Ting-Zhu Huang

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.

Chebyshev-type methods and preconditioning techniques

Recently, a Newton’s iterative method is attracting more and more attention from various fields of science and engineering. This method is generally quadratically convergent. In this paper, some Chebyshev-type methods with the third order convergence are analyzed in detail and used to compute approximate inverse preconditioners for solving the linear system Ax = b. Theoretic analysis and numerical experiments show that Chebyshev’s method is more effective than Newton’s one in the case of constructing approximate inverse preconditioners.

Modified Hermitian and skew‐Hermitian splitting methods for non‐Hermitian positive‐definite linear systems

To further study the Hermitian and non-Hermitian splitting methods for a non-Hermitian and positive-definite matrix, we introduce a so-called lopsided Hermitian and skew-Hermitian splitting and then establish a class of lopsided Hermitian/skew-Hermitian (LHSS) methods to solve the non-Hermitian and positive-definite systems of linear equations. These methods include a two-step LHSS iteration and its inexact version, the inexact Hermitian/skew-Hermitian (ILHSS) iteration, which employs some Krylov subspace methods as its inner process. We theoretically prove that the LHSS method converges to the unique solution of the linear system for a loose restriction on the parameter α. Moreover, the contraction factor of the LHSS iteration is derived. The presented numerical examples illustrate the effectiveness of both LHSS and ILHSS iterations. Copyright

Simple criteria for nonsingular H-matrices

In this paper, several new simple criteria for nonsingular H-matrices are obtained by making use of elements of matrices only. Advantages are illustrated by numerical examples. And a necessary condition for nonsingular H-matrices is also presented.

On some subclasses of P‐matrices

A matrix with positive row sums and all its off-diagonal elements bounded above by their corresponding row averages is called a B-matrix by J. M. Pena in References (SIAM J. Matrix Anal. Appl. 2001; 22:1027–1037) and (Numer. Math. 2003; 95:337–345). In this paper, it is generalized to more extended matrices—MB-matrices, which is proved to be a subclass of the class of P-matrices. Subsequently, we establish relationships between defined and some already known subclasses of P-matrices (see, References SIAM J. Matrix Anal. Appl. 2000; 21:67–78; Linear Algebra Appl. 2004; 393:353–364; Linear Algebra Appl. 1995; 225:117–123). As an application, some subclasses of P-matrices are used to localize the real eigenvalues of a real matrix. Copyright

Asymmetric Hermitian and skew-Hermitian splitting methods for positive definite linear systems

In this paper, efficient iterative methods for the large sparse non-Hermitian positive definite systems of linear equations, based on the Hermitian and skew-Hermitian splitting of the coefficient matrix, are studied. These methods include an asymmetric Hermitian/skew-Hermitian (AHSS) iteration and its inexact version, the inexact asymmetric Hermitian/skew-Hermitian (IAHSS) iteration, which employs some Krylov subspace methods as its inner process. We theoretically study the convergence properties of the AHSS method and the IAHSS method. Moreover, the contraction factor of the AHSS iteration is derived. Numerical examples illustrating the effectiveness of both AHSS and IAHSS iterations are presented.

Restarted weighted full orthogonalization method for shifted linear systems

It is known that the restarted full orthogonalization method (FOM) outperforms the restarted generalized minimum residual method (GMRES) in several circumstances for solving shifted linear systems when the shifts are handled simultaneously. On the basis of the Weighted Arnoldi process, a weighted version of the Restarted Shifted FOM is proposed, which can provide accelerating convergence rate with respect to the number of restarts. In the cases where our hybrid algorithm needs less enough number of restarts to converge than the Restarted Shifted FOM, the associated CPU consuming time is also reduced, as shown by the numerical experiments. Moreover, our algorithm is able to solve certain shifted systems which the Restarted Shifted FOM cannot handle sometimes.

Modified SOR-type iterative method for Z-matrices

Abstract In this paper, we present the convergence analysis for some modified SOR-type iterative method for solving linear systems and provide a comparison of spectral radius among the SOR iterative method and these modified methods. Comparison results and the numerical example show that the rate of convergence of the Gauss–Seidel method is faster than the rate of convergence of the SOR iterative method.

Quasi-Minimal Residual Variants of the COCG and COCR Methods for Complex Symmetric Linear Systems in Electromagnetic Simulations

The conjugate orthogonal conjugate gradient (COCG) method has been considered an attractive part of the Lanczos-type Krylov subspace method for solving complex symmetric linear systems. However, it is often faced with apparently irregular convergence behaviors in practical electromagnetic simulations. To avoid such a problem, the symmetric quasi-minimal residual (QMR) method has been developed. On the other hand, the conjugate A-orthogonal conjugate residual (COCR) method, which can be regarded as an extension of the conjugate residual method, also had been established. It shows that the COCR often gives smoother convergence behavior than the COCG method. The purpose of this paper is to apply the QMR approaches to the COCG and COCR to derive two new methods (including their preconditioned versions), and to report the benefits of the modified methods by some practical examples arising in electromagnetic simulations.

A comparative study of iterative solutions to linear systems arising in quantum mechanics

This study is mainly focused on iterative solutions with simple diagonal preconditioning to two complex-valued nonsymmetric systems of linear equations arising from a computational chemistry model problem proposed by Sherry Li of NERSC. Numerical experiments show the feasibility of iterative methods to some extent when applied to the problems and reveal the competitiveness of our recently proposed Lanczos biconjugate A-orthonormalization methods to other classic and popular iterative methods. By the way, experiment results also indicate that application specific preconditioners may be mandatory and required for accelerating convergence.