Terry D. Johnson
10.001 Fall 2000
 
 

In the problem below, we have order of magnitude differences between coefficients in the different rows.
 

Step 0a: Find the entry in the left column with the largest absolute value. This entry is called the pivot.

Step 0b: Perform row interchange (if necessary), so that the pivot is in the first row.
 

Step 1: Gaussian Elimination
 

Step 2: Find new pivot

Step 3: Switch rows (if necessary)
 

Step 4: Gaussian Elimination
 

Step 5: Find new pivot

Step 6: Switch rows (if necessary)
 

Step 7: Gaussian Elimination
 

Step 8: Back Substitute
 

-0.2x₄ = -0.05;    x₄ = 4

100x₃ + 200x₄ = 800;   x₃ = 0

x₂ + 2x₃ + x₄ =4;   x₂ = 0

x₁ + 2x₂ + x₃ = 1;   x₁ = 1
 

Pivoting helps reduce rounding errors; you are less likely to add/subtract with very small number (or very large) numbers.