Guang-Hui Cheng

Lanczos-type variants of the COCR method for complex nonsymmetric linear systems

Motivated by the celebrated extending applications of the well-established complex Biconjugate Gradient (CBiCG) method to deal with large three-dimensional electromagnetic scattering problems by Pocock and Walker [M.D. Pocock, S.P. Walker, The complex Bi-conjugate Gradient solver applied to large electromagnetic scattering problems, computational costs, and cost scalings, IEEE Trans. Antennas Propagat. 45 (1997) 140-146], three Lanczos-type variants of the recent Conjugate A-Orthogonal Conjugate Residual (COCR) method of Sogabe and Zhang [T. Sogabe, S.-L. Zhang, A COCR method for solving complex symmetric linear systems, J. Comput. Appl. Math. 199 (2007) 297-303] are explored for the solution of complex nonsymmetric linear systems. The first two can be respectively considered as mathematically equivalent but numerically improved popularizing versions of the BiCR and CRS methods for complex systems presented in Sogabes Ph.D. Dissertation. And the last one is somewhat new and is a stabilized and more smoothly converging variant of the first two in some circumstances. The presented algorithms are with the hope of obtaining smoother and, hopefully, faster convergence behavior in comparison with the CBiCG method as well as its two corresponding variants. This motivation is demonstrated by numerical experiments performed on some selective matrices borrowed from The University of Florida Sparse Matrix Collection by Davis.

Some inequalities for the minimum eigenvalue of the Hadamard product of an M -matrix and its inverse

In this paper, some new inequalities for the minimum eigenvalue of the Hadamard product of an M-matrix and its inverse are given. These inequalities are sharper than the well-known results. A simple example is shown.AMS Subject Classification:15A18, 15A42.

Block triangular preconditioners for the discretized time-harmonic Maxwell equations in mixed form

In this paper, we consider the solution of the saddle point linear systems arising from the finite element discretization of the time-harmonic Maxwell equations in mixed form. Two types of block triangular Schur complement-free preconditioners used with Krylov subspace methods are proposed, involving the choice of the parameter. Furthermore, we give the optimal parameter in practice. Theoretical analysis shows that all eigenvalues of the preconditioned matrices are strongly clustered. Finally, numerical experiments that validate the analysis are presented.

Solution to 3-D electromagnetic problems discretized by a hybrid FEM/MOM method

Abstract A hybrid finite-element method/method of moments (FEM/MOM) is used to solve 3-D electromagnetic problems. We are focused on the solution to the resulting system of linear equations. Because of the hybrid nature of the FEM/MOM method, the coefficient matrix of the linear system has a very special structure with sparse but indefinite sub-matrices generated by the FEM and dense sub-matrices generated by the MOM. Preconditioned Krylov subspace methods are employed to solve the linear systems. Three preconditioners are considered and analyzed regarding the special structure of the coefficient matrix. Numerical results obtained from scattering and radiation problems show that the BICGSTAB method accelerated by a preconditioner, which employs several GMRES iterations to approximate the Schur complement, is reliable and efficient for the solution of the linear system.

New Block Triangular Preconditioners for Saddle Point Linear Systems with Highly Singular (1,1) Blocks

We establish two types of block triangular preconditioners applied to the linear saddle point problems with the singular (1,1) block. These preconditioners are based on the results presented in the paper of Rees and Greif (2007). We study the spectral characteristics of the preconditioners and show that all eigenvalues of the preconditioned matrices are strongly clustered. The choice of the parameter is involved. Furthermore, we give the optimal parameter in practical. Finally, numerical experiments are also reported for illustrating the efficiency of the presented preconditioners.

A note on eigenvalues of perturbed 2x2 block Hermitian matrices

Letbe two Hermitian matrices. We propose new perturbation bounds on the differences between the eigenvalues of and by the bounds of the eigenvector components. These results extend those of Nakatsukasa.

The bounds of the smallest and largest eigenvalues for rank-one modification of the Hermitian eigenvalue problem

Abstract The bounds of the smallest and largest eigenvalues for rank-one modification of the Hermitian matrices are studied in this paper. The sharper bounds are obtained. Numerical examples illustrate that our bounds give accurate estimates.

LU-based Jacobi-like algorithms for non-orthogonal joint diagonalization

Abstract In this paper, based on the LU decomposition, we propose three non-orthogonal Jacobi-like alternating iterative algorithms with two strategies for solving the joint diagonalization problem of a set of Hermitian matrices. In this kind of algorithm, each transformation includes one upper triangular iterative step and one lower triangular iterative step, and each step involves one parameter. The optimal parameter of each step is derived analytically. The convergence of our proposed algorithms is proven. According to this convergence analysis, the existing GNJD algorithm is revisited. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed algorithms in comparison with existing ones.

Some inequalities for nonnegative tensors

Let A be a nonnegative tensor and x=(xi)>0 its Perron vector. We give lower bounds for xtm−1/∑xi2⋯xim and upper bounds for xsm−1/∑xi2⋯xim, where xs=max1≤i≤nxi and xt=min1≤i≤nxi.MSC:15A18, 15A69, 65F15, 65F10.