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Interval arithmetic facts for kids

Kids Encyclopedia Facts

Interval arithmetic is a specific type of computer arithmetic for (mathematical) intervals. It is mainly used for the automated detection of errors. There is a value, which isn't known exactly, but which can be given by an interval: the value is known to be inside the interval. There's a function to calculate the error. This function should be given the unknown value, but instead can only be given the interval. The result is a function which will map intervals.

Interval arithmetic can be used to treat rounding errors, or to treat insecurities with measurements: Each measurement has a certain error, which cannot be determined exactly.

Definition

For real intervals (interval of real numbers), interval arithmetic is defined as follows:

\frac{[x_1, x_2]}{[y_1, y_2]} = [x_1, x_2] \cdot \frac{1}{[y_1, y_2]},
where
\begin{align} \frac{1}{[y_1, y_2]} &= \left [\tfrac{1}{y_2}, \tfrac{1}{y_1} \right ] && 0 \notin [y_1, y_2] \\ \frac{1}{[y_1, 0]} &= \left [-\infty, \tfrac{1}{y_1} \right ] \\ \frac{1}{[0, y_2]} &= \left [\tfrac{1}{y_2}, \infty \right ] \\ \frac{1}{[y_1, y_2]} &= \left [-\infty, \tfrac{1}{y_1} \right ] \cup \left [\tfrac{1}{y_2}, \infty \right ] = [-\infty, \infty] && 0 \in (y_1, y_2) \end{align}

Applications

Interval arithmetic is mainly used in the field of validated numerics. It is also used in other technical areas.

Implementations

See also: Affine arithmetic

Since the birth of interval arithmetic, many experts have made interval arithmetic programs. The most famous works are INTLAB (made with MATLAB), arb, JuliaIntervals, and kv.

Community

There are several international conferences about interval arithmetic. One of the most largest meeting is the International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics (usually abbreviated as SCAN). There are also SWIM (Small Workshop on Interval Methods), PPAM (International Conference on Parallel Processing and Applied Mathematics), and REC (International Workshop on Reliable Engineering Computing).

Other links

Workshops

Libraries

Images for kids

  • Set of curves Outer approximation

    Tolerance function (turquoise) and interval-valued approximation (red)

  • Interval BMI Simple Example

    Body mass index for a person 1.80 m tall in relation to body weight m (in kilograms)

  • Interval BMI Example

    Body mass index for different weights in relation to height L (in metres)

  • Interval multiplication

    Multiplication of positive intervals

  • Value domain of monotonic function

    Values of a monotonic function

  • Meanvalue extension

    Mean value form

  • Illustration of outward rounding

    Outer bounds at different level of rounding

  • Interval-dependence problem-front view

    Approximate estimate of the value range

  • Interval-dependence problem

    Treating each occurrence of a variable independently

  • Interval-wrapping effect

    Wrapping effect

  • Interval Newton step

    Reduction of the search area in the interval Newton step in "thick" functions

  • Illustration of interval mincing

    Rough estimate (turquoise) and improved estimates through "mincing" (red)

  • Fuzzy arithmetic

    Approximation of the normal distribution by a sequence of intervals

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Interval arithmetic Facts for Kids. Kiddle Encyclopedia.