Weiwei Sun

MODIFIED GAUSS–SEIDEL TYPE METHODS AND JACOBI TYPE METHODS FOR Z-MATRICES

Abstract In this paper, we present the convergence analysis for some modified Gauss–Seidel and Jacobi type iterative methods and provide a comparison of spectral radius among the Gauss– Seidel iterative method and these modified methods. Some recent results are improved.

A Legendre--Petrov--Galerkin and Chebyshev Collocation Method for Third-Order Differential Equations

A Legendre--Petrov--Galerkin (LPG) method for the third-order differential equation is developed. By choosing appropriate base functions, the method can be implemented efficiently. Also, this new approach enables us to derive an optimal rate of convergence in L2-norm. The method is applied to some nonlinear problems such as the Korteweg--de Vries (KdV) equation with the Chebyshev collocation treatment for the nonlinear term. It is a Legendre--Petrov--Galerkin and Chebyshev collocation (LPG-CC) method. Numerical experiments are given to confirm the theoretical result.

New Perturbation Bounds for Unitary Polar Factors

In this paper, we present some new perturbation bounds for polar decompositions in the Frobenius norm. We prove that under any active condition of the perturbation being small, our bounds always improve previous bounds. Some perturbation bounds in the spectral norm and general unitarily invariant norms are also given.

Perturbation Bounds of Unitary and Subunitary Polar Factors

In this paper, we present some new perturbation bounds for (generalized) polar decompositions under the Frobenius norm for both complex and real matrices. For subunitary polar factors, we show that our bounds always improve the existing bounds. Based on some interesting properties of the matrix equation W+W*=W* W, some new bounds involving both the Frobenius norm and the spectral norm of the perturbation are given. The optimality of bounds is discussed.

Spectral Analysis of Hermite Cubic Spline Collocation Systems

In this paper, we present an eigenvalue analysis of the first-order and second-order Hermite cubic spline collocation differentiation matrices with arbitrary collocation points. Some important features are explored and compared with some other discrete methods, such as finite difference methods.

Preconditioning iterative algorithm for the electromagnetic scattering from a large cavity

A preconditioning iterative algorithm is proposed for solving electromagnetic scattering from an open cavity embedded in an infinite ground plane. In this iterative algorithm, a physical model with a vertically layered medium is employed as a preconditioner of the model of general media. A fast algorithm developed in (SIAM J. Sci. Comput. 2005; 27:553–574) is applied for solving the model of layered media and classical Krylov subspace methods, restarted GMRES, COCG, and BiCGstab are employed for solving the preconditioned system. Our numerical experiments on cavity models with large numbers of mesh points and large wave numbers show that the algorithm is efficient and the number of iterations is independent of the number of mesh points and dependent upon the wave number. Copyright

Some remarks on the perturbation of polar decompositions for rectangular matrices

In this article we focus on perturbation bounds of unitary polar factors in polar decompositions for rectangular matrices. First we present two absolute perturbation bounds in unitarily invariant norms and in spectral norm, respectively, for any rectangular complex matrices, which improve recent results of Li and Sun (SIAM J. Matrix Anal. Appl. 2003; 25:362–372). Secondly, a new absolute bound for complex matrices of full rank is given. When ‖A − A‖2 ≪ ‖A − A‖F, our bound for complex matrices is the same as in real case. Finally, some asymptotic bounds given by Mathias (SIAM J. Matrix Anal. Appl. 1993; 14:588–593) for both real and complex square matrices are generalized. Copyright

The perturbation bounds for eigenvalues of normal matrices

In this paper we present some new absolute and relative perturbation bounds of eigenvalues of normal matrices. The bounds depend upon the closeness of perturbed matrices to normal matrices and improve those previous results (Duke Math. J. 1953; 20:37–39, Linear Algebra Appl. 1996; 246:215–223). Copyright

Combined Perturbation Bounds: I. Eigensystems and Singular Value Decompositions

In this paper we present some new combined perturbation bounds of eigenvalues and eigensubspaces for a Hermitian matrix

Comparison results for parallel multisplitting methods with applications to AOR methods

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