Sceintific notation is convienent to express the two-place restriction.
We have .
The has no apparent effect.
This intersect-the-lines problem contrasts with the example
Illustrate that in this system
some small change in the numbers will produce only a
small change in the solution by changing the constant in the
bottom equation to and solving.
Compare it to the solution of the unchanged system.
Solve it by
rounding at each step to four significant digits.
The fully accurate solution is that and .
The four-digit conclusion is quite different.
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
as unequal to .
Thus, computer arithmetic is not associative.
Compare the computer's version of
Is twice the first equation the same as the second?
For the first one, first, is
For the other one, first
The first equation is
while the second is
Ill-conditioning is not only dependent on the matrix of
This example (Hamming 1971) shows that it can arise from an
interaction between the left and right sides of the system.
Let be a small real.
Solve the system by hand.
Notice that the 's divide out only because there is
an exact cancelation 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 .
gives a third equation of .
Substituting into the second equation gives
so and thus .
With those, the first equation says that .