Interval Arithmetic

Interval arithmetic is a mathematical structure that, for real intervals, defines operations similar to ordinary arithmetic. This area of mathematics is also called interval analysis or interval computing . This mathematical model is convenient for the study of various applied objects:

Values whose values are known only approximately , that is, a finite interval is defined in which these values are contained.
Values whose values during the calculation are distorted by rounding errors.
Random variables .

Objects and operations of interval arithmetic can be considered as a generalization of the model of real numbers, so intervals in a number of sources are called interval numbers . The practical importance of this model is due to the fact that the results of measurements and calculations almost always have some error, which must be taken into account and evaluated.

Background

Interval Operations

We will consider all possible finite real intervals $[a,b]\ (a\leqslant b)$ ${\ displaystyle [a, b] \ (a \ leqslant b)}$ $[a,b]\ (a \leqslant b)$ . Operations on them are defined as follows:

Addition: [ a , b ] + [ c , d ] = [ a + c , b + d ]
Subtraction: [ a , b ] - [ c , d ] = [ a - d , b - c ]
Multiplication: [ a , b ] × [ c , d ] = [min ( ac , ad , bc , bd ), max ( ac , ad , bc , bd )]
Division: [ a , b ] / [ c , d ] = [min ( a / c , a / d , b / c , b / d ), max ( a / c , a / d , b / c , b / d )]

It can be seen from the definition that the interval-sum contains all possible sums of numbers from the interval-terms and determines the boundaries of the set of such sums. Other actions are treated similarly. Note that the division operation is defined only if the interval-divider does not contain zero.

Degenerate intervals in which the beginning and the end coincide can be identified with ordinary real numbers. For them, the definitions given above coincide with classical arithmetic operations.

Operation Properties

Addition and multiplication of intervals are commutative and associative . The distribution property takes place in a weakened form:

X(Y+Z)\subset XY+XZ

{\ displaystyle X (Y + Z) \ subset XY + XZ}

Options and Extensions for Interval Arithmetic

IEEE 1788 Standard

The IEEE 1788-2015 standard standard for computer implementation of interval arithmetic was adopted in June 2015. ^{[1] The} working group for the preparation of the standard prepared two freely distributed reference implementations of it: ^[2] the C ++ library libieeep1788 ^[3] library for C ++, and the package that implements interval computing for the free mathematical software GNU Octave ^[4] .

The minimum subset of the standard, designed to simplify and accelerate its implementation - IEEE Std 1788.1-2017, was adopted in December 2017 and published in February 2018. ^[5]

Software

There are many realizations of interval arithmetic in various software packages ^[6] . Often they are made out as specialized libraries. A number of Fortran and C ++ compilers include support for interval values as a special data type.

Notes

↑ IEEE Standard for Interval Arithmetic
↑ Revol, Nathalie (2015). The (near-) future IEEE 1788 standard for interval arithmetic. 8th small workshop on interval methods. Slides (PDF)
↑ C ++ implementation of the preliminary IEEE P1788 standard for interval arithmetic
↑ GNU Octave interval package
↑ IEEE Std 1788.1-2017 - IEEE Standard for Interval Arithmetic (Simplified) (unspecified) . IEEE Standard . IEEE Standards Association. Date of treatment February 6, 2018.
↑ Software for Interval Computations collected by Vladik Kreinovich , University of Texas at El Paso

Literature

Alefeld G., Herzberger Yu . Introduction to interval computing . M.: Mir, 1987.356 s.
Dobronets B.S. Interval Mathematics . Krasnoyarsk: KSU Publishing House, 2004.
Shary S.P. Finite-dimensional interval analysis . M .: 2007.
Shokin Yu. I. Interval analysis . Novosibirsk: Siberian Branch of the Nauka Publishing House, 1981.

Links

Interval analysis and its applications .
Interval arithmetic (inaccessible link) .

[1] IEEE Standard for Interval Arithmetic

[2] Revol, Nathalie (2015). The (near-) future IEEE 1788 standard for interval arithmetic. 8th small workshop on interval methods. Slides (PDF)

[3] C ++ implementation of the preliminary IEEE P1788 standard for interval arithmetic

[4] GNU Octave interval package

[5] IEEE Std 1788.1-2017 - IEEE Standard for Interval Arithmetic (Simplified) (unspecified) . IEEE Standard . IEEE Standards Association. Date of treatment February 6, 2018.

[6] Software for Interval Computations collected by Vladik Kreinovich , University of Texas at El Paso