Question

Using concepts of SOR(Successive Over-Relaxation) and SSOR how to choose the relaxation parameter ω and then modify implementation of Gauss-Seidel to find its benefits.

Also explain the concepts of SOR and SSOR and Gauss-Seidel.

Answer 1

SOR (Successive Over-Relaxation) and SSOR (Symmetric Successive Over-Relaxation) are iterative methods used to solve a system of linear equations. The relaxation parameter ω is used to control the convergence rate.

Step-by-step explanation:

SOR (Successive Over-Relaxation) and SSOR (Symmetric Successive Over-Relaxation) are iterative methods used to solve a system of linear equations. These methods are often used when solving large-scale systems with sparse matrices. Both SOR and SSOR use the relaxation parameter ω to control the convergence rate of the iterative process.

Choosing the Relaxation Parameter ω

Choosing the optimal relaxation parameter ω for SOR or SSOR is important to ensure efficient convergence. The value of ω can be found experimentally or by using heuristics. Generally, a good value for ω lies between 0 and 2, with higher values leading to faster convergence in some cases.

Modifying Gauss-Seidel with SOR

Gauss-Seidel is an iterative method similar to SOR but without the relaxation parameter ω. To modify Gauss-Seidel to use SOR, we introduce the relaxation parameter ω into the updating equation. The updated equation becomes xn+1 = (1-ω)xn + ωx'.

Benefits of SOR

The benefits of SOR include faster convergence compared to Gauss-Seidel and improved stability. By carefully selecting the relaxation parameter ω, SOR can converge in fewer iterations and can handle a wider range of coefficient matrices that would otherwise lead to slow or non-convergence.