1. Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. by Lale Yurttas, Texas A&M University Chapter 111

2. Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. by Lale Yurttas, Texas A&M University Chapter 112 Gauss-Seidel Iterative or approximate methods provide an alternative to the elimination methods. The Gauss-Seidel method is the most commonly used iterative method. The system [A]{X}={B} is reshaped by solving the first equation for x 1, the second equation for x 2, and the third for x 3, …and n th equation for x n. For conciseness, we will limit ourselves to a 3x3 set of equations.

3. Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. by Lale Yurttas, Texas A&M University Chapter 113 Now we can start the solution process by choosing guesses for the x’s. A simple way to obtain initial guesses is to assume that they are zero. These zeros can be substituted into x 1 equation to calculate a new x 1 =b 1 /a 11.

4. Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. by Lale Yurttas, Texas A&M University Chapter 114 New x 1 is substituted to calculate x 2 and x 3. The procedure is repeated until the convergence criterion is satisfied: For all i, where j and j-1 are the present and previous iterations.

5. Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. by Lale Yurttas, Texas A&M University Chapter 115 Fig. 11.4

6. Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. by Lale Yurttas, Texas A&M University Chapter 116 Convergence Criterion for Gauss- Seidel Method The Gauss-Seidel method has two fundamental problems as any iterative method: –It is sometimes nonconvergent, and –If it converges, converges very slowly. Recalling that sufficient conditions for convergence of two linear equations, u(x,y) and v(x,y) are

7. Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. by Lale Yurttas, Texas A&M University Chapter 117 Similarly, in case of two simultaneous equations, the Gauss-Seidel algorithm can be expressed as

8. Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. by Lale Yurttas, Texas A&M University Chapter 118 Substitution into convergence criterion of two linear equations yield: In other words, the absolute values of the slopes must be less than unity for convergence:

9. Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. by Lale Yurttas, Texas A&M University Chapter 119 Figure 11.5