Definition:Calculation Rounding Error

Definition

Let $S$ be a set of continuous data.

Let $\map C S$ be a calculation which is to be made on $S$.

The calculation rounding error of $S$ is defined as:

$R = \size {\map C S - \overline {\map C {\overline S} } }$

where:

$\overline x$ denotes the rounded value of a given $x$

$\size {\, \cdot \,}$ denotes the absolute value function.

Examples

Arbitrary Example

Consider the equation:

$x = \dfrac 1 {1 - \cos 1 \degrees}$

Evaluating the calculation while rounding to $4$ decimal places gives:

$x = 5000$

but the true value is $6565.8$ to $1$ decimal place.

Hence the calculation rounding error of this calculation is $1.6565.8$, or some $24 \%$ or $31 \%$ relative error, depending on how the latter is calculated.

Also see

• Definition:Cumulative Rounding Error, an instance of a calculation rounding error

• Results about calculation rounding errors can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): error: 1. (in numerical computation) Rounding (or roundoff) errors
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): error: 1. (in numerical computation) Rounding (or roundoff) errors