Exercise 5

Ill-conditioning is not only dependent on the matrix of coefficients. This example shows that it can arise from an interaction between the left and right sides of the system. Let $\varepsilon$ be a small real.

• Solve the system by hand. Notice that the $\varepsilon$’s divide out only because there is an exact cancellation of the integer parts on the right side as well as on the left.

• Solve the system by hand, rounding to two decimal places, and with $\varepsilon=0.001$.