1. 17 Jul 2007 KKKQ 3013 PENGIRAAN BERANGKA Week 2 – Systems of Linear Equations 17 July 2007 8.00 am – 9.00 am

2. 17 Jul 2007 Week 2 Page 2 Topics Introduction Elimination Methods Decomposition Methods Matrix Inverse and Determinant Errors, Residuals and Condition Number Iteration Methods Incomplete and Redundant Systems

3. 17 Jul 2007 Week 2 Page 3 Tutorial Example 1 Refer exercise in Chapter 1 : Question #1 Forward elimination : (a) Solve [A].x = b [A]

4. 17 Jul 2007 Week 2 Page 4 Tutorial Example 1 Back substitution : Now, compare with Gauss Elimination Excel spreadsheet

5. 17 Jul 2007 Week 2 Page 5 Tutorial Example 1 (b) Determinant det (A) = (1).(1).(2).(0.5) = 1 det (A) = (-1) p.a 11. a 22 (1).a 33 (2).a 44 (3) = (-1) 0.(1).(1).(2).(0.5) = 1 For triangular matrix: det [A] = product of diagonal terms, a ii

6. 17 Jul 2007 Week 2 Page 6 Tutorial Example 1 (c) LU decomposition of [A] Now, check if [L].[U] = [A]

7. 17 Jul 2007 Week 2 Page 7 Tutorial Example 2 Solve using Gauss-Seidel iteration Hint: let z k = x k + y k i

8. 17 Jul 2007 Week 2 Page 8 Tutorial Example 2 Now, compare with Gauss Elimination Excel spreadsheet