Successive over-relaxation

Definition

Successive over-relaxation (SOR) is an iterative method used to accelerate the convergence of iterative solutions to linear systems by applying a relaxation factor that adjusts the update step. By combining the benefits of both the Jacobi and Gauss-Seidel methods, SOR aims to improve the efficiency of convergence in finding accurate solutions, particularly for large, sparse systems of equations. This technique involves selecting an optimal relaxation parameter that enhances the rate of convergence compared to standard iterative methods.

5 Must Know Facts For Your Next Test

1. The relaxation factor in successive over-relaxation is crucial; if chosen improperly, it can lead to divergence rather than convergence.
2. SOR can converge faster than both the Jacobi and Gauss-Seidel methods when applied appropriately, especially for large systems with favorable spectral properties.
3. The optimal relaxation factor for SOR can often be determined empirically or through theoretical analysis, typically falling between 1 and 2.
4. The SOR method is particularly useful in solving sparse linear systems that arise in various applications like engineering and physics.
5. Convergence of SOR can be guaranteed under certain conditions related to the matrix properties, such as being diagonally dominant or positive definite.

Review Questions

• How does successive over-relaxation improve upon traditional iterative methods like Jacobi and Gauss-Seidel?
  • Successive over-relaxation enhances traditional iterative methods by introducing a relaxation factor that accelerates convergence. Unlike Jacobi and Gauss-Seidel, which may converge slowly for certain systems, SOR optimizes the update step through this factor, allowing for quicker approaches to the solution. This makes SOR especially effective for large or sparse linear systems where efficiency is critical.
• What are the key conditions required for the successful application of successive over-relaxation on a linear system?
  • For successful application of successive over-relaxation, key conditions include ensuring that the matrix is either diagonally dominant or positive definite. These properties help guarantee convergence when using SOR. Additionally, choosing an optimal relaxation factor is essential; if it's too high or too low, it can hinder convergence rather than aid it.
• Evaluate the impact of the relaxation factor on the convergence behavior of successive over-relaxation methods in solving linear systems.
  • The relaxation factor significantly impacts how quickly successive over-relaxation converges to a solution. A properly chosen factor can drastically reduce the number of iterations needed compared to Jacobi or Gauss-Seidel methods. However, if this factor is not selected wisely, it may lead to slower convergence or even divergence. Therefore, understanding its influence is crucial for effectively utilizing SOR in practical applications.

Related terms

Relaxation Method: A class of iterative techniques used to solve linear systems, which involves updating the solution iteratively to achieve convergence.

Jacobi Method:

An iterative algorithm for solving a diagonally dominant system of linear equations by using the values from the previous iteration to compute new values.

Gauss-Seidel Method:

An improvement over the Jacobi method, where each new value is used immediately in subsequent calculations, leading to potentially faster convergence.