1. The Gauss Jordan method
Major difference - eliminate unknowns from all rows, not just subsequent ones
Normalize matrix so all entries are 1
Leads to identity matrix instead of upper triangular
Backsubstitution is easy

2. Example
First pivot

3. Normalize pivot row

4. Multiply 1st row by 3 and subtract from 2nd row

5. Do the other two rows
Now pivot again

6. Normalize

7. Multiply 2nd row by 0.75 and subtract from first row

8. For first row
and after all eliminated

9. No need to pivot, so normalize
Work on rows 1,2 and 4 with row 3

10. No rows below row 4 to pivot with, so normalize
and eliminate column 4

11. We now have our answer, since backsubstitution is trivial

12. LU decomposition - another method for solving matrix equations
Idea behind LU decomposition - start with or

13. We know (because we did it in G.E.) we can write
i.e or

14. Assume there exists [L]
such that

15. means that

16. LU method
1) factor (decompose) A into L and U
2) given b, determine d from Ld=b
3) using Ux=d and backsubstitution, solve for x
Advantage: Once you have L and U, can use many different b’s

17. How do you get L and U?
Gauss elimination gives you U.
It also gives you L.
The factors are the entries in L

18. Changes in algorithm for Gauss elimination for LU decomposition
loop over all the rows except the last one
loop over all the rows below the current one
get fik = aik/akk
multiply row k by f and subtract from row i
put fik in L at row i, column k
end loop
A is now upper triangular U
make all Lkk=1

19. A fancier way of storing L and U
Good if n is large
More overhead to sort out

20. Pivoting in LU decomposition
Still need it
Messes up order of L
What to do?
Need to pivot also both L and a permutation matrix P

21. Initialize P as identity matrix and pivot when A is pivoted
Initialize P as identity matrix and pivot when A is pivoted. Also pivot L

22. Example
Starting out

23. No pivot

24. Now exchange rows 2 and 4

25. The pivot factors are

26. No pivot again, factor

27. Now make the diagonal elements of L=1

28. Recall

30. LU method
1) factor (decompose) A into L and U
2) given b, determine d
3) using Ux=d and backsubstitution, solve for x
Advantage: Once you have L and U, can use many different b’s

31. Example (no pivoting):

32. Get d

33. Use Ux=d and backsubstitute

34. Now change b
We don’t have to do elimination again
Use the same L and U

35. Get d

36. Use Ux=d and backsubstitute