Rounding Errors

Given a real number $p$ and an approximation for it $p\ast$, we can find how much error there is in the approximation.

The actual error is $p-p\ast$, the absolute error is $\Vert p-p\ast \Vert ,$ and the relative error is:

$$\frac{|p-p\ast |}{|p|}.$$

When using computers to do floating-point arithmetic, rounding errors will occur because floating-point numbers have finite precision.

Nesting Technique

We can use a nesting technique to reduce rounding errors for some calculations.

For example, we can rewrite

$$f(x)=x^3-6.1x^2+3.2x+1.5$$

as

$$f(x)=(x-6.1)x+3.2)x+1.5.$$