Linear equations often appear in different disciplines, from physics, engineering, to economics. One of the main challenges in solving a large linear equation system is finding solutions efficiently. This is where iterative methods play an important role. Alih-instead of using direct methods such as Gauss elimination or LU decomposition, itative methods try to approach solutions by doing repeated calculations until solutions approach the desired level of accuracy.

1. pliant to Iterative Method

The iterative method is a numerical technique aimed at finding linear systems solutions by gradually redoing iterations. This technique is very useful for large-sized systems, where direct methods require computation and great memory. Two of the most common literacy methods are Jacobi Method and Gauss-Seidel Method.

2. Jacobi Method

Jacobi's method is one of the most basic literacy methods. The process consists of splitting every equation in the system into an expressive form of one variable, then using the temporary solution of each iteration to update the solution in the next iteration.

Jacobi's method steps can be summed up as follows:

• Separate AAA coefficient matrix into two parts: DDD (diagonal matrix) and RRR (rest).
• In each iteration, a new x (k + 1) x {(k + 1)} x (k + 1) is counted using the solution from previous iteration x (k) x (k) x (k) x (k) x (k) x (k) x (k).

The iteration equation can be written as: x (k + 1) = D religion 1 (b Rx (k)) x (k + 1)] x (b + 1)} x (k + 1) = D (b Berrx (k)

The excess of this method is its simplicity, but the lack of it lies in a slow-less convergence, especially for a system with a small diagonal element.

3. Gauss-Seidel Method

The GaussThe difference is that Gauss -Seidel uses a new renewable solution in the same iteration, so often these methods are faster than Jacobi in terms of convergence.

Step-step of the Gauss-Seidel method:

• Just like Jacobi, separate the AAA coefficient matrix into two parts: the diagonal matrix DDD and other parts L + UL + UL + U, where the LLL is the bottom triangle matrix, and the bill is the upper triangle matrix.
• The next incision is calculated using the updated new solution.

The equation is:

This method is generally faster than Jacobi in terms of convergence, but still depends on the structure of the matrix. The matrix that dominates diagonal is more likely to ensure convergence.

Four. An excess and Lack of Iterative Methods

The iterative method has several advantages compared to the direct method, among which:

• More memory efficient, because it doesn't require a matrix storage in decomposition form.
• Punctuality: easily applicable to a very large system.
• Parallelization(Laughter)

However, there is also a weakness that needs to be noticed:

• Slow convergence In some cases, especially if the coefficient matrix doesn't dominate the diagonal.
• Method Selection that is inaccurate can cause divergence, or solutions that are never achieved.

5. Current Application and Relevance

iterative methods are very relevant in many areas. In the world of modern scientific computing, these methods are used in numerical simulations, structure analysis, to image processing. For example, in fluid simulations or electromagnetic problems, linear equations systems with thousands to millions of variables are often resolved. Algorithm Conjugate Gradient Method and Method Multigrid is the development of more sophisticated and efficient literacy techniques.

With the increasing need for parallel computing, the italic method is also adopted in a high computing environment, where they can be applied massive in multiple processor simultaneously.

Conclusion

The itative method provides an effective and efficient alternative to completing a large linear system. By understanding its advantages and flaws, and knowing when to use it, it can help solve complex and challenging computing problems. Various development of modern literacy methods also continue to support applications in fields such as physics, engineering and data analysis.

source:Saad, Y. (2003). Iterative Method for Spare Linear Systems. Shit.