September 27

Numerical computation involves constant compromising between the theoretically perfect objects we model, and the finite resources of a computer. Consider the function:

$$e^x = 1 + x + x^2/2! + x^3/3! + \cdots$$

The infinite series on the right converges to the true value. In theory you can get as close as you want to $e^x$ by simply calculating enough terms. In practice, your wrist gets tired of the repeated multiplication and addition when you do it by hand. So use a computer.

It turns out that although computers do a fine job of calculating such a function, there are some surprising sources of error that must be guarded against.