nos. 2012MS0105 and 2013ZD01 )
文摘A necessary and sufficient condition is obtained for the generalized eigenfunction systems of 2 ×2 operator matrices to be a block Schauder basis of some Hilbert space, which offers a mathematical foundation of solving symplectic elasticity problems by using the method of separation of variables. Moreover, the theoretical result is applied to two plane elasticity problems via the separable Hamiltonian systems.
文摘A block representation of the BLU factorization for block tridiagonal matrices is presented. Some properties on the factors obtained in the course of the factorization are studied. Simpler expressions for errors incurred at the process of the factorization for block tridiagonal matrices are considered.
文摘In this paper, we present a unified approach to decomposing a special class of block tridiagonal matrices <i>K</i> (<i>α</i> ,<i>β</i> ) into block diagonal matrices using similarity transformations. The matrices <i>K</i> (<i>α</i> ,<i>β</i> )∈ <i>R</i><sup><i>pq</i>× <i>pq</i></sup> are of the form <i>K</i> (<i>α</i> ,<i>β</i> = block-tridiag[<i>β B</i>,<i>A</i>,<i>α B</i>] for three special pairs of (<i>α</i> ,<i>β</i> ): <i>K</i> (1,1), <i>K</i> (1,2) and <i>K</i> (2,2) , where the matrices <i>A</i> and <i>B</i>, <i>A</i>, <i>B</i>∈ <i>R</i><sup><i>p</i>× <i>q</i></sup> , are general square matrices. The decomposed block diagonal matrices <img src="Edit_00717830-3b3b-4856-8ecd-a9db983fef19.png" width="15" height="15" alt="" />(<i>α</i> ,<i>β</i> ) for the three cases are all of the form: <img src="Edit_71ffcd27-6acc-4922-b5e2-f4be15b9b8dc.png" width="15" height="15" alt="" />(<i>α</i> ,<i>β</i> ) = <i>D</i><sub>1</sub> (<i>α</i> ,<i>β</i> ) ⊕ <i>D</i><sub>2</sub> (<i>α</i> ,<i>β</i> ) ⊕---⊕ <i>D</i><sub>q</sub> (<i>α</i> ,<i>β</i> ) , where <i>D<sub>k</sub></i> (<i>α</i> ,<i>β</i> ) = <i>A</i>+ 2cos ( <i>θ<sub>k</sub></i> (<i>α</i> ,<i>β</i> )) <i>B</i>, in which <i>θ<sub>k</sub></i> (<i>α</i> ,<i>β</i> ) , k = 1,2, --- q , depend on the values of <i>α</i> and <i>β</i>. Our decomposition method is closely related to the classical fast Poisson solver using Fourier analysis. Unlike the fast Poisson solver, our approach decomposes <i>K</i> (<i>α</i> ,<i>β</i> ) into <i>q</i> diagonal blocks, instead of <i>p</i> blocks. Furthermore, our proposed approach does not require matrices <i>A</i> and <i>B</i> to be symmetric and commute, and employs only the eigenvectors of the tridiagonal matrix <i>T</i> (<i>α</i> ,<i>β</i> ) = tridiag[<i>β b</i>, <i>a</i>,<i>αb</i>] in a block form, where <i>a</i> and <i>b</i> are scalars. The transformation matrices, their inverses, and the explicit form of the decomposed block diagonal matrices are derived in this paper. Numerical examples and experiments are also presented to demonstrate the validity and usefulness of the approach. Due to the decoupled nature of the decomposed matrices, this approach lends itself to parallel and distributed computations for solving both linear systems and eigenvalue problems using multiprocessors.
文摘A fast algorithm FBTQ is presented which computes the QR factorization a block-Toeplitz matrix A (A∈R) in O(mns3) multiplications. We prove that the QR decomposition of A and the inverse Cholesky decomposition can be computed in parallel using the sametransformation.We also prove that some kind of Toeplltz-block matrices can he transformed into the corresponding block-Toeplitz matrices.
文摘In this article, the computation of μ-values known as Structured Singular Values SSV for the companion matrices is presented. The comparison of lower bounds with the well-known MATLAB routine mussv is investigated. The Structured Singular Values provides important tools to analyze the stability and instability analysis of closed loop time invariant systems in the linear control theory as well as in structured eigenvalue perturbation theory.
文摘Block matrices associated with discrete Trigonometric transforms （DTT＇s） arise in the mathematical modelling of several applications of wave propagation theory including discretizations of scatterers and radiators with the Method of Moments, the Boundary Element Method, and the Method of Auxiliary Sources. The DTT＇s are represented by the Fourier, Hartley, Cosine, and Sine matrices, which are unitary and offer simultaneous diagonalizations of specific matrix algebras. The main tool for the investigation of the aforementioned wave applications is the efficient inversion of such types of block matrices. To this direction, in this paper we develop an efficient algorithm for the inversion of matrices with U-diagonalizable blocks （U a fixed unitary matrix） by utilizing the U- diagonalization of each block and subsequently a similarity transformation procedure. We determine the developed method＇s computational complexity and point out its high efficiency compared to standard inversion techniques. An implementation of the algorithm in Matlab is given. Several numerical results are presented demonstrating the CPU-time efficiency and accuracy for ill-conditioned matrices of the method. The investigated matrices stem from real-world wave propagation applications.
