Rounding

Definition

Let $F \subseteq \R$ be unbounded above and below, with only finitely many elements in each bounded interval. An example is the set $\Z$ of integers. For $x \in \R$ let $x^- = \max \set{f \in F : f \le x}$ and $x^+ = \min \set{f \in F : f \ge x}$ . A rounding to $F$ is a map $\mathrm{rd} \colon \R \to F$ with $\mathrm{rd}(x) \in \set{x^-, x^+}$ for every $x$ . The standard roundings are:

rounding down: $\mathrm{rd}(x) = x^-$ ;
rounding up: $\mathrm{rd}(x) = x^+$ ;
rounding toward zero: $\mathrm{rd}(x) = x^-$ if $x \ge 0$ and $x^+$ if $x < 0$ ;
rounding to nearest: $\mathrm{rd}(x)$ is the element of $\set{x^-, x^+}$ closest to $x$ . When $x$ is exactly halfway, a tie rule decides, such as ties to even or ties away from zero.

Remarks

Example: for $F = \Z$ , rounding down is the floor function, rounding up is the ceiling function, and rounding to nearest with ties to even sends $2.5 \mapsto 2$ , $3.5 \mapsto 4$ and $2.7 \mapsto 3$ . Check for $2.7$ : $x^- = 2$ , $x^+ = 3$ , and $\abs{2.7 - 3} = 0.3 < 0.7 = \abs{2.7 - 2}$ . Non-example: the map $x \mapsto \floor{x} - 1$ is not a rounding to $\Z$ , since $\floor{2.7} - 1 = 1 \notin \set{2, 3}$ . Each of the four standard roundings is monotone: $x \le y$ implies $\mathrm{rd}(x) \le \mathrm{rd}(y)$ . An arbitrary rounding need not be monotone, since it may choose $x^+$ for one point and $x^-$ for a larger point of the same gap. For floating-point numbers, $F$ is extended by the subnormal numbers and by $\pm \infty$ to cover overflow.

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Floor function concept
Integers object
Rounding definition

Tag	`N6AFVYBY2Z`
URI	`m2://N6AFVYBY2Z`
Kind	definition
Writing	W0 stub
MSC	65G50, 65Y04
Defines	Rounding
Uses	Floor function, Integers

Rounding

Definition

Remarks

Reading path

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