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Monday, January 30, 2012

What are different aspects of iterative relaxation method?

This article explains relaxation in terms of iterative methods. This piece of writing is all about iterative methods or techniques for solving system of equations. Relaxation methods are iterative methods defined for numerical mathematics.

They are extensively used for solving system of equations which include the following types:

- Large sparse linear systems
Relaxation methods are used to solve large sparse linear systems which were like discretizations of finite difference of differential equations.

- Linear equations
Relaxation methods are used for solving linear equations for problems like that of linear least squares problems.

- System of linear inequalities
Iterative or relaxation methods effectively solve the system of linear inequalities similar to the problems that arise during linear programming.

- Non linear system of equations
These days, iterative methods or relaxation methods have been developed for solving non linear system of equations.


1.) Relaxation methods or iterative methods prove to be very effective and important methodology in providing solutions for linear system of equations and especially for the ones that are used to model elliptical partial differential equations such as Poisson’s equation and Laplace’s equation along with its generalization.

2.) These linear systems of equations are basically used to generally use to describe problems related to boundary values in which the value of the function in the solution is indicated or specified on the boundary of a specified domain.

The basic problem is to compute a solution within the boundaries. People often confuse between iterative methods for relaxation and relaxation methods for mathematical optimizations.

The iterative methods for relaxation techniques are not to be confused with relaxations for mathematical optimizations that are used to approximate a difficult problem by a comparative problem which is more simpler than the former one and whose relaxed or iterated solution provides information about the solution which can be taken in to account for the original problem.

The relaxation method for two dimensional problems is used to readily generalize the other numbers of the dimensions.

- The relaxation iterative methods converge under general conditions.
- But, these methods make slow progress as compared to the other competing methods.
- The study of the iterative relaxation methods constitute an essential part of linear algebra since the transformations of the relaxation methods provide pre conditioners for newer methods that are in a way quite excellent.
- In some cases multi grid methods can be used in order to accelerate the methods.
- It is a common problem in path oriented testing to generate the test data that is required to make the program follow a given path.
- This problem is over come using iterative relaxation method.

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