Linear Stationary Iterative Method Research Articles

Refinement of Extended Accelerated Over-Relaxation Method for Solution of Linear Systems

Given any linear stationary iterative methods in the form z^(i+1)=Jz^(i)+f, where J is the iteration matrix, a significant improvements of the iteration matrix will decrease the spectral radius and enhances the rate of convergence of the particular method while solving system of linear equations in the form Az=b. This motivates us to refine the Extended Accelerated Over-Relaxation (EAOR) method called Refinement of Extended Accelerated Over-Relaxation (REAOR) so as to accelerate the convergence rate of the method. In this paper, a refinement of Extended Accelerated Over-Relaxation method that would minimize the spectral radius, when compared to EAOR method, is proposed. The method is a 3-parameter generalization of the refinement of Accelerated Over-Relaxation (RAOR) method, refinement of Successive Over-Relaxation (RSOR) method, refinement of Gauss-Seidel (RGS) method and refinement of Jacobi (RJ) method. We investigated the convergence of the method for weak irreducible diagonally dominant matrix, matrix or matrix and presented some numerical examples to check the performance of the method. The results indicate the superiority of the method over some existing methods.

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 Cheaper Way to Compute Generalized Cross-Validation as a Stopping Rule for Linear Stationary Iterative Methods

We apply generalized cross-validation (GCV) as a stopping rule for general linear stationary iterative methods for solving very large-scale, ill-conditioned problems. We present a new general formula for the influence operator for these methods and, using this formula and a Monte Carlo approach, we show how to compute the GCV function at a cheaper cost. Then we apply our approach to a well known iterative method (ART) with simulated data in positron emission tomography (PET).

Optimum modified extrapolated jacobi method for some class of matrices

In this paper we study a two parametric linear stationary iterative method, called Modified Extrapolated Jacobi (MEJ) method, for solving linear systems Ax = b, where A is nonsingular consistently ordered 2-cyclic matrix. We give sufficient and necessary conditions for strong convergence of the MEJ method and we determine the optimum extrapolation parameters and the optimum spectral radius of it, in the case where all the eigenvalues of the block Jacobi iteration matrix associated with A are purely imaginary. In the last section, we compare the MEJ with other known methods.

Additive polynomial preconditioners for parallel computers

We present a polynomial preconditioner that can be used with the conjugate gradient method to solve symmetric and positive definite systems of linear equations. Each step of the preconditioning is achieved by simultaneously taking an iteration of the SOR method and an iteration of the reverse SOR method (equations taken in reverse order) and averaging the results. This yields a symmetric preconditioner that can be implemented on parallel computers by performing the forward and reverse SOR iterations simultaneously. We give necessary and sufficient conditions for additive preconditioners to be positive definite. We find an optimal parameter, ω, for the SOR-Additive linear stationary iterative method applied to 2-cyclic matrices. We show this method is asymptotically twice as fast as SSOR when the optimal ω is used. We compare our preconditioner to the SSOR polynomial preconditioner for a model problem. With the optimal ω, our preconditioner was found to be as effective as the SSOR polynomial preconditioner in reducing the number of conjugate gradient iterations. Parallel implementations of both methods are discussed for vector and multiple processors. Results show that if the same number of processors are used for both preconditioners, the SSOR preconditioner is more effective. If twice as many processors are used for the SOR-Additive preconditioner, it becomes more efficient than the SSOR preconditioner when the number of equations assigned to a processor is small. These results are confirmed by the Blue Chip emulator at the University of Washington.

On the optimization of an extrapolation method

We consider an extrapolation method, based on a linear stationary iterative method of first degree, for the numerical solution of the linear system A u = b . We obtain upper bounds for the minimum spectral radius of the extrapolation method as well as good optimum values for the extrapolation parameter. A number of applications and numerical examples are also presented which support the theory and show the efficiency of the obtained results.

Optimal Parameters for Linear Second-Degree Stationary Iterative Methods

In this paper we show that the optimal parameters for linear second-degree stationary iterative methods applied to nonsymmetric linear systems can be found by solving the same minimax problem used to find optimal parameters for the Chebyshev iteration. In fact, the Chebyshev iteration is asymptotically equivalent to a linear seconnd-degree stationary method. The method of finding optimal parameters for the Chebyshev iteration given in Manteuffel [Numer. Math., 28 (1977), pp. 307–327] can be used to find optimal parameters for the sitationary method as well.

On an accelerated procedure of extrapolation

This paper presents some theoretical results concerning an extrapolation method, based on a completely consistent linear stationary iterative method of first degree, for the numerical solution of the linear system Au = b.The main purpose of the paper is to find ranges for the extrapolation parameter, such that the extrapolation method converges independently of whether the original iterative method is convergent or not.

Convergent nonnegative matrices and iterative methods for consistent linear systems

In this paper we study linear stationary iterative methods with nonnegative iteration matrices for solving singular and consistent systems of linear equationsAx=b. The iteration matrices for the schemes are obtained via regular and weak regular splittings of the coefficients matrixA. In certain cases when only some necessary, but not sufficient, conditions for the convergence of the iterations schemes exist, we consider a transformation on the iteration matrices and obtain new iterative schemes which ensure convergence to a solution toAx=b. This transformation is parameter-dependent, and in the case where all the eigenvalues of the iteration matrix are real, we show how to choose this parameter so that the asymptotic convergence rate of the new schemes is optimal. Finally, some applications to the problem of computing the stationary distribution vector for a finite homogeneous ergodic Markov chain are discussed.

On Complex Second-Degree Iterative Methods

An optical second degree iterative method for accelerating a basic linear stationary iterative method of first degree with zeal eigenvalues has been studied by Young [18], [19] and Young and Kincaid [20]. The derivation of this method is now established for the case when the eigenvalues are contained in an elliptical region in the complex plane. By extending results developed by Kublanowskaya (Faddeev–Faddeeva [1]) and N iethammer [8], this second-degree method is obtained through an application of conformal mapping and summation techniques.

On the Consistency of Linear Stationary Iterative Methods

For a given matrix G and a given vector k the linear stationary iterative method $u^{(n + 1)} = Gu^{(n)} + k$ is consistent with the linear system $Au = b$ (where A is a given matrix and b is a given vector) if $\mathcal {S}(A,b) \subseteq \mathcal {S}(I - G,k)$. Here, in general, $\mathcal {S}(A,b)$ denotes the set of solutions of $Au = b$. The iterative method is reciprocally consistent if $\mathcal {S}(1 - G,k) \subseteq \mathcal {S}(A,b)$ and is completely consistent if $\mathcal {S}(1 - G,k) = \mathcal {S}(A,b)$. If $\mathcal {S}(A,b)$ is nonempty, then the method is consistent if and only if $G = I - MA$ and $k = Mb$ for some matrix M. The method is completely consistent if $G = I - MA$ and $k = Mb$ for some nonsingular matrix M. This generalizes the elementary result that if A is nonsingular, then the method is completely consistent if and only if $I - G$ is nonsingular and $k = (I - G)A^{ - 1} b$. It is also shown that if $S(I - G,k)$ is nonempty, then the method is reciprocally consistent if and only if $A = Q(I - G)$ and $b = Qk$ for some matrix Q.