SSOR Method Research Articles

Improved SSOR-based signal detector by using based on gauss-seidel method for large-scale MIMO systems

As the core technology in large-scale MIMO system, the reliability of signal detection has an important impact on the whole system. Traditional linear detectors, such as the zero-forcing (ZF), often involve very complex matrix inverse calculations when the number of antennas in large-scale MIMO systems is too large. In this paper, we first introduce the traditional signal detector based on SSOR method, the iterative operation is used to replace the complex matrix inverse calculation, thus reducing the computational complexity. Then, the Gauss-Seidel method is proposed to calculate the initial iteration value, and the simulation results show that the bit error rate (BER) performance of the improved SSOR method is greatly improved.

Analysis of modified SSOR method for augmented linear systems

In this paper, the modified SSOR (MSSOR) method is further investigated for solving augmented linear systems, including both the nonsingular and singular cases. The selection of quasi-optimal iteration parameters are discussed to minimize the radius of the MSSOR iteration matrix for the nonsingular case. The obtained optimality result is unified, which includes those for some existing methods as special cases. Besides, semi-convergence analysis of the MSSOR method for singular case is given using a singular preconditioning technique, which results in much better convergence behavior than some existing nonsingular preconditioning techniques. Numerical experiments are presented to confirm the theoretical results.

Numerical Solution of Mixed Problems of the Theory of Elasticity with One-Sided Constraints

В данной работе рассмотрены особенности применения конечно-элементной технологии для решения задач теории упругости с односторонними связями. Тематика данного исследования с одной стороны определяется тем, что многие ответственные детали и узлы машиностроительных и энергомашиностроительных конструкций имеют выраженный контакт в пределах некоторой заданной поверхности. Для оценки прочности и ресурса таких деталей и узлов необходимо располагать надежной информацией о напряженно-деформированном состоянии. Данные о напряженно-деформированном состоянии можно получить, используя современный аппарат математического моделирования, например, конечно-элементную технологию.

A general preconditioner for linear complementarity problem with an [formula omitted]-matrix

In this paper, we first present a general preconditioner P for solving linear complementarity problem (LCP) associated with an M-matrix A and a vector f, and prove that the LCP(A,f) is equivalent to the LCP(PA,Pf). Then based on this general preconditioner P, two preconditioned SSOR methods for solving the linear complementarity problems are proposed. We show that this general preconditioner P accelerates the convergence of two SSOR methods under the assumption that PA is a Z-matrix. In addition, we also give a practically concrete choice for the preconditioner P satisfying aforementioned assumption. Numerical examples are used to illustrate the theoretical results obtained.

The saddle point problem and the Manteuffel algorithm

In the last two decades, the augmented linear systems and the saddle point problems have been solved by many researchers who have used the conjugate gradient method or the generalized SOR iterative method and variants of them. In the latter class of methods, when the block \(A \in {\mathrm{I\!R}}^{m\times m}\) of the matrix coefficient \(\mathcal {A} = \left[ \begin{array}{cc} A &{} B \\ -B^T &{} 0\end{array}\right] \in {\mathrm{I\!R}}^{(m+n) \times (m+n)}\), \(m \ge n,\) of the linear system to be solved, is symmetric positive definite and \({\mathrm{rank}}(B) =r \le n\), convergence regions and optimal values of the parameters involved have been determined. In this work, we consider the block A to be nonsymmetric positive definite, \({\mathrm{rank}}(B) =r < n \), and use a two-level stationary iterative method whose main step is the linear second-order stationary iterative method for the solution of this class of problems. This method leads to the singular Manteuffel algorithm and the determination of its optimal parameters. As a byproduct, the optimal parameters of the Generalized Modified SSOR method in a particular case are also determined. Numerical examples verify our theoretical findings.

A triple-parameter modified SSOR method for solving singular saddle point problems

In this paper, we study a triple-parameter modified SSOR (TMSSOR) method for solving singular saddle point problems. We prove the semi-convergence of the TMSSOR method under suitable restrictions on the iteration parameters, and obtain the local optimal parameters which minimize the pseudo-spectral radii of the associated iteration matrices. Finally, numerical experiments demonstrate the effectiveness of the TMSSOR method for solving singular saddle point problems.

On the modified symmetric successive over-relaxation method for augmented systems

Recently, Darvishi and Hessari (Comput Math Appl 61:3128–3135, 2011) introduced a modified SSOR method for augmented systems. In this paper, we establish an improvement SSOR method for solving the large sparse augmented systems, which is the extension of the modified SSOR method. The convergence analysis of our method is studied under suitable restrictions on iteration parameters. Furthermore, the functional equation between the parameters and the eigenvalues of the iteration matrix of this method is obtained. Finally, numerical computations are presented based on a particular linear system, which clearly show the effectiveness of our algorithm.

