Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 Part 3- Chapter 12 Iterative Methods
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2 Iterative Methods Iterative or approximate methods provide an alternative to the elimination methods – like Gaussian Elimination (Chapter 9). The Gauss-Seidel method is the most commonly used iterative method for linear systems
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Gauss - Seidel 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. 3
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 4 Next, choose guesses for the x’s. (A simple way to obtain initial guesses is to assume that they are zero.) Substitute the guesses into the x 1 equation to calculate a new x 1. (If we choose zeros for our guesses x 1 =b 1 /a 11 )
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 5 New x 1 is substituted to calculate x 2 …. The updated x 1 and x 2 are used to calculate x 3 The procedure is repeated until the convergence criterion is satisfied for each x For all i, where j and j-1 are the present and previous iterations.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Gauss Seidel Jacobi
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Reformulate 9
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Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 11 Gauss-Seidel Left-Division
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Convergence and diagonal dominance The Gauss-Seidel technique does not always converge. The following criteria is sufficient to guarantee convergence, but not necessary 12 If you meet this criteria you converge If don’t, it still might converge
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 13 The absolute value of the numbers on the diagonal should be bigger than the sum of the absolute value of the other numbers on the same row.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Consider the system of equations from Example The big numbers go on the diagonal
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Relaxation A weighting factor can enhance the rate of convergence 15 λ=0 to 2 λ=0 to 1 underrelaxation λ=1 to 2 overrelaxation Used to make a nonconvergent system converge Used to make a slowly converging system converge more quickly
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Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 12.2 with Relaxation 17 Is not diagonally dominant
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 18 Result from Gauss Seidel Result from Left Division
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 19 diagonally dominant
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 20 Result from Gauss Seidel Result from Left Division
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Non-linear systems We can approach non-linear systems in a very similar manner Successive Substitution 21
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 12.3 Use successive substitution to solve the following set of non-linear simultaneous equations. 22
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Solve for x’s 23 It turns out that for initial guesses of 1.5 and 3.5 this technique diverges If you set it up differently, you can get it to converge This technique is so tempermental that it has limited utility
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Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. An alternative is to use a variation on the Newton Raphson technique Uses partial differential equations We do not have time to cover this approach 27