Determination of the optimal value of relaxation parameter in symmetric SOR method for rectangular coefficient matrices

doi:10.1016/j.amc.2006.01.069

Applied Mathematics and Computation

Volume 181, Issue 2, 15 October 2006, Pages 1018-1025

https://doi.org/10.1016/j.amc.2006.01.069 Get rights and content

Abstract

In this paper, we obtain the optimal value of relaxation parameter in symmetric SOR (SSOR) method. By SSOR method we find the least square solution of minimal norm to the linear system Ax = b where A is an m by n matrix.

Introduction

Consider the following linear system: $Ax = b,$ where A is an m × n real matrix, x and b are real n-vector and m-vector, respectively. Tian [8] presented the problem of finding the least square solution of minimal norm to the linear system (1). Iterative methods for solving the problem using splittings of the coefficient matrix A and extending results for the case in which A is a square invertible matrix have been studied by some authors, such as Berman and Plemmons [2], Neumann [7], Miller and Neumann [6] and Tian [9]. Chen [3] considered the case that rank(A) = n in system (1). He suggested a combined direct-iterative method by augmenting the system (1) to a square nonsingular system to solve the least square problem. Miller and Neumann [6] have shown that Chen’s augmentation scheme for the full column rank case can be extended to the case where rank(A) = r and r < min{m, n}. They developed the theory of successive overrelaxation iterative method to solve the augmented and consistent linear system. They augmented the system to a block 4 × 4 consistent system and splitted the augmented coefficient matrix by a subproper SOR splitting. They also determined an interval for the relaxation parameter in which the subproper SOR iteration matrix is semiconvergent. The optimal relaxation parameter which minimizes the modulus of the controlling eigenvalue of the SOR matrix and a method to transform a solution to the augmented system to the solution of minimal two-norm were given. Hadjidimos and Yeyios [5] introduced symmetric SOR (SSOR) for solving of square system Ax = b where A is a nonsingular matrix. Tian [8] developed the theory of accelerated overrelaxation iterations to solve the least square of minimal norm to the system (1).

In [4] we developed SSOR method to find the least square solution of minimal norm to the system (1). In this paper, we obtain the optimal value of relaxation parameter in SSOR method in a special case. Numerical examples show the experimental results coincide the analytical optimal value of the parameter.

This paper is organized as follows. In the next section the SSOR method for rectangular matrices is given. In Section 3, we compute the optimal value of relaxation parameter for SSOR method. We give some numerical examples in Section 4. Finally, Section 5 gives the paper conclusion.

Section snippets

SOR for rectangular matrices

Consider a system of n linear equations with n unknowns written in matrix form $Ax = b,$ where matrix A has nonzero diagonal elements. We consider the following splitting of A: $A = D - L - U,$ where D is a diagonal matrix and L and U are strictly lower and upper triangular matrices, respectively. Let x^(k) be kth approximation of solution of (2) by SOR method using splitting (3). In symmetric SOR we obtain $x^{(k + \frac{1}{2})}$ as follows [5]: $x^{(k + \frac{1}{2})} = (D - ω L)^{- 1} ((1 - ω) D + ω U) x^{(k)} + ω (D - ω L)^{- 1} b$ or $x^{(k + \frac{1}{2})} = L_{ω} x^{(k)} + c,$ where $L_{ω} = (D - ω L)^{- 1} ((1 - ω) D + ω U$

Finding the optimal value of the relaxation parameter

Let the eigenvalues of T_ω are real. In this section, we obtain the optimum value of parameter ω in this special case of T_ω. If the eigenvalues of T_ω are real, then as T_ω is a real matrix its eigenvectors are real vectors, too. From Eq. (22) we can write $t = [\begin{matrix} v_{1}^{t} & v_{2}^{t} \end{matrix}] [\begin{matrix} 0 & - B \\ B^{t} & 0 \end{matrix}] [\begin{matrix} v_{1} \\ v_{2} \end{matrix}] = {(v_{1}^{t} {Bv}_{2})}^{t} - v_{1}^{t} {Bv}_{2},$ where $v = [\begin{matrix} v_{1} \\ v_{2} \end{matrix}]$ . As by our hypothesis $v_{1}^{t} {Bv}_{2}$ is a real value, hence for every vector v we have t = 0. Thus, Eq. (21) changes to $λ = 1 - \frac{ω (2 - ω)}{1 - ω^{2} x^{2}} .$ Eq. (23) shows λ is a function of x, so we can obtain the extreme values

Numerical examples

In this section we give some numerical examples. They are two consistent and one inconsistent linear systems. For each example, we solve the problem by SSOR iterative method for some values of ω. These values are optimal value of ω or ω_opt and two arbitrary values, one less than ω_opt and the other one greater than it. We show the number of iterations for all examples in Table 1.

Example 1

Consider the following system: $[\begin{matrix} 2 & 3 & - 5 \\ 4 & 5 & 3 \\ 7 & 6 & - 9 \\ 6 & 8 & - 2 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}] = [\begin{matrix} 0 \\ 12 \\ 4 \\ 12 \end{matrix}]$ this is a consistent system, the rank of coefficient matrix

Conclusion

In this paper, we obtained the optimal value for relaxation parameter of SSOR method in a special case. This can be very useful to apply SSOR method for solving the least square solution of minimal norm of linear systems with rectangular coefficient matrices.

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