non-Hermitian Positive Definite Systems Research Articles

Minimum residual shift-splitting iteration method for non-Hermitian positive definite and positive semidefinite linear systems

By applying the minimum residual technique to the shift-splitting (SS) iteration scheme, we introduce a non-stationary iteration method named minimum residual SS (MRSS) iteration method to solve non-Hermitian positive definite and positive semidefinite systems of linear equations. Theoretical analyses show that the MRSS iteration method is unconditionally convergent for both of the two kinds of systems of linear equations. Numerical examples are employed to verify the feasibility and effectiveness of the MRSS iteration method.

On Non-Hermitian Positive (Semi)Definite Linear Algebraic Systems Arising from Dissipative Hamiltonian DAEs

We discuss different cases of dissipative Hamiltonian differential-algebraic equations and the linear algebraic systems that arise in their linearization or discretization. For each case we give examples from practical applications. An important feature of the linear algebraic systems is that the (non-Hermitian) system matrix has a positive definite or semidefinite Hermitian part. In the positive definite case we can solve the linear algebraic systems iteratively by Krylov subspace methods based on efficient three-term recurrences. We illustrate the performance of these iterative methods on several examples. The semidefinite case can be challenging and requires additional techniques to deal with the “singular part,” while the “positive definite part” can still be treated with the three-term recurrence methods.

Semi-local convergence of the Newton-HSS method under the center Lipschitz condition

Newton-type methods have gained much attention in the past decades, especially for the semilocal convergence based on no information around the solution $x_*$ of the target nonlinear equation. For large sparse non-Hermitian positive definite systems of nonlinear equation, assuming that the nonlinear operator satisfies the center Lipschitz condition, which is wider than usual Lipschtiz condition and H$o$lder continuous condition, we establish a new Newton-Kantorovich convergence theorem for the Newton-HSS method. Once the convergence criteria is satisfied, the iteration sequence $\{x_k\}_{k = 0}^∞$ generated by the Newton-HSS method is well defined, and converges to the solution $x_*$. Numerical results illustrate the effect.

The Analysis of Dynamic Buckling of an Impacted Column using Difference Methods

In this paper, the object of our study is two coupled partial differential equation which have a rich physical background. With a reasonable change to the initial value, we use central difference method and compact finite difference method to get the numerical results of the governing equations. In the analysis of the latter, we use the Hermitian and skew-Hermitian splitting (HSS) method to solve a large sparse non-Hermitian positive definite system of linear equations. In this way, the computational efficiency can be improved and the convergence is guaranteed which is proved by Bai [Bai et al. [2003] “Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems,” SIAM J. Matrix Anal. Appl. 24, 603–626] and Chen [Chen et al. [2014] “Convergence analysis of the modified Newton-HSS method under the Hölder continuous condition,” J. Comput. Appl. Math. 264, 115–130]. And we also pay attention to the effect of initial conditions, on which we add a small perturbation to observe its influence by comparing numerical results.

Multi-step modified Newton-HSS methods for systems of nonlinear equations with positive definite Jacobian matrices

The Hermitian and skew-Hermitian splitting (HSS) method is an unconditionally convergent iterative method for solving large sparse non-Hermitian positive definite system of linear equations. In this paper, we establish a class of multi-step modified Newton-HSS (MMN-HSS) methods for solving large sparse system of nonlinear equations with positive definite Jacobian matrices. The MMN-HSS methods use the multi-step modified Newton methods to solve the nonlinear equations, and the HSS method to approximately solve the Newton equation. Local and semilocal convergence theorems are proved under proper conditions. Also, we present the global multi-step modified Newton-HSS methods with a backtracking strategy and analyse its global convergence. Finally, numerical results are given to confirm the feasibility and effectiveness of our method.

Preconditioning of GMRES by skew-Hermitian iterations

A class of preconditioners for solving non-Hermitian positive definite systems of linear algebraic equations is proposed and investigated. It is based on Hermitian and skew-Hermitian splitting of the initial matrix. A generalization for saddle point systems having semidefinite or singular (1, 1) blocks is given. Our approach is based on an augmented Lagrangian formulation. It is shown that such preconditioners can be efficiently used for the iterative solution of systems of linear algebraic equations by the GMRES method.

