Symmetric Linear Systems Of Equations Research Articles

Minimum residual NDSS iteration method for a class of complex symmetric linear systems

The double-step scale splitting and the minimum residual modified HSS iteration method are effective methods for a class of large sparse complex symmetric systems of linear equations. In this paper, we give an efficient method to solve the linear equations related to complex symmetric systems. Firstly, we modify the double-step scale splitting (DSS) iteration methods, obtaining the new double-step scale splitting (NDSS) iteration methods. Furthermore, we improve the algorithm by taking use of minimum residual technique. We call the modified algorithm minimum residual NDSS (MRNDSS) iteration method. For the new presented method, we study some properties of the algorithm carefully. Compared with the NDSS method, MRNDSS has two different parameters in each alternating iteration step and two additional parameters depending on the residual error from previous step. In our numerical experiments, our new method performs more efficient than four other iteration methods which are usually applied to solve the same problems.

Novel minimum residual MHSS iteration method for solving complex symmetric linear systems

For solving complex symmetric systems of linear equations, based on the modified Hermitian and skew-Hermitian splitting (MHSS) iteration scheme, a minimum residual MHSS (MRMHSS) iteration method was proposed by applying the minimum residual technique to the MHSS iteration process. We have known that the MRMHSS iteration method is very efficient, while the convergence condition is difficult to verify in practice. In this work, we improve the MRMHSS iteration method by determining the involved iteration parameters using a new norm and prove that the so obtained novel variant can be unconditionally convergent. Numerical results are reported to demonstrate the applicability and effectiveness of the proposed method.

On the preconditioned conjugate gradient method for complex symmetric systems

In this note, to solve a class of complex symmetric system of linear equations, we construct a new block triangular (NBT) preconditioner which leads to a symmetric positive definite preconditioned matrix. The spectral properties of the preconditioned matrix are studied. We use conjugate gradient method to solve the preconditioned system and compare the convergence with some existing methods. Numerical results verify the effectiveness of the preconditioned conjugate gradient method with NBT preconditioner.

On preconditioned Euler-extrapolated single-step Hermitian and skew-Hermitian splitting method for complex symmetric linear systems

In this paper, we introduce a preconditioned Euler-extrapolated single-step Hermitian and skew-Hermitian splitting (PE-SHSS) iteration method for solving a class of complex symmetric system of linear equations. The convergence properties of the PE-SHSS iteration method are investigated under suitable restrictions. In addition, the spectral properties of the corresponding preconditioned matrix are discussed. Finally, three numerical examples are also used to verify the effectiveness of the PE-SHSS iteration method.

Spectral estimates for saddle point matrices arising in weak constraint four‐dimensional variational data assimilation

SummaryWe consider the large sparse symmetric linear systems of equations that arise in the solution of weak constraint four‐dimensional variational data assimilation, a method of high interest for numerical weather prediction. These systems can be written as saddle point systems with a 3 × 3 block structure but block eliminations can be performed to reduce them to saddle point systems with a 2 × 2 block structure, or further to symmetric positive definite systems. In this article, we analyse how sensitive the spectra of these matrices are to the number of observations of the underlying dynamical system. We also obtain bounds on the eigenvalues of the matrices. Numerical experiments are used to confirm the theoretical analysis and bounds.

Minimum residual modified HSS iteration method for a class of complex symmetric linear systems

For solving a class of complex symmetric system of linear equations, we apply the minimum residual technique to the modified Hermitian and skew-Hermitian splitting (MHSS) iteration scheme and propose an iteration method referred to as minimum residual MHSS (MRMHSS) iteration method. Compared with the classical MHSS method, the MRMHSS method has two more iteration parameters, which can be automatically and easily computed. Then, some properties of the MRMHSS iteration method are carefully studied. Finally, we use four examples to test the performance of the MRMHSS iteration method by comparing its numerical results with three other iteration methods.

Modified Newton-DSS method for solving a class of systems of nonlinear equations with complex symmetric Jacobian matrices

Double-step scale splitting (DSS) iteration method is proved to be an unconditionally convergent iteration method, which is also efficient and robust for solving a class of large sparse complex symmetric systems of linear equations. In this paper, by making use of the DSS iteration technique as the inner solver to approximately solve the Newton equations, we establish a new modified Newton-DSS method for solving systems of nonlinear equations whose Jacobian matrices are large, sparse, and complex symmetric. Subsequently, we investigate the local and semilocal convergence properties of our method under some proper assumptions. Finally, numerical results on some problems illustrate the superiority of our method over some previous methods.

Parameterized GSOR Method for a Class of Complex Symmetric Systems of Linear Equations

Parameterized GSOR Method for a Class of Complex Symmetric Systems of Linear Equations

A preconditioned SSOR iteration method for solving complex symmetric system of linear equations

We present a preconditioned version of the symmetric successive overrelaxation (SSOR) iteration method for a class of complex symmetric linear systems. The convergence results of the proposed method are established and conditions under which the spectral radius of the iteration matrix of the method is smaller than that of the SSOR method are analyzed. Numerical experiments illustrate the theoretical results and depict the efficiency of the new iteration method.

An immersed-isogeometric model: Application to linear elasticity and implementation with THBox-splines

We investigate the application of immersed boundary approaches in isogeometric analysis for the treatment of flexible domains by suitably incorporating trimming operations and geometry mappings. The considered immersed-isogeometric model is framed in the context of an automatic adaptive scheme to solve linear elasticity problems. The proposed method leads to a symmetric system of linear equations, and it is essentially free of user-defined penalty and stabilization parameters. Adaptivity is achieved by employing hierarchically nested spline spaces. In particular, we focus on truncated hierarchical box splines (THBox-splines) defined over regular triangulations. Several numerical examples demonstrate the optimal convergence of the adaptive scheme.

