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

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

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 r2 r2 – 500 r1 r2 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

4.8 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

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 r1 r2

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 r1 r2 r1 r2 r2 r2 – 0.002 r1 0.00 5.00 1.02 2.03 4.32 1.21 y = 4.24 and x= -1.47

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 r1 r2 r1 r2 r2 r2 – 0.004 r1 r2 r2 – 0.002 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

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