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Interval arithmetic (also known as interval mathematics; interval analysis or interval computation) is a mathematical technique used to mitigate rounding and measurement errors in mathematical computation by computing function bounds. Numerical methods involving interval arithmetic can guarantee relatively reliable and mathematically correct results. Instead of representing a value as a single number, interval arithmetic or interval mathematics represents each value as a range of possibilities.

Mathematically, instead of working with an uncertain real-valued variable ${\displaystyle x}$, interval arithmetic works with an interval ${\displaystyle }$ that defines the range of values that ${\displaystyle x}$ can have. In other words, any value of the variable ${\displaystyle x}$ lies in the closed interval between ${\displaystyle a}$ and ${\displaystyle b}$. A function ${\displaystyle f}$, when applied to ${\displaystyle x}$, produces an interval ${\displaystyle }$ which includes all the possible values for ${\displaystyle f(x)}$ for all ${\displaystyle x\in }$.

Interval arithmetic is suitable for a variety of purposes; the most common use is in scientific works, particularly when the calculations are handled by software, where it is used to keep track of rounding errors in calculations and of uncertainties in the knowledge 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 helps find guaranteed solutions to equations (such as differential equations) and optimization problems.

Introduction

The main objective of interval arithmetic is to provide a simple way of calculating upper and lower bounds of a function's range in one or more variables. These endpoints are not necessarily the true supremum or infimum of a range since the precise calculation of those values can be difficult or impossible; the bounds only need to contain the function's range as a subset.

This treatment is typically limited to real intervals, so quantities in the form

${\displaystyle =\{x\in \mathbb {R} \mid a\leq x\leq b\},}$

where ${\displaystyle a={-\infty }}$ and ${\displaystyle b={\infty }}$ are allowed. With one of ${\displaystyle a}$, ${\displaystyle b}$ infinite, the interval would be an unbounded interval; with both infinite, the interval would be the extended real number line. Since a real number ${\displaystyle r}$ can be interpreted as the interval ${\displaystyle ,}$ intervals and real numbers can be freely combined.

Example

Consider the calculation of a person's body mass index (BMI). BMI is calculated as a person's body weight in kilograms divided by the square of their height in meters. Suppose a person uses a scale that has a precision of one kilogram, where intermediate values cannot be discerned, and the true weight is rounded to the nearest whole number. For example, 79.6 kg and 80.3 kg are indistinguishable, as the scale can only display values to the nearest kilogram. It is unlikely that when the scale reads 80 kg, the person has a weight of exactly 80.0 kg. Thus, the scale displaying 80 kg indicates a weight between 79.5 kg and 80.5 kg, or the interval ${\displaystyle 79.5,80.5)}$.

The BMI of a man who weighs 80 kg and is 1.80m tall is approximately 24.7. A weight of 79.5 kg and the same height yields a BMI of 24.537, while a weight of 80.5 kg yields 24.846. Since the body mass is continuous and always increasing for all values within the specified weight interval, the true BMI must lie within the interval ${\displaystyle 24.537,24.846}$. Since the entire interval is less than 25, which is the cutoff between normal and excessive weight, it can be concluded with certainty that the man is of normal weight.

The error in this example does not affect the conclusion (normal weight), but this is not generally true. If the man were slightly heavier, the BMI's range may include the cutoff value of 25. In such a case, the scale's precision would be insufficient to make a definitive conclusion.

The range of BMI examples could be reported as ${\displaystyle 24.5,24.9}$ since this interval is a superset of the calculated interval. The range could not, however, be reported as ${\displaystyle 24.6,24.8}$, as the interval does not contain possible BMI values.

Multiple intervalsedit

Height and body weight both affect the value of the BMI. Though the example above only considered variation in weight, height is also subject to uncertainty. Height measurements in meters are usually rounded to the nearest centimeter: a recorded measurement of 1.79 meters represents a height in the interval ${\displaystyle 1.785,1.795)}$. Since the BMI uniformly increases with respect to weight and decreases with respect to height, the error interval can be calculated by substituting the lowest and highest values of each interval, and then selecting the lowest and highest results as boundaries. The BMI must therefore exist in the interval

${\displaystyle {\frac {79.5,80.5)}{1.785,1.795)^{2}}}\subseteq 24.673,25.266.}$

In this case, the man may have normal weight or be overweight; the weight and height measurements were insufficiently precise to make a definitive conclusion.

Interval operatorsedit

A binary operation ${\displaystyle \star }$ on two intervals, such as addition or multiplication is defined by

${\displaystyle x_{1},x_{2}{\,\star \,}y_{1},y_{2}=\{x\star y\,|\,x\in x_{1},x_{2}\,\land \,y\in y_{1},y_{2}\}.}$

In other words, it is the set of all possible values of ${\displaystyle x\star y}$, where ${\displaystyle x}$ and ${\displaystyle y}$ are in their corresponding intervals. If ${\displaystyle \star }$ is monotone for each operand on the intervals, which is the case for the four basic arithmetic operations (except division when the denominator contains ${\displaystyle 0}$), the extreme values occur at the endpoints of the operand intervals. Writing out all combinations, one way of stating this is

${\displaystyle x_1,x_2\star <!--\; \star \; -->y_1,y_2=min\{x_1\star <!--\; \star \; -->y_1,x_1\star <!--\; \star \; -->y_2,x_2\star <!--\; \star \; -->y_1,x_2\star <!--\; \star \; -->y_2\},max\{x_1}$Zdroj:https://en.wikipedia.org?pojem=Extensions_for_Scientific_Computation
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