Round-off error

Builds on Rounding Floating-point arithmetic Machine epsilon

Definition

Let $F$ be a set of floating-point numbers and let $\mathrm{fl} \colon \R \to F$ be a rounding. The round-off error of representing $x \in \R$ is $\mathrm{fl}(x) - x$ , and for $x \ne 0$ its relative round-off error is $(\mathrm{fl}(x) - x)/x$ . Let an algorithm compute $y = f(x)$ in exact real arithmetic, and let it return $\hat y$ when each operation is done in floating-point arithmetic. Then the round-off error of the computation is $\hat y - y$ .

Remarks

Example: with base $10$ , precision $3$ and rounding to nearest, $\mathrm{fl}(2/3) = 0.667$ , so the round-off error is $0.667 - 2/3 = 1/3000$ and the relative round-off error is $1/2000 = 0.0005$ . This is within the unit roundoff $u = \tfrac12 \cdot 10^{-2}$ of the system. Non-example: replacing $e = \sum_{k \ge 0} 1/k!$ by $\sum_{k=0}^{5} 1/k!$ in exact arithmetic gives an error, but it is a truncation error, not a round-off error. If $x$ lies in the range of normalized numbers and $\mathrm{fl}$ rounds to nearest, then $\abs{\mathrm{fl}(x) - x} \le u \abs{x}$ , where $u = \tfrac12 \beta^{1-p}$ is the unit roundoff, see machine epsilon.

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Tag	`PA1B7DJ0C4`
URI	`m2://PA1B7DJ0C4`
Kind	definition
Writing	W0 stub
MSC	65G50
Defines	Round-off error
Uses	Rounding, Floating-point arithmetic, Machine epsilon

Round-off error

Definition

Remarks

Reading path

Further reading

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