1. 1 Gauss-Seidel Lecture Notes Dr. Rakhmad Arief Siregar Universiti Malaysia Perlis Applied Numerical Method for Engineers Chapter 11

2. 2 Introduction In the previous discussion of Gauss elimination an elimination has been introduced In this chapter, focused on Gauss-Seidel method, iterative method will be disscussed. Gauss-Seidel method is particularly well suited for large numbers of equation, in which round-off error may occurred if using gauss elimination

3. 3 Gauss-Seidel Iterative or approximate methods provide an alternative to elimination methods. This method almost similar to the techniques we developed to obtain the roots of single equation in Chap. 6. Gauss-Seidel will utilize guessing value.

4. 4 Gauss-Seidel Assume that we are given a set of n equations: Suppose we limit ourselves to a 3 x 3 set of equations If the diagonal element are all nonzero, the first equation can be solved for x 1, the second for x 2 and the third x 3 to yield:

5. 5 Gauss-Seidel Value for x 1, x 2 and x 3 :

6. 6 Convergent Gauss-Seidel iterative Methods The number of significant figures

7. 7 Gauss-Seidel Guessing x 2 and x 3 =0 Guessing x 3 = 0 First iteration:

8. 8 Gauss-Seidel Insert previous x 2 and x 3 Insert previous x 1 and x 3 Second iteration:

9. 9 Ex. 11.3 Use Gauss-Seidel technique to solve: Solution: (recalled: x 1 =3, x 2 =-2.5, x 3 =7)

10. 10 Solution First iteration:

11. 11 Solution Guessing x 3 = 0 First iteration: Guessing x 2 and x 3 =0

12. 12 Solution Second iteration:

13. 13 Problem 11.10 Use Gauss-Seidel method to solve following system until the percent relative error falls below  s = 5%: Solution: x 1 = 0.500253, x 2 = 8.000112 and x 3 =  6.00007.

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