SE301: Numerical Methods Topic 3: Solution of Systems of Linear Equations Lectures 12-17:

1. SE301: Numerical MethodsTopic 3:Solution of Systems of Linear EquationsLectures 12-17: KFUPM Read Chapter 9 of the textbook KFUPM

2. Lecture 12Vector, Matrices, andLinear Equations KFUPM

3. VECTORS KFUPM

4. MATRICES KFUPM

5. MATRICES KFUPM

6. Determinant of a MATRICES KFUPM

7. Adding and Multiplying Matrices KFUPM

8. Systems of Linear Equations KFUPM

9. Solutions of Linear Equations KFUPM

10. Solutions of Linear Equations • A set of equations is inconsistent if there exists no solution to the system of equations: KFUPM

11. Solutions of Linear Equations • Some systems of equations may have infinite number of solutions KFUPM

12. Graphical Solution of Systems ofLinear Equations solution KFUPM

13. Cramer’s Rule is Not Practical KFUPM

14. Lecture 13Naive Gaussian Elimination Naive Gaussian Elimination Examples KFUPM

15. Naive Gaussian Elimination • The method consists of two steps: • Forward Elimination: the system is reduced to upper triangular form. A sequence of elementary operations is used. • Backward Substitution: Solve the system starting from the last variable. KFUPM

16. Elementary Row Operations • Adding a multiple of one row to another • Multiply any row by a non-zero constant KFUPM

17. ExampleForward Elimination KFUPM

18. ExampleForward Elimination KFUPM

19. ExampleForward Elimination KFUPM

20. ExampleBackward Substitution KFUPM

21. Forward Elimination KFUPM

22. Forward Elimination KFUPM

23. Backward Substitution KFUPM

24. Lecture 14Naive Gaussian Elimination Summary of the Naive Gaussian Elimination Example How to check a solution Problems with Naive Gaussian Elimination Failure due to zero pivot element Error KFUPM

25. Naive Gaussian Elimination • The method consists of two steps • Forward Elimination: the system is reduced to upper triangular form. A sequence of elementary operations is used. • Backward Substitution: Solve the system starting from the last variable. Solve for xn ,xn-1,…x1. KFUPM

26. Example 1 KFUPM

27. Example 1 KFUPM

28. Example 1Backward Substitution KFUPM

29. How Do We Know If a Solution is Good or Not • Given AX=B • X is a solution if AX-B=0 • Due to computation error AX-B may not be zero • Compute the residuals R=|AX-B| • One possible test is ????? KFUPM

30. Determinant KFUPM

31. How Many Solutions Does a System of Equations AX=B Have? KFUPM

32. Examples KFUPM

33. Lectures 15-16:Gaussian Elimination with Scaled Partial Pivoting Problems with Naive Gaussian Elimination Definitions and Initial step Forward Elimination Backward substitution Example KFUPM

34. Problems with Naive Gaussian Elimination • The Naive Gaussian Elimination may fail for very simple cases. (The pivoting element is zero). • Very small pivoting element may result in serious computation errors KFUPM

35. Example 2 KFUPM

36. Example 2Initialization step Scale vector: disregard sign find largest in magnitude in each row KFUPM

37. Why Index Vector? • Index vectors are used because it is much easier to exchange a single index element compared to exchanging the values of a complete row. • In practical problems with very large N, exchanging the contents of rows may not be practical since they could be stored at different locations. KFUPM

38. Example 2Forward Elimination--Step 1: eliminate x1 KFUPM

39. Example 2Forward Elimination--Step 1: eliminate x1 First pivot equation KFUPM

40. Example 2Forward Elimination--Step 2: eliminate x2 KFUPM

41. Example 2Forward Elimination--Step 2: eliminate x2 KFUPM

42. Example 2Forward Elimination--Step 3: eliminate x3 Third pivot equation KFUPM

43. Example 2Backward Substitution KFUPM

44. Example 3 KFUPM

45. Example 3Initialization step KFUPM

46. Example 3Forward Elimination--Step 1: eliminate x1 KFUPM

47. Example 3Forward Elimination--Step 1: eliminate x1 KFUPM

48. Example 3Forward Elimination--Step 2: eliminate x2 KFUPM

49. Example 3Forward Elimination--Step 2: eliminate x2 KFUPM

50. Example 3Forward Elimination--Step 2: eliminate x2 KFUPM