Round-off error

Round-off error

:"For the acrobatic movement, roundoff, see Roundoff. "

A round-off error, also called rounding error, is the difference between the calculated approximation of a number and its exact mathematical value. Numerical analysis specifically tries to estimate this error when using approximation equations and/or algorithms, especially when using finite digits to represent infinite digits of real numbers. This is a form of quantization error.

Example


Notation Represent Approximate Error
1/7 0.142 857 0.142 857 0.000 000 142 857
ln 2 0.693 147 180 559 945 309 41... 0.693 147 0.000 000 180 559 945 309 41...
log10 2 0.301 029 995 663 981 195 21... 0.3010 0.000 029 995 663 981 195 21...
1.259 921 049 894 873 164 76... 1.25992 0.000 001 049 894 873 164 76...
1.414 213 562 373 095 048 80... 1.41421 0.000 003 562 373 095 048 80...
e 2.718 281 828 459 045 235 36... 2.718 281 828 459 045 0.000 000 000 000 000 235 36...
π 3.141 592 653 589 793 238 46... 3.141 592 653 589 793 0.000 000 000 000 000 238 46...

Increasing the number of digits allowed in a representation reduces the magnitude of possible roundoff errors, but any representation limited to finitely many digits will still cause some degree of roundoff error for uncountably many real numbers. This kind of error is unavoidable for conventional representations of numbers, but can be reduced by the use of guard digits.

Double-rounding can increase the round-off error. For example, if the numeral 9.945309 is rounded to two decimal places (9.95) for data entry purposes, and then rounded again to one decimal place (10.0) for display purposes, the apparent round-off error is 0.054691. If the original number was rounded to one decimal place in one step (9.9), the round-off error is only 0.045309.

There are at least two ways of performing the termination at the limited digit place:
*"'truncation: simply chop off the remaining digits.:0.142857 ≈ 0.142 (dropping all significant digits after 3rd)
*rounding: add 5 to the next digit and then chop it. The result may round up or round down.:0.142857 ≈ 0.143 (rounding the 4th significant digit. This is rounded up because 8ge5):0.142857 ≈ 0.14 (rounding the 3rd significant digit. This is rounded down because 2 < 5 ! )

See also

*Precision (arithmetic)
*Truncation
*Rounding
*Floating point
*Machine epsilon
*Wilkinson's polynomial

External links

* [http://mathworld.wolfram.com/RoundoffError.html Roundoff Error] at MathWorld.
* D. Goldberg. [http://www.validlab.com/goldberg/paper.pdf What Every Computer Scientist Should Know About Floating-Point Arithmetic]


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • round-off error — apvalinimo paklaida statusas T sritis Standartizacija ir metrologija apibrėžtis Paklaida, atsirandanti apvalinant skaičiavimų rezultatus. atitikmenys: angl. error of roundness; rounding error; round off error vok. Rundungsfehler, m rus.… …   Penkiakalbis aiškinamasis metrologijos terminų žodynas

  • round-off error — apvalinimo paklaida statusas T sritis fizika atitikmenys: angl. error of roundness; rounding error; round off error vok. Abrundungsfehler, m; Rundungsabweichung, f; Rundungsfehler, m rus. погрешность округления, f pranc. erreur d’arrondi, f …   Fizikos terminų žodynas

  • round-off error — /raʊnd ɒf ˈɛrə/ (say rownd of eruh) noun any computational error resulting from calculations being carried out only to a specified number of decimal (or binary) digits …  

  • round off — verb a) To change the shape of an object to make it more circular. b) To change a number into an approximation having fewer significant digits. See Also: round off error, round to, round down, round out, round up …   Wiktionary

  • error — errorless, adj. errorlessly, adv. /er euhr/, n. 1. a deviation from accuracy or correctness; a mistake, as in action or speech: His speech contained several factual errors. 2. belief in something untrue; the holding of mistaken opinions. 3. the… …   Universalium

  • error of roundness — apvalinimo paklaida statusas T sritis Standartizacija ir metrologija apibrėžtis Paklaida, atsirandanti apvalinant skaičiavimų rezultatus. atitikmenys: angl. error of roundness; rounding error; round off error vok. Rundungsfehler, m rus.… …   Penkiakalbis aiškinamasis metrologijos terminų žodynas

  • error of roundness — apvalinimo paklaida statusas T sritis fizika atitikmenys: angl. error of roundness; rounding error; round off error vok. Abrundungsfehler, m; Rundungsabweichung, f; Rundungsfehler, m rus. погрешность округления, f pranc. erreur d’arrondi, f …   Fizikos terminų žodynas

  • Quantization error — The difference between the actual analog value and quantized digital value due is called quantization error. This error is due either to rounding or truncation.Many physical quantities are actually quantized by physical entities. Examples of… …   Wikipedia

  • Discretization error — In numerical analysis, computational physics, and simulation, discretization error is error resulting from the fact that a function of a continuous variable is represented in the computer by a finite number of evaluations, for example, on a… …   Wikipedia

  • Truncation error — or discretization error is error made by numerical algorithms that arises from taking finite number of steps in computation. It is present even with infinite precision arithmetic, because it is caused by truncation of the infinite Taylor series… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”