Successive Overrelaxation Iteration Research Articles

Domains of divergence of the USSOR method applied onp-cyclic matrices

Using a recently derived classical type general functional equation, relating the eigenvalues of a weakly cyclic Jacobi iteration matrix to the eigenvalues of its associated Unsymmetric Successive Overrelaxation (USSOR) iteration matrix, we obtain bounds for the convergence of the USSOR method, when applied to systems with ap-cyclic coefficient matrix.

Convergence domains of the SSOR method for a class of generalized consistently ordered matrices

The functional equation relating the eigenvalues of the block Symmetric Successive Overrelaxation (SSOR) iteration matrix with those of the block Jacobi iteration matrix found by Chong and Cai (1985) is used in order to obtain precise domains of convergence of the block SSOR iteration method associated with a class of generalized consistently ordered (GCO) ( k, p− k)-matrices A( p⩾2, k=1, …, p−1). We show that the domain of convergence depends only on the relaxation parameter ω, the spectral radius of the block Jacobi iteration matrix and the value of l = k p . Unlike the case ( k, p− k) = (1, p−1), p⩾3, which was studied in an earlier paper by the authors (1989), beyond certain critical values of l the domain of convergence does not grow monotonically with l and manifests various sorts of behavior. However, as in the case ( k, p− k) = (1, p−1), it is shown that the intersection of the convergence domains taken over all pairs ( p, k) coincides with the exact domain of convergence of the point SSOR iteration method associated with nonsingular H-matrices A, that is, matrices which can be diagonally scaled to being strictly diagonally dominant.

Successive overrelaxation iteration for the stability of large scale systems

The successive overrelaxation iteration is used to investigate the stability of large scale systems, and some sufficient conditions such that the zero solution of an overall system is asymptotically stable are obtained, which only depend upon the fundamental matrices of the variational isolated subsystems and the interconnected matrices.

The SOR method on parallel computers

The Jacobi Overrelaxation (JOR) method is usually cited as a "perfect" parallel algorithm, whereas the Successive Overrelaxation (SOR) method is considered as quite the opposite. For linear systems with dense matrices, an algorithm for the SOR method is presented which is suited for parallelization nearly in the same way as JOR. For systems with band matrices, an algorithm is described which, using a pipeline principle, yields a speedupS=p(1+(p?1)p/m) ?1, wherep denotes the number of processors andm the number of SOR iterations. Thus form=p, there is no speedup whereas the speedup tends to its maximal valuep ifm tends to infinity.

A note on the SSOR convergence domain due to Neumaier and Varga

In a recent paper A. Neumaier and R. S. Varga show that if A is an n ×ncomplex H-matrix with 12 <ϱ(∣JA∣) < 1, then ϱ(SAω) < 1 for all 0 < ω <g̃w:=2/ (1 + √2ϱ(∣JA∣) − 1), where SAω and JA are, respectively, the symmetric successive overrelaxation (SSOR) iteration matrix and the Jacobi iteration matrix associated with A. We show that ϱ(SAg̃w) < 1. This allows us to also sharpen further results on the SSOR method for H-matrices due to Varga, Niethammer, and Cai.

Successive overrelaxation methods for solving the rank deficient linear squares problem

We develop successive overrelaxation (SOR) methods for finding the least squares solution of minimal norm to the system Ax = b , (∗) where A is an m × n matrix of rank r. The methods are obtained by first augmenting the system (∗) to a block 4 × 4 consistent system of linear equations. The augmented coefficient matrix is then split by a subproper SOR splitting. An interval for the relaxation parameter in which the subproper SOR iteration matrix is semiconvergent is determined along with the optimal relaxation parameter which minimizes the modulus of the controlling eigenvalue of the SOR matrix. Since the scheme computes at first only a solution to the augmented system, it is subsequently shown how to transform such a solution to the unique solution of minimal 2-norm. Analysis of the practical implementation of the algorithms developed here will be given elsewhere.

On the analysis of the unsymmetric successive overrelaxation method when applied top-cyclic matrices

A Determinantal Invariance, associated with consistently ordered weakly cyclic matrices, is given. The DI is then used to obtain a new functional equation which relates the eigenvalues of a particular block Jacobi iteration matrix to the eigenvalues of its associated Unsymmetric Successive Overrelaxation (USSOR) iteration matrix. This functional equation as well as the theory of nonnegative matrices and regular splittings are used to obtain convergence and divergence regions of the USSOR method.

An Orthogonal‐Upstream Finite Element Approach to Modeling Aquifer Contaminant Transport

An orthogonal‐upstream weighting (OUW) finite element scheme, which will result in a matrix amenable to point successive overrelaxation (SOR) solution strategies, is presented for approximating the aquifer contaminant‐transport equation. This scheme differs from the standard Galerkin weighting (GW) and nonorthogonal‐upstream weighting (NUW) schemes in that the set of weighting functions is required to be orthogonal to the set of basis functions. These weighting functions are referred to as OUW functions and are developed for line, quadrilateral, and triangular elements. Numerical results have been obtained for two examples and are compared with results obtained through analytical solutions (AS) and the Galerkin and/or NUW schemes. The direct elimination solutions of the OUW finite element equations yield results comparable to those obtained by the direct solution of the Galerkin and/or NUW finite element equations. The SOR computations of the OUW finite element scheme generate convergent solutions for all cases. In contrast, the SOR calculations of the Galerkin and/or NUW finite element schemes result in convergent solutions for dispersion‐dominant cases but produce divergent results for advection‐dominant cases. For small problems when the central processing unit (CPU) memory is not a consideration, the direct elimination solutions of the Galerkin and NUW finite element methods are the most efficient schemes. For large problems, when one wish to use point SOR iteration to solve the matrix equation because of CPU memory limitations, the OUW scheme provides an alternative because it gives convergent SOR computations for all Peclet numbers. Even when CPU memory is not a consideration, one may still prefer OUW to Galerkin or NUW because of its susceptibility to the SOR iteration for all grid Peclet numbers, especially for large problems when the SOR iteration will be much more efficient than the direct elimination solution.

The convergence of accelerated overrelaxation iterations

Accelerated Overrelaxation Iterations extrapolate the standard Successive Overrelaxation Iterations. This paper provides conditions for the convergence of the Accelerated Overrelaxation process; and a conditionally optimal version is derived for solving an arbitrary real linear equation system.

Numerical simulation of free convective heat transfer from a sphere

A numerical study of time-dependent free convective heat transfer from a solid sphere to an incompressible Newtonian fluid has been carried out for Grashof numbers between 0.05 and 12 500 and for Prandtl numbers of 0.72, 10 and 100. The energy and vorticity transport equations were solved using Peaceman and Rachford's ADI method and the stream function equation was solved using point iterative successive over-relaxation. From the late-time steady-state solutions it was observed that even at extremely low Grashof numbers, weak convection processes were present in the region close to the outer boundary. However, it was found that even at moderate Grashof numbers the dominant mode of vorticity transport close to the surface was by diffusion.

On the row sum criterion and the convergence of some iterative processes

The well known row sum criterion (in German "Zeilensummen-kriterium") is reformulated for transformations on Banach spaces with cones, and some sufficient conditions guaranteeing the convergence ofJacobi, Gauss-Seidel and successive over-relaxation iterations are derived, similar to those given byWalter [12] andKulisch [3] for linear algebraic systems.

The S.S.O.R. Iteration Scheme for Equations with 1 Ordering

The symmetric successive over-relaxation iteration scheme is compared with successive over-relaxation for equations with σ1 ordering.