Part 3 - Chapter 9 Linear Systems of Equations: Gauss Elimination

1. Part 3 - Chapter 9 Linear Systems of Equations: Gauss Elimination

2. Gauss EliminationChapter 9 Solving Small Numbers of Equations • There are many ways to solve a system of linear equations: • Graphical method • Cramer’s rule • Method of elimination • Computer methods are necessary for large numbers of equations For n ≤ 3

3. Graphical Method • For two equations: • Solve both equations for x2:

4. Fig. 9.1 • Plot x2 vs. x1 on rectilinear paper, the intersection of the lines present the solution.

6. Figure 9.2 No Solutions Infinite Solutions Ill conditioned

7. Determinants and Cramer’s Rule • Cramer’s rule uses determinants to solve for the vector of unknowns, x.

8. The technique for calculating a determinant is similar to that for calculating the cross product (chapter 8)

9. Is the same as:

10. Cramer’s rule expresses the solution of a system of linear equations in terms of ratios of determinants of the array of coefficients of the equations. For example, x1 would be computed as:

12. Solve Example 9.2 using the det function in MATLAB

15. Method of Elimination • The basic strategy is to • successively solve one of the equations of the set for one of the unknowns • and to eliminate that variable from the remaining equations by substitution. • The elimination of unknowns can be extended to systems with more than two or three equations; however, the method becomes extremely tedious to solve by hand.

16. For example…page 230 3rd edition

17. Naive Gauss Elimination • Extension of method of elimination to large sets of equations by developing a systematic scheme or algorithm to eliminate unknowns and to back substitute. • As in the case of the solution of two equations, the technique for n equations consists of two phases: • Forward elimination of unknowns • Back substitution

19. Pivot row

20. Pivot row

21. Pivot row

24. Example 9.3

25. %% Example 9.3

26. Pitfalls of Elimination Methods • Division by zero. It is possible that during both elimination and back-substitution phases a division by zero can occur. • Round-off errors. • Ill-conditioned systems. Systems where small changes in coefficients result in large changes in the solution.

27. Singular systems. When two equations are identical, we would loose one degree of freedom. Use the fact that the determinant of a singular system is zero to test for singular systems.

28. Techniques for Improving Solutions • Use of more significant figures. • Pivoting. If a pivot element is zero, normalization step leads to division by zero. The same problem may arise, when the pivot element is close to zero. Problem can be avoided: • Partial pivoting. Switching the rows so that the largest element is the pivot element. • Complete pivoting. Searching for the largest element in all rows and columns then switching.

30. Example 9.4Partial Pivoting Use Gauss elimination with and without partial pivoting to solve:

31. Multiply the first equation by 1/.0003

32. Subtract

33. Back Substitute

34. The answer is highly dependent on the number of significant figures you carry

35. Implement Pivoting Make the equation with the largest value of aI,1 the pivot row

36. Multiply the first equation by .0003/1

37. Subtract

38. Back Substitute

39. The answer is much less dependent on the number of significant figures you carry

40. Tridiagonal Systems • A tridiagonal system is a banded system with a bandwidth of 3: • Tridiagonal systems can be solved using the same method as Gauss elimination, but with much less effort because most of the matrix elements are already 0.

41. Tridiagonal System Solver

42. Summary • For small systems of linear equations we can use • Graphical techniques • Cramer’s rule • Method of elimination • For large systems of linear equations we need a computer technique such as gaussian elimination

43. Gaussian Elimination • Naïve Gaussian elimination • Pivoting • Partial Pivoting • Complete Pivoting • Left Division implements Gaussian elimination