1. 4.8 Pivoting
Implementing Gaussian Elimination computationally can lead to problems
Happens if the element used to create zeros is small relative to other elements its column.
E.g.
0.01
5.00
1.02
2.03
Small compared to 5.00

2. 4.8 Pivoting
Solve these two systems by G.E.
Keep working to 3 s.f.
0.01
5.00
1.02
2.03
4.32
1.21
0.02
5.00
1.01
2.02
4.31
1.22
Determinant is roughly -5 in both cases
Hence, get unique solution in both cases

3. 4.8 Pivoting
0.01
5.00
1.02
2.03
4.32
1.21
0.02
5.00
1.01
2.02
4.31
1.22
r r2 – 500 r1
r r2 – 250r1
0.01
0.00
1.02
-508
4.32
-2160
0.02
0.00
1.01
-250
4.31
-1080
Big difference in x-value
y = 4.25 and x= -1.5
y = 4.32 and x= -2.66

4. 4.8 Pivoting
0.01
5.00
1.02
2.03
4.32
1.21
0.02
5.00
1.01
2.02
4.31
1.22
Get rid of small entries by pivoting
Because circled entries are small we swap rows 1 and 2

5. 4.8 Pivoting
0.01
5.00
1.02
2.03
4.32
1.21
0.02
5.00
1.01
2.02
4.31
1.22
r r2

6. 4.8 Pivoting
0.01
5.00
1.02
2.03
4.32
1.21
0.02
5.00
1.01
2.02
4.31
1.22
r r2
r r2
r r2 – r1
0.00
5.00
1.02
2.03
4.32
1.21
y = 4.24 and x= -1.47

7. 4.8 Pivoting
0.01
5.00
1.02
2.03
4.32
1.21
0.02
5.00
1.01
2.02
4.31
1.22
r r2
r r2
r r2 – r1
r r2 – r1
0.00
5.00
1.02
2.03
4.32
1.21
0.00
5.00
1.00
2.02
4.31
1.22
x-values comparable
y = 4.24 and x= -1.47
y = 4.31 and x= -1.50

8. 4.8 Pivoting Pivoting reduces numerical error by swapping rows
Use when our chosen element is small compared to other entries in its column