Block Preconditioned SSOR Methods for -Matrices Linear Systems

We present a block preconditioner and consider block preconditioned SSOR iterative methods for solving linear system . When is an -matrix, the convergence and some comparison results of the spectral radius for our methods are given. Numerical examples are also given to illustrate that our methods are valid.

Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes

This paper is concerned with the development of general-purpose algebraic flux correction schemes for continuous (linear and multilinear) finite elements. In order to enforce the discrete maximum principle (DMP), we modify the standard Galerkin discretization of a scalar transport equation by adding diffusive and antidiffusive fluxes. The result is a nonlinear algebraic system satisfying the DMP constraint. An estimate based on variational gradient recovery leads to a linearity-preserving limiter for the difference between the function values at two neighboring nodes. A fully multidimensional version of this scheme is obtained by taking the sum of local bounds and constraining the total flux. This new approach to algebraic flux correction provides a unified treatment of stationary and time-dependent problems. Moreover, the same algorithm is used to limit convective fluxes, anisotropic diffusion operators, and the antidiffusive part of the consistent mass matrix.The nonlinear algebraic system associated with the constrained Galerkin scheme is solved using fixed-point defect correction or a nonlinear SSOR method. A dramatic improvement of nonlinear convergence rates is achieved with the technique known as Anderson acceleration (or Anderson mixing). It blends a number of last iterates in a GMRES fashion, which results in a Broyden-like quasi-Newton update. The numerical behavior of the proposed algorithms is illustrated by a grid convergence study for convection-dominated transport problems and anisotropic diffusion equations in 2D.

Convergence of a generalized MSSOR method for augmented systems

Recently, Wu et al. [S.-L. Wu, T.-Z. Huang, X.-L. Zhao, A modified SSOR iterative method for augmented systems, J. Comput. Appl. Math. 228 (1) (2009) 424–433] introduced a modified SSOR (MSSOR) method for augmented systems. In this paper, we establish a generalized MSSOR (GMSSOR) method for solving the large sparse augmented systems of linear equations, which is the extension of the MSSOR method. Furthermore, the convergence of the GMSSOR method for augmented systems is analyzed and numerical experiments are carried out, which show that the GMSSOR method with appropriate parameters has a faster convergence rate than the MSSOR method with optimal parameters.

Convergence of Generalized Relaxed SSOR Method for Augmented Systems

In this paper, we present the generalized relaxed SSOR method (GRSSOR) for solving the large sparse augmented systems of linear equations, which is the extension of SOR-like, GSOR and GSSOR methods. Furthermore, convergence of GRSSOR method for augmented systems is analyzed, which show that GRSSOR method with appropriate parameters will have very good effectiveness

Convergence of SSOR methods for linear complementarity problems

In this paper, by applying the SSOR splitting, we propose two new iterative methods for solving the linear complementarity problem LCP ( M , q ) . Convergence results for these two methods are presented when M is an H -matrix (and also an M -matrix). Finally, two numerical examples are given to show the efficiency of the presented methods.

The impact of eigenvalue locality on the convergence behavior of the PSD method for two-cyclic matrices

In this paper, we analyse the convergence of the preconditioned simultaneous displacement (PSD) method applied to linear systems of the form Au = b where A is a two-cyclic matrix. Convergence conditions and optimum values of the parameters of the method are determined in the cases where the eigenvalues of the associated Jacobi iteration matrix are either all real or all imaginary. It is shown that the convergence behavior of the PSD method is greatly affected by the locality of the eigenvalues of the associated Jacobi iteration matrix. In particular, it is shown that when these eigenvalues are real the PSD method degenerates into the extrapolated Gauss–Seidel method whereas when they are imaginary its convergence is increased by an order of magnitude and becomes equivalent to the extrapolated SOR method. Finally, a comparison with the SSOR method reveals that the PSD method possesses a better convergence behavior in all cases.

