Accuracy of Computations

Using two decimal places, add $253$ and $2/3$.

Consider this system ().

• (a)  Solve it.     (b)  Solve it by rounding at each step to four digits.

Rounding inside the computer often has an effect on the result. Assume that your machine has eight significant digits.

• Show that the machine will compute $(2/3)+((2/3)-(1/3))$ as unequal to $((2/3)+(2/3))-(1/3)$. Thus, computer arithmetic is not associative.

• Compare the computer’s version of $(1/3)x+y=0$ and $(2/3)x+2y=0$. Is twice the first equation the same as the second?

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, roundin…