Finite integration methods for isospectral flows

In this paper we consider the approximate computation of isospectral flows based on finite integration methods( FIM) with radial basis functions( RBF) interpolation,a new algorithm is developed. Our method ensures the symmetry of the solutions. Numerical experiments demonstrate that the solutions have higher accuracy by our algorithm than by the second order Runge- Kutta( RK2) method

On the eigenstructure of hermitian Toeplitz matrices with prescribed eigenpairs

In this paper we concern the spectral properties of hermitian Toeplitz matrices. Based on the fact that every centrohermitian matrix can be reduced to a real matrix by a simple similarity transformation, we first consider the eigenstructure of hermitian Toeplitz matrices and then discuss a related inverse eigenproblem. We show that the dimension of the subspace of hermitian Toeplitz matrices with two given eigenvectors is at least two and independent of the size of the matrix, the solution of the inverse hermitian Toeplitz eigenproblem with two given eigenpairs is unique.FC

Computing the square roots of matrices with central symmetry

For computing square roots of a nonsingular matrix A, which are functions of A, two well known fast and stable algorithms, which are based on the Schur decomposition of A, were proposed by Bj¨ork and Hammarling [3], for square roots of general complex matrices, and by Higham [10], for real square roots of real matrices. In this paper we further consider (the computation of) the square roots of matrices with central symmetry. We first investigate the structure of the square roots of these matrices and then develop several algorithms for computing the square roots. We show that our algorithms ensure significant savings in computational costs as compared to the use of standard algorithms for arbitrary matrices.Fundação para a Ciência e a Tecnologia (FCT

The reconstruction of an hermitian Toeplitz matrices with prescribed eigenpairs

In this paper we concern the reconstruction of an hermitian Toeplitz matrices with prescribed eigenpairs. Based on the fact that every centrohermitian matrix can be re- duced to a real matrix by a simple similarity transformation, we rst consider the eigenstructure of hermitian Toeplitz matrices and then discuss a related reconstruction problem. We show that the di- mension of the subspace of hermitian Toeplitz matrices with two given eigenvectors is at least two and independent of the size of the matrix, and the solution of the reconstruction problem of an hermitian Toeplitz matrix with two given eigenpairs is unique.Fundação para a Ciência e a Tecnologia (FCT) - Research Programme POCTINational Natural Science Foundation of China - nº 10771022, 10571012Scienti c Research Foundation for the Returned Overseas Chinese ScholarsState Education Ministry of China - nº 890 (2008)Major Foundation of Educational Committee of Hunan Province - nº 09A002 (2009

Structure-preserving schur methods for computing square roots of real skew-hamiltonian matrices

The contribution in this paper is two-folded. First, a complete characterization is given of the square roots of a real nonsingular skew-Hamiltonian matrix W. Using the known fact that every real skew-Hamiltonian matrix has infinitely many real Hamiltonian square roots, such square roots are described. Second, a structure-exploiting method is proposed for computing square roots of W, skew-Hamiltonian and Hamiltonian square roots. Compared to the standard real Schur method, which ignores the structure, this method requires significantly less arithmetic.National Natural Science Foundations of China, the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry,Fundação para a Ciência e a Tecnologia (FCT

Inverse eigenvalue problems and their associated approximation problems for matrices with J-(Skew) Centrosymmetry

The inverse problems play an important role in MEG reconstructions. In this paper, a partially described inverse eigenvalue problem and an associated optimal approximation problem for J-centrosymmetric matrices are considered respectively. It is shown under which conditions the inverse eigenproblem has a solution. An expression of its general solution is given. In case a solution of the inverse eigenproblem exists, the optimal approximation problem can be solved. The formula of its unique solution is given. Also, the case for J-skew centrosymmetric matrices is considered.Fundação para a Ciência e a Tecnologia (FCT)National Natural Science Foundations of Chin

On inverse eigenvalue problems for block Toeplitz matrices with Toeplitz blocks

We propose an algorithm for solving the inverse eigenvalue problem for real symmetric block Toeplitz matrices with symmetric Toeplitz blocks. It is based upon an algorithm which has been used before by others to solve the inverse eigenvalue problem for general real symmetric matrices and also for Toeplitz matrices. First we expose the structure of the eigenvectors of the so-called generalized centrosymmetric matrices. Then we explore the properties of the eigenvectors to derive an efficient algorithm that is able to deliver a matrix with the required structure and spectrum. We have implemented our ideas in a Matlab code. Numerical results produced with this code are included.Fundação para a Ciência e a Tecnologia (FCT

m-step preconditioners for nonhermitian positive definite Toeplitz systems

It is known that if A is a Toeplitz matrix，then A enjoys a circulant and skew circulant splitting(de— noted by CSCS)，i．e．，A=C+ S with C a circulant matrix and S a skew circulant matrix． Based on the CSCS iteration，we give m-step preconditioners P for certain classes of Toeplitz matrices in this paper．We show that if both C and S are positive definite，then the spectrum of the preconditioned matrix(PA)^* PA are clustered around one for some moderate size ． Experimental results show that the proposed preconditioners perform slightly better than T．Chan’S preconditioners for some moderate size m．info:eu-repo/semantics/publishedVersio

Minimization problems for certain structured matrices

For given $Z,B\in \mathbb{ C}^{n\times k}$, the problem of finding $A\in \mathbb{C}^{n\times n}$, in some prescribed class ${\cal W}$, that minimizes $\|AZ-B\|$ (Frobenius norm) has been considered by different authors for distinct classes ${\cal W}$. Here, we study this minimization problem for two other classes which include the symmetric Hamiltonian, symmetric skew-Hamiltonian, real orthogonal symplectic and unitary conjugate symplectic matrices. We also consider (as others have done for other classes ${\cal W}$) the problem of minimizing $\|A-\tilde{A}\|$ where $\tilde{A}$ is given and $A$ is a solution of the previous problem. The key idea of our contribution is the reduction of each one of the above minimization problems to two independent subproblems in orthogonal subspaces of $\mathbb{C}^{n\times n}$. This is possible due to the special structures under consideration. We have developed MATLAB codes and present the numerical results of some tests.National Natural Science Foundation of China, no. 11371075

The shifted classical circulant and skew circulant splitting iterative methods for Toeplitz matrices

It is known that every Toeplitz matrix T enjoys a circulant and skew circulant splitting (denoted by CSCS), i.e., T=C-S with C a circulant matrix and S a skew circulant matrix. Based on the variant of such a splitting (also referred to as CSCS), we first develop classical CSCS iterative methods and then introduce shifted CSCS iterative methods for solving hermitian positive definite Toeplitz systems in this paper. The convergence of each method is analyzed. Numerical experiments show that the classical CSCS iterative methods work slightly better than the Gauss-Seidel (GS) iterative methods if the CSCS is convergent and that there is always a constant $\alpha$ such that the shifted CSCS iteration converges much faster than the Gauss-Seidel iteration, no matter whether the CSCS itself is convergent or not.National Natural Science Foundation of China No. 11371075The authors would like to thank the supports of the National Natural Science Foundation of China under Grant No. 11371075, the research innovation program of Hunan province for postgraduate students under Grant No. CX2015B374, the Portuguese Funds through FCT–Fundac˜ao para a Ciˆencia, within the Project UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio