Symmetric SOR method for augmented systems

doi:10.1016/j.amc.2006.05.094

Applied Mathematics and Computation

Volume 183, Issue 1, 1 December 2006, Pages 409-415

https://doi.org/10.1016/j.amc.2006.05.094 Get rights and content

Abstract

In this paper, we develop symmetric successive overrelaxation (SSOR) method to find solution of augmented systems. These systems have appeared in many different applications of scientific computing, e.g., the finite element approximation to solve the Navier–Stokes equation and the constrained optimization. The convergence of SSOR method are studied. Finally, we apply the new method to solve augmented systems. The iteration number of this method to receive the solution is very better than SOR-like method.

Introduction

An augmented system is taken as $(\begin{matrix} A & B \\ B^{T} & 0 \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} b \\ q \end{matrix}),$ where A is an m × m real symmetric and positive definite matrix and B is an m × n real matrix. Systems like (1) appears in many different applications of scientific computing, such as the finite element approximation to solve the Navier–Stokes equation [4], [5], the constrained least squares problems and generalized least squares problems (see, e.g., [1], [2], [8], [9], [12], [13]) and constrained optimization [10]. There are several studies to solve augmented system (1). Santos et al. [8], and Santos and Yuan [9] have studied preconditioned iterative method to solve the augmented system (1) with A = I and rank deficient B arising from rank deficient least squares problems by considering a full rank submatrix of B as a preconditioner. Yuan [11], [12] and so, Yuan and Iusem [13] have presented several variants of the SOR method and preconditioned conjugate gradient methods to solve general augmented systems like (1) arising from generalized least squares problems where A can be symmetric and positive semidefinite and B can be rank deficient. Golub et al. [6] have presented several SOR-like algorithms to solve augmented system (1).

Hadjidimos and Yeyios [7] introduced symmetric AOR (SAOR) for solving of square system Ax = b where A is a nonsingular matrix. Darvishi and Khosro-Aghdam [3] developed the symmetric successive overrelaxation (SSOR) method to find the least square solution of minimal norm to system Ax = b where A is an m × n complex matrix and rank (A) ⩽ min{m, n}.

In this study, we develop symmetric successive overrelaxation (SSOR) method to solve augmented system (1). In the following section the outline of SSOR method to solve system (1) is provided. In Section 3 we obtain the convergence region for the method. In Section 4, we apply the SSOR method to solve an augmented system. Finally, the paper is concluded in Section 5.

Section snippets

Symmetric SOR method

For the sake of simplicity, we rewrite augmented system (1) as $(\begin{matrix} A & B \\ - B^{T} & 0 \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} b \\ - q \end{matrix}),$ where $A \in R^{m \times m}$ is symmetric and positive definite and $B \in R^{m \times n}$ . For the coefficient matrix of the augmented system (2), we consider the following splitting: $A = (\begin{matrix} A & B \\ - B^{T} & 0 \end{matrix}) = D - A_{l} - A_{u},$ where $D = (\begin{matrix} A & 0 \\ 0 & Q \end{matrix}), A_{l} = (\begin{matrix} 0 & 0 \\ B^{T} & 0 \end{matrix}), A_{u} = (\begin{matrix} 0 & - B \\ 0 & Q \end{matrix})$ and $Q \in R^{n \times n}$ is nonsingular and symmetric. We set $L = D^{- 1} A_{l} = (\begin{matrix} 0 & 0 \\ Q^{- 1} B^{T} & 0 \end{matrix}), U = D^{- 1} A_{u} = (\begin{matrix} 0 & - A^{- 1} B \\ 0 & I \end{matrix}) .$ Let $z^{(n)} = (\begin{matrix} x^{(n)} \\ y^{(n)} \end{matrix})$ be the nth approximation of solution (2) by SOR method using splitting (3). In symmetric SOR we obtain $z^{(n + \frac{1}{2})}$ as follows: $z^{(n + \frac{1}{2})} = ℓ_{ω}$

Convergence region

We study the convergence region for parameter ω, in symmetric SOR method to solve augmented system (2).

Theorem 1

Suppose that μ is an eigenvalue of Q⁻¹B^TA⁻¹B. If λ satisfies $(1 - ω) (1 - λ) (λ - (ω - 1)^{2}) = ω^{2} (ω - 2)^{2} λ μ$ then λ is an eigenvalue of ȷ_ω. Conversely, if λ is an eigenvalue of ȷ_ω such that λ ≠ (ω − 1)², λ ≠ 1 and μ satisfies (11), then μ is a nonzero eigenvalue of Q⁻¹B^TA⁻¹B.

Proof

Suppose that λ and x are eigenvalue and eigenvector of ȷ_ω, respectively. Then we have $_{ω} x = λ x$ or $[(I - ω L) (I - ω U)]^{- 1} [(I - ω L) (I - ω U) - ω (2 - ω) D^{- 1} A] x = λ x$ hence $[(I - ω L$

Numerical example

In this section, we give an example to illustrate the SSOR method to find the solution of the related augmented system. The results are compared with the results obtained by algorithms of SOR-like method that presented by Golub et al. [6]. We report the number of iterations (denoted by IT) and norm of absolute value of error vectors (denoted by “RES”). Here, “RES” is defined to be $RES = \sqrt{‖ b - {Ax}^{(k)} - {By}^{(k)} ‖_{2}^{2} + ‖ q - B^{T} x^{(k)} ‖_{2}^{2}}$ with $(x^{(k)^{T}}, y^{(k)^{T}})^{T}$ the final approximate solution. All results show our method is

Conclusion

The SSOR method is a simple and powerful technique for solving the augmented systems. Its simplicity lies in the fact that one parameter is presented. Full exploitation of the presence of this parameter will provide us with methods which will converge faster then SOR-like method. The determination of optimum value of the parameter needs further studies.

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View full text

Symmetric SOR method for augmented systems

Abstract

Introduction

Section snippets

Symmetric SOR method

Convergence region

Numerical example

Conclusion

Appl. Math. Comput.

Math. Comput. Simulat. XXIV

J. Comput. Appl. Math.

J. Comput. Appl. Math.

On the augmented system approach to sparse least squares problems

Numer. Math.

Solution of augmented linear systems using orthogonal factorization

BIT

Fast nonsymmetric iteration and preconditioning for Navier–Stokes equations

SIAM J. Sci. Comput.