Positive Definite and Semi-Definite Splitting Methods for Non-Hermitian Positive Definite Linear Systems

In this paper, we further generalize the technique for constructing the normal (or positive definite) and skew-Hermitian splitting iteration method for solving large sparse nonHermitian positive definite system of linear equations. By introducing a new splitting, we establish a class of efficient iteration methods, called positive definite and semi-definite splitting (PPS) methods, and prove that the sequence produced by the PPS method converges unconditionally to the unique solution of the system. Moreover, we propose two kinds of typical practical choices of the PPS method and study the upper bound of the spectral radius of the iteration matrix. In addition, we show the optimal parameters such that the spectral radius achieves the minimum under certain conditions. Finally, some numerical examples are given to demonstrate the effectiveness of the considered methods.

Semilocal convergence analysis for the modified Newton-HSS method under the Hölder condition

The present paper is concerned with theoretical properties of the modified Newton-HSS method for large sparse non-Hermitian positive definite systems of nonlinear equations. Assuming that the nonlinear operator satisfies the H?lder continuity condition, a new semilocal convergence theorem for the modified Newton-HSS method is established. The H?lder continuity condition is milder than the usual Lipschitz condition. The semilocal convergence theorem is established by using the majorizing principle, which is based on the concept of majorizing sequence given by Kantorovich. Two real valued functions and two real sequences are used to establish the convergence criterion. Furthermore, a numerical example is given to show application of our theorem.

The generalized HSS method with a flexible shift-parameter for non-Hermitian positive definite linear systems

Based on the Hermitian and skew-Hermitian splitting (HSS), we come up with a generalized HSS iteration method with a flexible shift-parameter for solving the non-Hermitian positive definite system of linear equations. This iteration method utilizes the optimization technique to obtain the optimal value of the flexible shift-parameter at iteration process. Both theory and experiment have shown that the new strategy is efficient.

A new splitting and preconditioner for iteratively solving non-Hermitian positive definite systems

A new iteration method for solving a linear system with coefficient matrix being non-Hermitian positive definite is presented in this note. We study the spectral radius and contraction properties of the iteration matrix and then analyze the best possible choice of the parameter. With the results obtained, we show that the new method is convergent for a non-Hermitian positive definite linear system and propose a preconditioner to improve the condition number of the system. The numerical examples show that the new method is much more efficient than the HSS (or PSS) iteration method.

Convergence analysis of modified Newton-HSS method for solving systems of nonlinear equations

Hermitian and skew-Hermitian splitting(HSS) method has been proved quite successfully in solving large sparse non-Hermitian positive definite systems of linear equations. Recently, by making use of HSS method as inner iteration, Newton-HSS method for solving the systems of nonlinear equations with non-Hermitian positive definite Jacobian matrices has been proposed by Bai and Guo. It has shown that the Newton-HSS method outperforms the Newton-USOR and the Newton-GMRES iteration methods. In this paper, a class of modified Newton-HSS methods for solving large systems of nonlinear equations is discussed. In our method, the modified Newton method with R-order of convergence three at least is used to solve the nonlinear equations, and the HSS method is applied to approximately solve the Newton equations. For this class of inexact Newton methods, local and semilocal convergence theorems are proved under suitable conditions. Moreover, a globally convergent modified Newton-HSS method is introduced and a basic global convergence theorem is proved. Numerical results are given to confirm the effectiveness of our method.

Practical convergent splittings and acceleration methods for non-Hermitian positive definite linear systems

We present two practical convergent splittings for solving a non-Hermitian positive definite system. By these new splittings and optimization models, we derive three new improved Chebyshev semi-iterative methods and discuss convergence of these methods. Finally, the numerical examples show that the acceleration methods can reduce evidently the amount of work in computation.

The Self-Adaptive Multi-Splitting Parallel Methods for Non-Hermitian Positive Definite Systems

We present the convergent splitting and convergent multisplitting for linear system of algebraic equations Ax = b when the coefficient matrix is a non-hermitian positive definite matrix. Furthermore, we also establish the comparison theorems of different splittings or multisplittings based on numerical radius. Mainly, we propose two new selfadaptive multisplitting parallel methods which the weighting matrices are self-adaptive. Finally, we give an application to solve the two- dimensional advection-diffusion equations.

Modified parallel multisplitting iterative methods for non-Hermitian positive definite systems

In this paper we present three modified parallel multisplitting iterative methods for solving non-Hermitian positive definite systems Ax?=?b. The first is a direct generalization of the standard parallel multisplitting iterative method for solving this class of systems. The other two are the iterative methods obtained by optimizing the weighting matrices based on the sparsity of the coefficient matrix A. In our multisplitting there is only one that is required to be convergent (in a standard method all the splittings must be convergent), which not only decreases the difficulty of constructing the multisplitting of the coefficient matrix A, but also releases the constraints to the weighting matrices (unlike the standard methods, they are not necessarily be known or given in advance). We then prove the convergence and derive the convergent rates of the algorithms by making use of the standard quadratic optimization technique. Finally, our numerical computations indicate that the methods derived are feasible and efficient.