A necessary and sufficient condition for semiconvergence and optimal parameter of the SSOR method for solving the rank deficient linear least squares problem

Let A ∈ C r m × n be partitioned as A = A 11 A 12 A 21 A 22 , where A 11 ∈ C r r × r . Write B = A 21 A 11 - 1 and C = A 11 - 1 A 12 . Suppose that B ≠ 0. For finding the minimum norm least squares solution A + b of the linear systems Ax = b , many authors studied the SOR, AOR, and SSOR methods for solving the augmented systems (1.1) A ^ z = b ˆ , and obtained many results. In this paper we deeply study the SSOR method, whose iteration matrix is written as J ω , and prove the following new conclusions: (1) If ∥ B∥ < 1, then J ω is semiconvergent ⇔ ω ∈ ( 0 , 2 ) . If ∥ B∥ ⩾ 1, then J ω is semiconvergent ⇔ ω ∈ ( 0 , ω ˆ 2 ) ∪ ( ω ˆ 1 , 2 ) , where ω ˆ 2 = 1 - ‖ B ‖ - 1 ‖ B ‖ + 1 and ω ˆ 1 = 1 + ‖ B ‖ - 1 ‖ B ‖ + 1 . (2) The optimal parameters of J ω are ω ˜ 2 = 1 - ‖ B ‖ 1 + 1 + ‖ B ‖ 2 and ω ˜ 1 = 1 + ‖ B ‖ 1 + 1 + ‖ B ‖ 2 , and min ω δ ( J ω ) = min ω max { | λ | : λ ∈ σ ( J ω ) , λ ≠ 1 } = ( 1 - ω ˜ 1 ) 2 = ( 1 - ω ˜ 2 ) 2 = ‖ B ‖ 2 1 + 1 + ‖ B ‖ 2 2 . In addition, we obtain other results concerning the SOR, AOR and SSOR methods.

Aor type iterative methods for solving least squares problems

In order to solve least squares problems, the block SOR, block SSOR and block AOR iterative methods are proposed and investigated by some authors. In this paper, we generalize the usual block AOR method. Other five AOR type iterative methods are defined. Then the convergence and divergence regions for the AOR methods are discussed.

Comparisons of Block Iterative Methods

Comparison theorems for spectral radii of two block USSOR methods, two block SSOR methods, two block AOR methods, two block SOR methods with respect to a partition π and the corresponding finer partition π* are given, respectively, whenever A is an M-matrix or an H-matrix.

On nonlinear SOR-like methods, III — Global convergence of SOR, SSOR and USSOR methods for convex problems

This paper gives global convergence theorems for the SOR, SSOR and USSOR methods applied to a system of nonlinear equations F(x) = 0 inRn, whereF is the gradient of a twice differentiable elliptic functional.

Error Bounds for some overrelaxation methods

Suppose Ax = b is a system of linear equations where the matrix A is symmetric positive definite and consistently ordered. Some bounds for ∥ϵ k :∥2 in terms of the norms of and and their inner product for the SOR, SSOR, USSOR, MSOR and AOR methods have been given by many authors. Recently, Li presents a uniform bound for the overrelaxation methods. In this paper, an error bound for the USMAOR method is derived. As special cases of the USMAOR method, bounds of ∥ϵ k :∥2 for the SMAOR, USMSOR, SMSOR, USAOR, SAOR, USSOR, SSOR, MAOR, MSOR, AOR, EAOR, EGS I, EGS II, SOR, JOR, Gauss-Seidel and Jacobi methods are obtained

“Potential Inaccuracies in MODFLOW Simulations Involving the SIP and SSOR Methods for Matrix Solution,” by James L. Osiensky and Roy E. Williams, March‐April 1997 issue, v. 35, no. 2: 229–232.

“Potential Inaccuracies in MODFLOW Simulations Involving the SIP and SSOR Methods for Matrix Solution,” by James L. Osiensky and Roy E. Williams, March‐April 1997 issue, v. 35, no. 2: 229–232.

A uniform error bound for the overrelaxation methods

Let Ax = b be a system of linear equations where A is symmetric and positive definite. Suppose that the associated block Jacobi matrix B is consistently ordered, weekly cyclic of index 2, and convergent [i.e., μ 1 ≔ ϱ( B) < 1]. Consider using the overrelaxation methods (SOR, AOR, MSOR, SSOR, or USSOR), x n + 1 = T ω x n + c ω for n ⩾ 0, to solve the system. We derive a uniform error bound for the overrelaxation methods, ∥x−x n∥ 2⩽ 1 [1+s(μ 1 2) + t(μ 1 2)] 2 x (t 0 + |t 1|μ 1 2) 2∥δ n∥ 2 − 2t 0〈δ n,δ n+1〉 +|t 1|μ 1 2∥δ n∥∥δ n+1∥+∥δ n+1∥ 2 where ∥ · ∥ = ∥ · ∥ 2, δ n = x n − x n − 1 , and s( μ 2) and t( μ 2) ≔ t 0 + t 1 μ 2 are two coefficients of the corresponding functional equation connecting the eigenvalues λ of T ω to the eigenvalues μ of B. As special cases of the uniform error bound, we will give two error bounds for the SSOR and USSOR methods.