Method for bounding the errors of numerical computations
This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages)
This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Interval arithmetic" – news · newspapers · books · scholar · JSTOR(January 2020) (Learn how and when to remove this template message)
This article's tone or style may not reflect the encyclopedic tone used on Wikipedia. See Wikipedia's guide to writing better articles for suggestions.(February 2020) (Learn how and when to remove this template message)
(Learn how and when to remove this template message)
Tolerance function (turquoise) and interval-valued approximation (red)
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 , interval arithmetic works with an interval that defines the range of values that can have. In other words, any value of the variable lies in the closed interval between and . A function , when applied to , produces an interval which includes all the possible values for for all .
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.
and 25 Related for: Interval arithmetic information
Intervalarithmetic (also known as interval mathematics; interval analysis or interval computation) is a mathematical technique used to mitigate rounding...
1. Some arithmetic systems operate on mathematical objects other than numbers, such as intervalarithmetic and matrix arithmetic. Arithmetic operations...
half-open intervals are defined by removing the respective endpoints. This redefinition is useful in intervalarithmetic when dividing by an interval containing...
of intervals approximating r {\displaystyle r} . This allows the basic arithmetic operations on the real numbers to be defined in terms of interval arithmetic...
example, the sequence "ABBCAB" could become 0.0112013, in arithmetic coding as a value in the interval [0, 1). The next step is to encode this ternary number...
The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic established in 1985 by the Institute...
alpha theory, while a typical example of a priori certification is intervalarithmetic. A certificate for a root is a computational proof of the correctness...
approaches: (1) intervalarithmetic approach; and (2) the extension principle approach. A fuzzy number is equal to a fuzzy interval. The degree of fuzziness...
computation. Affine arithmetic is meant to be an improvement on intervalarithmetic (IA), and is similar to generalized intervalarithmetic, first-order Taylor...
extensions of Heyting arithmetic by types including N N {\displaystyle {\mathbb {N} }^{\mathbb {N} }} , constructive second-order arithmetic, or strong enough...
numerical analysis. For computation, intervalarithmetic is used, and all results are represented by intervals. Validated numerics were used by Warwick...
real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the nth root...
yielding so-called intervalarithmetics. Let us now consider the following very simple program: y = x; z = x - y; With reasonable arithmetic types, the result...
Informally, in frequentist statistics, a confidence interval (CI) is an interval which is expected to typically contain the parameter being estimated....
rounding in floating point number systems. This value characterizes computer arithmetic in the field of numerical analysis, and by extension in the subject of...
fundamental theorem of arithmetic (unique prime factorization). A stronger result is the following: Every rational number in the interval ( ( 1 / e ) 1 / e...
same limiting value (0, +∞, or −∞). Directed rounding is used in intervalarithmetic and is often required in financial calculations. If x is positive...
best one found so far by the algorithm. Intervalarithmetic, interval mathematics, interval analysis, or interval computation, is a method developed by...
case an affine line). The mode, median, and arithmetic mean are allowed to measure central tendency of interval variables, while measures of statistical...
scientist known for her research on computer arithmetic, including floating-point arithmetic and intervalarithmetic. She is a researcher for the French Institute...
In mathematics and statistics, the arithmetic mean ( /ˌærɪθˈmɛtɪk ˈmiːn/ arr-ith-MET-ik), arithmetic average, or just the mean or average (when the context...
of seconds elapsed since 1970-01-01T00:00:10 TAI. This makes time intervalarithmetic much easier. Time values from these systems do not suffer the ambiguity...
example, the interval class between pitch classes 4 and 9 is 5 because 9 − 4 = 5 is less than 4 − 9 = −5 ≡ 7 (mod 12). See modular arithmetic for more on...
Global Information
![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
![{\displaystyle [c,d]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d85b3b21d6d891d97f85e263d394e3c90287586f)
![{\displaystyle x\in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/026357b404ee584c475579fb2302a4e9881b8cce)