On Newton-HSS Methods for Systems of Nonliear Equations with Positive-Definite Jacobian Matrices

The Hermitian and skew-Hermitian splitting (HSS) method is an unconditionally convergent iteration method for solving large sparse non-Hermitian positive definite system of linear equations. By making use of the HSS iteration as the inner solver for the Newton method, we establish a class of Newton-HSS methods for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points. For this class of inexact Newton methods, two types of local convergence theorems are proved under proper conditions, and numerical results are given to examine their feasibility and effectiveness. In addition, the advantages of the Newton-HSS methods over the Newton-USOR, the Newton-GMRES and the Newton-GCG methods are shown through solving systems of nonlinear equations arising from the finite difference discretization of a two-dimensional convection-diffusion equation perturbed by a nonlinear term. The numerical implementations also show that as preconditioners for the Newton-GMRES and the Newton-GCG methods the HSS iteration outperforms the USOR iteration in both computing time and iteration step. Mathematics subject classification: 65F10, 65W05.

Several splittings for non-Hermitian linear systems

For large sparse non-Hermitian positive definite system of linear equations, we present several variants of the Hermitian and skew-Hermitian splitting (HSS) about the coefficient matrix and establish correspondingly several HSS-based iterative schemes. Theoretical analyses show that these methods are convergent unconditionally to the exact solution of the referred system of linear equations, and they may show advantages on problems that the HSS method is ineffective.

On the HSS iteration methods for positive definite Toeplitz linear systems

We study the HSS iteration method for large sparse non-Hermitian positive definite Toeplitz linear systems, which first appears in Bai, Golub and Ng’s paper published in 2003 [Z.-Z. Bai, G.H. Golub, M.K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl. 24 (2003) 603–626], and HSS stands for the Hermitian and skew-Hermitian splitting of the coefficient matrix A . In this note we use the HSS iteration method based on a special case of the HSS splitting, where the symmetric part H = 1 2 ( A + A T ) is a centrosymmetric matrix and the skew-symmetric part S = 1 2 ( A − A T ) is a skew-centrosymmetric matrix for a given Toeplitz matrix. Hence, fast methods are available for computing the two half-steps involved in the HSS and IHSS iteration methods. Some numerical results illustrate their effectiveness.

Splitting iteration methods for non-Hermitian positive definite systems of linear equations

For large sparse system of linear equations with a non-Hermitian positive definite coefficient matrix, we review the recently developed Hermitian/skew-Hermitian splitting (HSS) iteration, normal/skew-Hermitian splitting (NSS) iteration, positive-definite/skew-Hermitian splitting (PSS) iteration, and block triangular/skew-Hermitian splitting (BTSS) iteration. These methods converge unconditionally to the exact solution of the linear system, with the upper bounds of their convergence factors being only dependent on the spectrum of the Hermitian (or normal, or positive-definite) splitting matrix, but independent of the spectrum of the skew-Hermitian splitting matrix as well as the eigenvectors of all matrices involved.

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.

On inexact hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems

We study theoretical properties of two inexact Hermitian/skew-Hermitian splitting (IHSS) iteration methods for the large sparse non-Hermitian positive definite system of linear equations. In the inner iteration processes, we employ the conjugate gradient (CG) method to solve the linear systems associated with the Hermitian part, and the Lanczos or conjugate gradient for normal equations (CGNE) method to solve the linear systems associated with the skew-Hermitian part, respectively, resulting in IHSS(CG, Lanczos) and IHSS(CG, CGNE) iteration methods, correspondingly. Theoretical analyses show that both IHSS(CG, Lanczos) and IHSS(CG, CGNE) converge unconditionally to the exact solution of the non-Hermitian positive definite linear system. Moreover, their contraction factors and asymptotic convergence rates are dominantly dependent on the spectrum of the Hermitian part, but are less dependent on the spectrum of the skew-Hermitian part, and are independent of the eigenvectors of the matrices involved. Optimal choices of the inner iteration steps in the IHSS(CG, Lanczos) and IHSS(CG, CGNE) iterations are discussed in detail by considering both global convergence speed and overall computation workload, and computational efficiencies of both inexact iterations are analyzed and compared deliberately.