Absolute and relative error

What is it?

Formulas

$$e_{\text{abs}} = |x - \hat x|, \qquad e_{\text{rel}} = \frac{|x - \hat x|}{|x|}$$

$$\operatorname{fl}(a \circ b) = (a \circ b)(1 + \delta), \quad |\delta| \le \varepsilon_{\text{mach}} = 2^{-53}$$

the IEEE 754 model of arithmetic

Where it shows up in computing

• Floating point (IEEE 754)★★★★★fundamentalScientific computing and algorithms

  Floating point is designed around a guaranteed relative error per operation.

• Scientific computing★★★★★fundamentalScientific computing and algorithms

  Every numerical result should come with an error estimate; truncation and rounding errors are budgeted separately.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

What depends on it

• Conditioning
• Numerical stability