1. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 Chapter 11

2. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2 Special Matrices and Gauss-Seidel Chapter 11 Certain matrices have particular structures that can be exploited to develop efficient solution schemes. –A banded matrix is a square matrix that has all elements equal to zero, with the exception of a band centered on the main diagonal. These matrices typically occur in solution of differential equations. –The dimensions of a banded system can be quantified by two parameters: the band width BW and half-bandwidth HBW. These two values are related by BW=2HBW+1. Gauss elimination or conventional LU decomposition methods are inefficient in solving banded equations because pivoting becomes unnecessary.

3. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 3 Figure 11.1

4. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 4 Tridiagonal Systems A tridiagonal system has a bandwidth of 3: An efficient LU decomposition method, called Thomas algorithm, can be used to solve such an equation. The algorithm consists of three steps: decomposition, forward and back substitution, and has all the advantages of LU decomposition.

5. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 5 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.

6. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 6 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.

7. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 7 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.

8. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 8 Fig. 11.4

9. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 9 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

10. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 10 Similarly, in case of two simultaneous equations, the Gauss-Seidel algorithm can be expressed as

11. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 11 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:

12. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 12 Figure 11.5