Gauss-Seidel

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

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. 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. 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 x1, the second for x2 and the third x3 to yield:

5. Gauss-Seidel • Value for x1, x2 and x3:

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

7. Gauss-Seidel • First iteration: • Guessing x2 and x3 =0 • Guessing x3 = 0

8. Gauss-Seidel • Second iteration: • Insert previous x2 and x3 • Insert previous x1 and x3

9. Ex. 11.3 • Use Gauss-Seidel technique to solve: • Solution: (recalled: x1=3, x2=-2.5, x3=7)

10. Solution • First iteration:

11. Solution • First iteration: • Guessing x2 and x3 =0 • Guessing x3 = 0

12. Solution • Second iteration:

13. Problem 11.10 • Use Gauss-Seidel method to solve following system until the percent relative error falls below s = 5%: • Solution: • x1 = 0.500253, x2 = 8.000112 and x3 = 6.00007.