Numerical Methods نس ...

Introduction. Iterative methods are powerful numerical techniques used to approximate solutions to large systems of linear equations. Unlike direct methods, which aim to find the exact solution in a finite number of steps, iterative methods produce a sequence of approximations that converge to the solution. In this chapter, we will explore various iterative methods for solving linear systems. We will begin by discussing general considerations for iterative methods, including convergence criteria and stopping conditions. Next, we will delve into the Jacobi and relaxation methods, which are classic iterative techniques. We will then examine the Gauss-Seidel and successive relaxation methods, which often exhibit faster convergence rates. Finally, we will discuss practical aspects of implementing iterative methods, such as convergence analysis and choosing appropriate parameters. By understanding these iterative methods, you will gain valuable tools for solving large-scale linear systems that are computationally challenging to solve using direct methods.

Chapter 3  Iterative Methods for Solving Linear Systems

3.1 General Considerations. 

3.2  Jacobi and Relaxation Methods. 

       3.2.1 Jacobi  Method. 

       3.2.2 Relaxation Method. 

3.3 Gauss-Seidel and Successive Relaxation Methods. 

      3.3.1 Gauss-Seidel Method. 

      3.3.2 Successive Relaxation Method (SOR). 

3.4 Remarks on the Implementation of Iterative Methods. 

3.5 Convergence of Jacobi and Gauss-Seidel Methods. 

      3.5.1 Sufficient Conditions for Convergence of Iterative Methods.