Rounding error
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Rounding_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

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 3^rd)

• 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 4^th significant digit. This is rounded up because )

0.142857 ≈ 0.14 (rounding the 3^rd significant digit. This is rounded down because )

See also

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

External links

• Roundoff Error at MathWorld.
• D. Goldberg. What Every Computer Scientist Should Know About Floating-Point Arithmetic

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