An Alternative Lopsided PMHSS Iteration Method for Complex Symmetric Systems of Linear Equations

An Alternative Lopsided PMHSS Iteration Method for Complex Symmetric Systems of Linear Equations

Two-parameter TSCSP method for solving complex symmetric system of linear equations

We introduce a two-parameter version of the two-step scale-splitting iteration method, called TTSCSP, for solving a broad class of complex symmetric system of linear equations. We present some conditions for the convergence of the method. An upper bound for the spectral radius of the method is presented and optimal parameters which minimize this bound are given. Inexact version of the TTSCSP iteration method (ITTSCSP) is also presented. Some numerical experiments are reported to verify the effectiveness of the TTSCSP iteration method and the numerical results are compared with those of the TSCSP, the SCSP and the PMHSS iteration methods. Numerical comparison of the ITTSCSP method with the inexact version of TSCSP, SCSP and PMHSS are presented. We also compare the numerical results of the BiCGSTAB method in conjunction with the TTSCSP and the ILU preconditioners.

On symmetric block triangular splitting iteration method for a class of complex symmetric system of linear equations

For solving a class of complex symmetric linear system, we first transform the system into a block two-by-two real formulation and construct a symmetric block triangular splitting (SBTS) iteration method based on two splittings. Then, eigenvalues of iterative matrix are calculated, convergence conditions with relaxation parameter are derived, and two optimal parameters are obtained. Besides, we present the optimal convergence factor and test two numerical examples to confirm theoretical results and to verify the high performances of SBTS iteration method compared with two classical methods.

複素対称行列に対する前処理の評価方法について(行列・固有値問題の解法とその応用, 平成18年研究部会連合発表会)

複素対称行列に対する前処理の評価方法について(行列・固有値問題の解法とその応用, 平成18年研究部会連合発表会)

Accelerated GPMHSS Method for Solving Complex Systems of Linear Equations

AbstractPreconditioned modified Hermitian and skew-Hermitian splitting method (PMHSS) is an unconditionally convergent iteration method for solving large sparse complex symmetric systems of linear equations, and uses one parameter α. Adding another parameter β, the generalized PMHSS method (GPMHSS) is essentially a twoparameter iteration method. In order to accelerate the GPMHSS method, using an unexpected way, we propose an accelerated GPMHSS method (AGPMHSS) for large complex symmetric linear systems. Numerical experiments show the numerical behavior of our new method.

On the Symmetric Componentwise Relative Backward Error for Linear Systems of Equations

We derive an upper bound on the symmetric componentwise relative backward error for symmetric linear systems of equations. Since the bound can be computed efficiently and, except for some artificial examples, seems to be of the same order of magnitude as the true symmetric componentwise backward error, we believe that it is suitable for practical use. Our results also provide new insight into the relationship between the symmetric and unsymmetric backward errors.

AOR–Uzawa iterative method for a class of complex symmetric linear system of equations

In this paper, based on accelerated overrelaxation (AOR) method and Uzawa method, we present AOR–Uzawa iterative method for solving a broad class of complex symmetric linear systems. We investigate its convergence properties. In addition, the resulting AOR–Uzawa preconditioner is used to precondition Krylov subspace methods such as the restarted generalized minimal residual (GMRES) method for solving the real equivalent formulation of the complex symmetric system. Finally, two numerical examples are given to show the effectiveness of the proposed method.

A parameterized SHSS iteration method for a class of complex symmetric system of linear equations

In this paper, we present a parameterized variant of the single-step Hermitian and skew-Hermitian (SHSS) iteration method for solving a class of complex symmetric system of linear equations. We study the convergence properties of the parameterized SHSS (P-SHSS) iteration method, the choices of the parameters and the quasi-optimal parameters. Numerical experiments are given to verify the effectiveness of the P-SHSS iteration method.

An orthogonally accumulated projection method for symmetric linear system of equations

A type of iterative orthogonally accumulated projection methods for solving linear system of equations are proposed in this paper. This type of methods are applications of accumulated projection(AP) technique proposed recently by authors. Instead of searching projections in a sequence of subspaces as done in the original AP approach, these methods try to efficiently construct a sequence of orthonormal vectors while the inner-product between the solution to the system and each vector in the sequence can be easily calculated, thus the solution can be retrieved in finite number of iterations in case of exact arithmetic operations. We also discuss the strategies to handle loss-of-orthogonality during the process of constructing orthonormal vectors. Numerical experiments are provided to demonstrate the efficiency of these methods.

A new iterative method for solving a class of complex symmetric system of linear equations

We present a new stationary iterative method, called Scale-Splitting (SCSP) method, and investigate its convergence properties. The SCSP method naturally results in a simple matrix splitting preconditioner, called SCSP-preconditioner, for the original linear system. Some numerical comparisons are presented between the SCSP-preconditioner and several available block preconditioners, such as PGSOR (Hezari et al. Numer. Linear Algebra Appl. 22, 761---776, 2015) and rotate block triangular preconditioners (Bai Sci. China Math. 56, 2523---2538, 2013), when they are applied to expedite the convergence rate of Krylov subspace iteration methods for solving the original complex system and its block real formulation, respectively. Numerical experiments show that the SCSP-preconditioner can compete with PGSOR-preconditioner and even more effective than the rotate block triangular preconditioners.