Computer arithmetic: Difference between revisions
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{{Short description|Implementation of arithmetic operations}} |
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'''Computer arithmetic''' is an area that belongs to both [[computer science]] and [[mathematics]]. |
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'''Computer arithmetic''' is the scientific field that deals with representation of [[number]]s on [[computer]]s and corresponding implementations of the [[arithmetic operation]]s.<ref>{{citation|contribution=Number Representation and Computer Arithmetic|title=Encyclopedia of Information Systems|first=Behrooz|last=Parhami|year=2003|pages=217–333|publisher=Elsevier|contribution-url=https://web.ece.ucsb.edu/Faculty/Parhami/pubs_folder/parh02-arith-encycl-infosys.pdf}}</ref> |
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<ref>{{citation|contribution=Computer Arithmetic in Practice|title=Taylor & Francis Group|first=Slawomir|last=Grys|year=2023|pages=212|publisher=CRC Press|doi=10.1201/9781003363286 |isbn=978-1-003-36328-6 |contribution-url=https://www.taylorfrancis.com/books/mono/10.1201/9781003363286/computer-arithmetic-practice-s%C5%82awomir-gry%C5%9B}}</ref> |
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It includes: |
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Historically, computer arithmetic dealt with the design of [[arithmetic logic unit]]s. |
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*[[Fixed-point arithmetic]] |
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*[[Floating-point arithmetic]] |
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*[[Interval arithmetic]] |
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*[[Arbitrary-precision arithmetic]] |
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*[[Modular arithmetic]] |
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**[[Multi-modular arithmetic]] |
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**[[p-adic arithmetic|''p''-adic arithmetic]], consisting of computing modulo a single [[prime number]] and retrieving the [[integer]] or [[rational number|rational]] result by using [[Hensel lifting]] |
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**[[Finite field arithmetic]] |
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*[[Matrix multiplication|Matrix arithmetic]] |
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In the cases where the size of the representation of a number is fixed (fixed-point, floating-point and interval arithmetic), the main concern is the control the computational error, as far as possible; see, for example [[IEEE 754]]. |
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Presently, computer arithmetic deals mainly with the problem of having computer [[implementation]]s of [[arithmetic operations]] that are as efficient as possible and introduce the smallest possible [[rounding error]]s, including [[integer overflow|overflow]]s and [[underflow]]s. |
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In the other cases, where an exact result should be provided, the main concern is the practical efficiency, which is optimized by combining improvements of [[computational complexity]] with [[hardware (computing)|hardware]] specificities. |
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Rounding errors are unavoidable since numbers with infinitely many digits must be represented with a finite number of [[bit]]s |
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[[ARITH Symposium on Computer Arithmetic]] is an international symposium devoted to computer arithmetic. |
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The most used representation of numbers in computers is [[floating-point arithmetic]]. The norm [[IEEE 754]] specifies how floating numbers must be represented, and establish that floating-point operations must be implemented in such a way that the result of an operation is always the exact rounding of the exact mathematical result. This could seem to be an evidence, but it is far to be simple (see [[FDIV bug]], for example). |
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==References== |
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Another way for limiting the rounding errors consist of using [[multiple-precision arithmetic]]. The [[software library]] [[GNU GMP]] is a ''de facto'' standard for that. |
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{{reflist}} |
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[[Category:Computer arithmetic]] |
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The annual international conference [[ARITH]] is devoted to computer arithmetic. |
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{{comp-sci-stub}} |
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Latest revision as of 16:59, 27 December 2024
Computer arithmetic is the scientific field that deals with representation of numbers on computers and corresponding implementations of the arithmetic operations.[1] [2]
It includes:
- Fixed-point arithmetic
- Floating-point arithmetic
- Interval arithmetic
- Arbitrary-precision arithmetic
- Modular arithmetic
- Multi-modular arithmetic
- p-adic arithmetic, consisting of computing modulo a single prime number and retrieving the integer or rational result by using Hensel lifting
- Finite field arithmetic
- Matrix arithmetic
In the cases where the size of the representation of a number is fixed (fixed-point, floating-point and interval arithmetic), the main concern is the control the computational error, as far as possible; see, for example IEEE 754.
In the other cases, where an exact result should be provided, the main concern is the practical efficiency, which is optimized by combining improvements of computational complexity with hardware specificities.
ARITH Symposium on Computer Arithmetic is an international symposium devoted to computer arithmetic.
References
[edit]- ^ Parhami, Behrooz (2003), "Number Representation and Computer Arithmetic" (PDF), Encyclopedia of Information Systems, Elsevier, pp. 217–333
- ^ Grys, Slawomir (2023), "Computer Arithmetic in Practice", Taylor & Francis Group, CRC Press, p. 212, doi:10.1201/9781003363286, ISBN 978-1-003-36328-6
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