An over-determined system is a set of equations that has more independent equations than unknowns. For such a system the matrix inverse method and Cramer’s method will not work because the A matrix is not square. However, some over-determined systems have exact solutions, and they can be obtained with the left-division method x = A/b.For other over-determined systems, no exact solution exists. In some of these cases, the left-division method does not yield an answer, while in other cases the left-division method gives an answer that satisfies the equation set only in a “least squares” sense, as explained in Example 6.5-1. When MATLAB gives an answer to an over-determined set, it does not tell us whether the answer is the exact solution. We must determine this information ourselves, as shown in Example 6.5-2.
Some over-determined systems have an exact solution. The left-division method sometimes gives an answer for over-determined systems, but it does not indicate whether the answer is the exact solution. We need to check the ranks of A and [A b] to know whether the answer is the exact solution. The next example illustrates this situation.
To interpret MATLAB answers correctly for an over-determined system, first check the ranks of A and [A b] to see whether an exact solution exists; if one does not exist, then you know that the left-division answer is a least squares solution.
Test Your Understanding
T6.5-1 Use MATLAB to solve the following set:
x – 3y = 2
3x + 5y = 7
70x – 28y = 153
(Answer: The unique solution, x = 2.2143, y = 0.0714, is given by the
left-division method.) .
T6.5-2 Use MATLAB to solve the following set:
x – 3y = 2
3x +5y = 7
5x – 2y =-4
(Answer: No exact solution.)