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 real numbers (which in theory have infinite digits). This is a form of quantization error.

Representation error

The error introduced by attempting to represent a number on the computer is called representation error. Some examples:

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 five standard ways of performing the rounding in IEEE standard arithmetic:

• truncation: simply chop off the remaining digits; also called rounding to zero.

0.142857 ≈ 0.142 (dropping all significant digits after 3^rd)

• round to nearest: round to the nearest value, with ties broken in one of two ways. 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 )

• round to -∞: always round to the left on the number line
• round to ∞: always round to the right on the number line

Examples

• in January 25, 1990 during the First Gulf War, a Patriot missile defense system in Al-Dhahran, Saudi Arabia, was rendered ineffective due to a round-off error. The system's computer only had 24-bit precision, so it chopped 0.0001% of timing values, while the accuracy threshold is 20 hours. After the system had been running for 100 hours, an error of 0.3422 seconds had accumulated. This discrepancy caused the Patriot system to continuously recycle itself instead of targeting properly. As a result, an Iraqi Scud missile could not be targeted and was allowed to detonate on a barracks, killing 28 people.^{[1] [2]}

• In the 37th second of flight of the Ariane rocket (launched on June 4, 1996), the inertial reference system attempted to convert a 64-bit floating-point number to a 16-bit number, but instead triggered an overflow error which was interpreted by the guidance system as flight data, causing the rocket to veer off course and be destroyed.^[1]

• The Vancouver stock exchange devised a short-lived index. At its inception in 1982, the index was given a value of 1000.000. After 22 months of recomputing the index and truncating to three decimal places at each change in market value, the index stood at 524.881, despite the fact that its "true" value should have been 1009.811.^[1]

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
• 20 Famous Software Disasters

Notes and references

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