Efficient Solutions for Linear Equations with Numerical Methods

1. Numerical MethodsSolution of Systems of Linear Equations

2. Vector, Matrices, andLinear Equations

3. VECTORS

4. MATRICES

5. MATRICES

6. Determinant of a MATRICES

7. Adding and Multiplying Matrices

8. Systems of Linear Equations

9. Solutions of Linear Equations

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

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

12. Graphical Solution of Systems ofLinear Equations Solution x1=1, x2=2

13. Cramer’s Rule is Not Practical

14. Lecture 13Naive Gaussian Elimination Naive Gaussian Elimination Examples

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.

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

17. ExampleForward Elimination

18. ExampleForward Elimination

19. ExampleForward Elimination

20. ExampleBackward Substitution

21. Forward Elimination

22. Forward Elimination

23. Backward Substitution

24. Naive Gaussian Elimination Summary of the Naive Gaussian Elimination Example Problems with Naive Gaussian Elimination Failure due to zero pivot element Error Pseudo-Code

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.

26. Example 1

27. Example 1

28. Example 1Backward Substitution

29. Determinant

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

31. Examples

32. Pseudo-Code: Forward Elimination Do k = 1 to n-1 Do i = k+1 to n factor = ai,k / ak,k Do j = k+1 to n ai,j = ai,j – factor * ak,j End Do bi = bi – factor * bk End Do End Do

33. Pseudo-Code: Back Substitution xn = bn / an,n Do i = n-1 downto 1 sum = bi Do j = i+1 to n sum = sum – ai,j * xj End Do xi = sum / ai,i End Do

34. Gaussian Elimination with Scaled Partial Pivoting Problems with Naive Gaussian Elimination Definitions and Initial step Forward Elimination Backward substitution Example

35. 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

36. Example 2

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

38. 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.

39. Example 2Forward Elimination--Step 1: eliminate x1

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

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

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

43. Example 2Backward Substitution

44. Example 3

45. Example 3Initialization step

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

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

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

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

50. Example 3Forward Elimination--Step 3: eliminate x3