Interval arithmetic
Interval arithmetic also so-called as interval mathematics, interval analysis, or interval computation is the mathematical technique used to put bounds on rounding errors as alive as measurement errors in mathematical computation. Numerical methods using interval arithmetic canreliable, mathematically correct results. Instead of representing a good as the single number, interval arithmetic represents each value as a range of possibilities. For example, instead of estimating the height of someone as exactly 2.0 metres, using interval arithmetic one might bethat the adult is somewhere between 1.97 & 2.03 metres.
Mathematically, instead of working with an uncertain real 
, one working with the ends of an interval ![[a,b]](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
that contains . In interval arithmetic, any variable lies in the closed interval between 
in addition to 
. A function 
, when applied to , yields an uncertain result; produces an interval ![[c,d]](https://wikimedia.org/api/rest_v1/media/math/render/svg/d85b3b21d6d891d97f85e263d394e3c90287586f)
which includes any the possible values for 
for all ![x\in [a,b]](https://wikimedia.org/api/rest_v1/media/math/render/svg/026357b404ee584c475579fb2302a4e9881b8cce)
.
Interval arithmetic is suitable for a race of purposes. The near common usage is in software, to keep track of rounding errors in calculations and of uncertainties in the cognition of the exact values of physical and technical parameters. The latter often arise from measurement errors and tolerances for components or due to limits on computational accuracy. Interval arithmetic also enables find guaranteed solutions to equations such(a) as differential equations and optimization problems.
Introduction
The leading objective of interval arithmetic is a simple way to calculate upper and lower bounds for the range of a function in one or more variables. These endpoints are not necessarily the true supremum or infimum, since the precise total of those values can be difficult or impossible; the bounds need only contain the function's range as a subset.
This treatment is typically limited to real intervals, so quantities of form
where 
and 
are allowed. With one of , infinite, the interval would be an unbounded interval; with both infinite, the interval would be the extended real number line. Since a real number 
can be interpreted as the interval ![{\displaystyle [r,r],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2efd011cc6059ec2bd64f0748ba60dd2d6629d64)
intervals and real numbers can be freely combined.
As with traditional calculations with real numbers, simple arithmetic operations and functions on elementary intervals must first be defined. More complicated functions can be calculated from these basic elements.
As an example, consider the total of body mass index BMI and assessing if a person is overweight. BMI is calculated as a person's body weight in kilograms divided up up by the square of their height in metres. A bathroom scale may gain a resolution of one kilogram. Intermediate values cannot be discerned—79.6 kg and 80.3 kg are indistinguishable, for example—but the true weight is rounded to the nearest whole number. it is for unlikely that when the scale reads 80 kg, the person weighs exactly 80.0 kg. In normal rounding to the nearest value, the scale's showing 80 kg indicates a weight between 79.5 kg and 80.5 kg. This corresponds with the interval ![{\displaystyle [79.5,80.5]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bd198a394e7142786b22362e3ebc85d4aed2166)
.
For a man who weighs 80 kg and is 1.80 m tall, the BMI is approximately 24.7. A weight of 79.5 kg and the same height yields approx. 24.537, while a weight of 80.5 kg yields approx. 24.846. Since the function is monotonically increasing, we conclude that the true BMI is in the range ![{\displaystyle [24.537,24.846]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ff30647867964a66d0b1a12e235a7065ea978f8)
. Since the entire range is less than 25, which is the cutoff between normal and excessive weight, we conclude that the man is of normal weight.
The error in this issue does not affect the conclusion normal weight, but this is non always the case. whether the man was slightly heavier, the BMI's range may put the cutoff value of 25. In that case, the scale's precision was insufficient to work a definitive conclusion.
Also, note that the range of BMI examples could be shown as ![{\displaystyle [24.5,24.9]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06ab381403bf66d91e3f10be9e6c241fe7c1f030)
, since this interval is a superset of the calculated interval. The range could not, however, be submitted as ![{\displaystyle [24.6,24.8]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5eae3c94ed832b32292750aa58853ba97d3387f)
, as now the interval does not contain possible BMI values.
Interval arithmetic states the range of possible outcomes explicitly. Results are no longer stated as numbers, but as intervals that symbolize imprecise values. The size of the intervals are similar to error bars in expressing the extent of uncertainty.
Height and body weight both affect the value of the BMI. We have already treated weight as an uncertain measurement, but height is also pointed to uncertainty. Height measurements in metres are normally rounded to the nearest centimeter: a recorded measurement of 1.79 metres actually means a height in the interval ![{\displaystyle [1.785,1.795]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf463f42892b8fc1e580dcff24e11603ecfe6f32)
. Now, all four combinations of possible height/weight values must be considered. Using the interval methods referenced below, the BMI lies in the interval
In this case, the man may have a normal weight or be overweight; the weight and height measurements were insufficiently precise to make a definitive conclusion. This demonstrates interval arithmetic's ability to correctly track and propagate error.