Gaussian Elimination

1. Gaussian Elimination Chemical Engineering Majors Author(s): Autar Kaw http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates

2. Naïve Gauss Eliminationhttp://numericalmethods.eng.usf.edu

3. Naïve Gaussian Elimination A method to solve simultaneous linear equations of the form [A][X]=[C] Two steps 1. Forward Elimination 2. Back Substitution

4. Forward Elimination The goal of forward elimination is to transform the coefficient matrix into an upper triangular matrix

5. Forward Elimination A set of n equations and n unknowns . . . . . . (n-1) steps of forward elimination

6. Forward Elimination Step 1 For Equation 2, divide Equation 1 by and multiply by .

7. Forward Elimination Subtract the result from Equation 2. − _________________________________________________ or

8. Forward Elimination Repeat this procedure for the remaining equations to reduce the set of equations as . . . . . . . . . End of Step 1

9. Forward Elimination Step 2 Repeat the same procedure for the 3rd term of Equation 3. . . . . . . End of Step 2

10. Forward Elimination At the end of (n-1) Forward Elimination steps, the system of equations will look like . . . . . . End of Step (n-1)

11. Matrix Form at End of Forward Elimination

12. Back Substitution Solve each equation starting from the last equation Example of a system of 3 equations

13. Back Substitution Starting Eqns . . . . . .

14. Back Substitution Start with the last equation because it has only one unknown

15. Back Substitution

16. THE END http://numericalmethods.eng.usf.edu

17. Naïve Gauss EliminationExamplehttp://numericalmethods.eng.usf.edu

18. Example: Liquid-Liquid Extraction A liquid-liquid extraction process conducted in the Electrochemical Materials Laboratory involved the extraction of nickel from the aqueous phase into an organic phase. A typical set of experimental data from the laboratory is given below: Assuming g is the amount of Ni in organic phase and a is the amount of Ni in the aqueous phase, the quadratic interpolant that estimates g is given by

19. Example: Liquid-Liquid Extraction The solution for the unknowns x1, x2, and x3 is given by Find the values of x1, x2, and x3 using Naïve Gauss Elimination. Estimate the amount of nickel in organic phase when 2.3 g/l is in the aqueous phase using quadratic interpolation.

20. Number of Steps of Forward Elimination Number of steps of forward elimination is (n-1)=(3-1)=2

21. Example: Liquid-Liquid Extraction Solution Forward Elimination: Step 1 Yields

22. Example: Liquid-Liquid Extraction Forward Elimination: Step 1 Yields

23. Example: Liquid-Liquid Extraction Forward Elimination: Step 2 Yields This is now ready for Back Substitution

24. Example: Liquid-Liquid Extraction Back Substitution: Solve for x3 using the third equation

25. Example: Liquid-Liquid Extraction Back Substitution: Solve for x2 using the second equation

26. Example: Liquid-Liquid Extraction Back Substitution:Solve for x1 using the first equation

27. Example: Liquid-Liquid Extraction The solution vector is The polynomial that passes through the three data points is then Where g is the amount of nickel in the organic phase and a is the amount of in the aqueous phase.

28. Example: Liquid-Liquid Extraction When 2.3 g/l is in the aqueous phase, using quadratic interpolation, the estimated amount of nickel in the organic phase is

29. THE END http://numericalmethods.eng.usf.edu

30. Naïve Gauss EliminationPitfallshttp://numericalmethods.eng.usf.edu

31. Pitfall#1. Division by zero

32. Is division by zero an issue here?

33. Is division by zero an issue here? YES Division by zero is a possibility at any step of forward elimination

34. Pitfall#2. Large Round-off Errors Exact Solution

35. Pitfall#2. Large Round-off Errors Solve it on a computer using 6 significant digits with chopping

36. Pitfall#2. Large Round-off Errors Solve it on a computer using 5 significant digits with chopping Is there a way to reduce the round off error?

37. Avoiding Pitfalls • Increase the number of significant digits • Decreases round-off error • Does not avoid division by zero

38. Avoiding Pitfalls • Gaussian Elimination with Partial Pivoting • Avoids division by zero • Reduces round off error

39. THE END http://numericalmethods.eng.usf.edu

40. Gauss Elimination with Partial Pivotinghttp://numericalmethods.eng.usf.edu

41. Pitfalls of Naïve Gauss Elimination • Possible division by zero • Large round-off errors

42. Avoiding Pitfalls • Increase the number of significant digits • Decreases round-off error • Does not avoid division by zero

43. Avoiding Pitfalls • Gaussian Elimination with Partial Pivoting • Avoids division by zero • Reduces round off error

44. What is Different About Partial Pivoting? At the beginning of the kth step of forward elimination, find the maximum of If the maximum of the values is in the p th row, then switch rows p and k.

45. Matrix Form at Beginning of 2nd Step of Forward Elimination

46. Example (2nd step of FE) Which two rows would you switch?

47. Example (2nd step of FE) Switched Rows

48. Gaussian Elimination with Partial Pivoting A method to solve simultaneous linear equations of the form [A][X]=[C] Two steps 1. Forward Elimination 2. Back Substitution

49. Forward Elimination Same as naïve Gauss elimination method except that we switch rows before each of the (n-1) steps of forward elimination.

50. Example: Matrix Form at Beginning of 2nd Step of Forward Elimination