Complex Symmetric Systems Research Articles

On SMSNSSOR iteration method for solving complex symmetric linear systems

In this paper, we extend SMSNS [Pourbagher M, Salkuyeh DK. On the solution of a class of complex symmetric linear systems. Appl Math Lett. 2018;76:14–20.] iteration method for solving a class of complex symmetric system of linear equations. We propose a successive-overrelaxation (SOR) acceleration scheme for SMSNS (SMSNSSOR), discuss the convergence conditions of it and give the optimal parameters which make the fast convergence. Numerical results demonstrate that SMSNSSOR iteration method is feasible and effective for solving complex symmetric systems, and performs better than some other usually used iteration methods.

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.

Modified Newton-PBS method for solving a class of complex symmetric nonlinear systems

Modified Newton-PBS method for solving a class of complex symmetric nonlinear systems

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.

Fast Solution of 3-D Eddy-Current Problems in Multiply Connected Domains by a, v-φ and t-φ Formulations With Multigrid-Based Algorithm for Cohomology Generation

The fast solution of three-dimensional eddy current problems is still an open problem, especially when real-size finite element models with millions of degrees of freedom are considered. In order to lower the number of degrees of freedom a magnetic scalar potential can be used in the insulating parts of the model. This may become difficult when the model geometry presents some conductive parts which are multiply connected. In this work a multigrid-based algoritm is proposed that allows for a calculation in linear-time of cohomology, which is needed to introduce the scalar potential without cuts. This algorithm relies on an algebraic multigrid solver for curl-curl field problems, which ensures optimal computational complexity. Numerical results show that the novel algorithm outperforms state-of-the-art methods for cohomology generation based on homological algebra. In addition, based on this algoritm, novel <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$a,v$ </tex-math></inline-formula> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varphi $ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varphi $ </tex-math></inline-formula> formulations to analyze three-dimensional eddy current problems in multiply connected domains are proposed. Both formulations, after discretization by the cell method, lead to a complex symmetric system of linear equations amenable to fast iterative solution by Krylov-subspace solvers. These formulations are able to provide very accurate numerical results, with a minimum amount of degrees of freedoms to represent the eddy current model. In this way the computational performance is improved compared to the classical <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$A,V$ </tex-math></inline-formula> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$A$ </tex-math></inline-formula> formulation typically implemented in finite element software for electromagnetic design.

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.

A COMINRES‐QLP method for solving complex symmetric linear systems

AbstractSolving complex symmetric systems efficiently is a significantly important issue in the applications of electromagnetics since the discretization of the numerical model using the finite element method results in complex symmetric systems when computing the electromagnetic field. However, such complex symmetric systems are usually much more difficult to be solved because of the property of non‐Hermitian. Therefore, efficient numerical methods for the solution of complex symmetric systems are demanded. In this study, we proposed an iterative method called COMINRES‐QLP which is specially designed for complex symmetric linear systems. Moreover, we proposed a new stopping criterion which could be very useful in practical computations when the condition number of the coefficient matrix is unknown.

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.

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.

Auto-Gauging of Vector Potential by Parallel Sparse Direct Solvers—Numerical Observations

When using magnetic vector potential (MVP)-based formulations for magnetostatic or eddy-current problems, either gauge conditions specifying the divergence of the MVP or tree gauging by eliminating redundant degrees of freedom of the MVP is usually imposed to ensure uniqueness of solutions. Explicit gauging of the MVP is not always necessary since classical iterative solvers can automatically and implicitly fix the gauge as long as the right-hand side vectors are consistent. Besides iterative solvers, implicit gauging is also observed when using state-of-the-art parallel sparse direct solvers (PSDSs), thanks to the built-in functions of handling null-spaces of either real symmetric positive semi-definite matrix systems or those complex symmetric systems from eddy-current problems. Both static and eddy-current examples are solved by PSDS to demonstrate results of local physical quantities or global quantities such as magnetic energy or joule losses. High-order edge/nodal elements are also considered in our numerical examples and it is observed that PSDS can also easily and correctly handle the delicate discrete null spaces.

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.

CHARMM-GUI Membrane Builder for Complex Biological Membrane Simulations with Glycolipids and Lipoglycans.

Glycolipids (such as glycoglycerolipids, glycosphingolipids, and glycosylphosphatidylinositol) and lipoglycans (such as lipopolysaccharides (LPS), lipooligosaccharides (LOS), mycobacterial lipoarabinomannan, and mycoplasma lipoglycans) are typically found on the surface of cell membranes and play crucial roles in various cellular functions. Characterizing their structure and dynamics at the molecular level is essential to understand their biological roles, but systematic generation of glycolipid and lipoglycan structures is challenging because of great variations in lipid structures and glycan sequences (i.e., carbohydrate types and their linkages). To facilitate the generation of all-atom glycolipid/LPS/LOS structures, we have developed Glycolipid Modeler and LPS Modeler in CHARMM-GUI ( http://www.charmm-gui.org ), a web-based interface that simplifies building of complex biological simulation systems. In addition, we have incorporated these modules into Membrane Builder so that users can readily build a complex symmetric or asymmetric biological membrane system with various glycolipids and LPS/LOS. These tools are expected to be useful in innovative and novel glycolipid/LPS/LOS modeling and simulation research by easing tedious and intricate steps in modeling complex biological systems and shall provide insight into structures, dynamics, and underlying mechanisms of complex glycolipid-/LPS-/LOS-containing biological membrane systems.

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.

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

Delayed over-relaxation in iterative schemes to solve rank deficient linear system of (matrix) equations

Recently in [Journal of Computational Physics, 321 (2016), 829-907], an approach has been developed for solving linear system of equations with nonsingular coefficient matrix. The method is derived by using a delayed over-relaxation step (DORS) in a generic (convergent) basic stationary iterative method. In this paper, we first prove semi-convergence of iterative methods with DORS to solve singular linear system of equations. In particular, we propose applying the DORS in the Modified HSS (MHSS) to solve singular complex symmetric systems and in the Richardson method to solve normal equations. Moreover, based on the obtained results, an algorithm is developed for solving coupled matrix equations. It is seen that the proposed method outperforms the relaxed gradient-based (RGB) method [Comput. Math. Appl. 74 (2017), no. 3, 597-604] for solving coupled matrix equations. Numerical results are examined to illustrate the validity of the established results and applicability of the presented algorithms.

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.

On the solution of a class of complex symmetric linear systems

We present an iterative method for solving the complex symmetric system ( W + i T ) x = b , where T ∈ R n × n is a positive definite matrix and W ∈ R n × n is indefinite. Convergence of the method is investigated. The induced preconditioner is applied to accelerate the convergence rate of the GMRES( ℓ ) method and the numerical results are compared with those of the Hermitian normal splitting (HNS) preconditioner.

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)