Generalized Saddle Point Systems Research Articles

Structured backward errors for two kinds of generalized saddle point systems

ABSTRACT In this paper, we investigate structured backward errors for two kinds of generalized saddle point systems. The explicit formulas of structured backward errors are derived, which are useful for testing the stability of practical algorithms.

The alternate direction iterative methods for generalized saddle point systems

The paper studies two splitting forms of generalized saddle point matrix to derive two alternate direction iterative schemes for generalized saddle point systems. Some convergence results are established for these two alternate direction iterative methods. Meanwhile, a numerical example is given to show that the proposed alternate direction iterative methods are much more effective and efficient than the existing one.

Condition numbers of generalized saddle point systems

In this paper, we consider the normwise, mixed and componentwise condition numbers of generalized saddle point system. The explicit expressions and the easily computable upper bounds for these condition numbers are presented. Numerical experiments are given to illustrate the effectiveness of the proposed upper bounds.

New Perturbation Bounds Analysis of a Kind of Generalized Saddle Point Systems

AbstractIn this paper we consider new perturbation bounds analysis of a kind of generalized saddle point systems. We provide perturbation upper bounds for the solutions of generalized saddle point systems, which extend the corresponding results in [W.-W. Xu, W. Li, New perturbation analysis for generalized saddle point systems, Calcolo., 46(2009), pp. 25-36] to more general cases.

On normwise structured backward errors for the generalized saddle point systems

In this paper, we present the explicit and computable formula of the structured backward errors of the generalized saddle point systems. Simple numerical examples show that the expressions are useful for testing the stability of practical algorithms.

Spectrum analysis of a more general augmentation block preconditioner for generalized saddle point matrices

In this paper, some errors are pointed out and the correct results are given for the paper Axelsson et al. (J Comput Appl Math 280:141–157, 2015). Then, a class of general augmentation block preconditioners for solving generalized saddle point systems with singular (1, 1) blocks are considered. Results concerning the eigenvalue distribution and forms of the eigenvectors of the preconditioned generalized saddle point matrix are presented. These results extend previous one in the literature.

Generalized Block Triangular Preconditioner for Saddle Point Systems

In this paper, based on the preconditioner for saddle point systems in Benzi and Liu [M. Benzi, J. Liu, Block preconditioning for saddle point systems with indefinite (1,1) block, International Journal of Computer Mathematics, 84(8) (2007) 1117-1129], we present the generalized block preconditioner for generalized saddle point systems, which uses some more parameters and is a generalization of Benzi and Liu's block preconditioner. Moreover, spectral properties of our generalized block triangular preconditioner are given.

A note on spectrum analysis of augmentation block preconditioned generalized saddle point matrices

In this note we consider a set of augmentation block preconditioners for solving generalized saddle point systems whose coefficient matrices have singular (1,1) blocks. Results concerning the eigenvalue distribution and forms of the eigenvectors of the augmentation block preconditioned generalized saddle point matrix and its minimal polynomial are given, and an optimal augmentation block preconditioner of the set is derived. These results extend previous ones in the literature. Numerical experiments that show the very effective performance of the optimal augmentation block preconditioner are reported.

Structured backward errors for generalized saddle point systems

In this paper we investigate structured backward errors for three kinds of generalized saddle point systems where the matrix is not symmetric and its (2,2) block is not zero and has perturbations. Computable formula of backward errors are derived. The expressions are useful for testing the stability of practical algorithms.

A note for preconditioning nonsymmetric matrices

In this paper, we consider preconditioners for generalized saddle point systems with a nonsymmetric coefficient matrix. A constraint preconditioner for this systems is constructed, and some spectral properties of the preconditioned matrix are given. Our results extend the corresponding ones in [3] and [4].

On convergence of double splitting methods for non-Hermitian positive semidefinite linear systems

Some convergence results for double splitting iterations for (possibly non-Hermitian) positive semidefinite linear systems are established. Furthermore, the convergence of double splitting methods for generalized saddle point systems is studied, and a convergence condition for double splitting methods applied to this type of system is given.

New perturbation analysis for generalized saddle point systems

In this paper we present perturbation bounds and condition numbers for the generalized saddle point system. Numerical results are also given.

A mortar method for energy‐momentum conserving schemes in frictionless dynamic contact problems

AbstractIn the present work the mortar method is applied to planar large deformation contact problems without friction. In particular, the proposed form of the mortar contact constraints is invariant under translations and rotations. These invariance properties lay the foundation for the design of energy‐momentum time‐stepping schemes for contact–impact problems. The iterative solution procedure is embedded into an active set algorithm. Lagrange multipliers are used to enforce the mortar contact constraints. The solution of generalized saddle point systems is circumvented by applying the discrete null space method. Numerical examples demonstrate the robustness and enhanced numerical stability of the newly developed energy‐momentum scheme. Copyright © 2008 John Wiley & Sons, Ltd.

Perturbation analysis of generalized saddle point systems

We discuss the perturbation analysis of generalized saddle point systems in this paper. We give the nonlinear perturbation bounds, then derive the condition numbers, and analyze the sensitivity of the computed solutions.