Roundoff Error

Roundoff error is the difference between an approximation of a number used in computation and its exact (correct) value. In certain types of computation, roundoff error can be magnified as any initial errors are carried through one or more intermediate steps.

An egregious example of roundoff error is provided by a short-lived index devised at the Vancouver stock exchange (McCullough and Vinod 1999). 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.

Other sorts of roundoff error can also occur. A notorious example is the fate of the Ariane rocket launched on June 4, 1996 (European Space Agency 1996). In the 37th second of flight, 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.

The Patriot missile defense system used during the Gulf War was also rendered ineffective due to roundoff error (Skeel 1992, U.S. GAO 1992). The system used an integer timing register which was incremented at intervals of 0.1 s. However, the integers were converted to decimal numbers by multiplying by the binary approximation of 0.1,

As a result, after 100 hours ( ticks), an error of

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.

REFERENCES:

Chartier, T. "Devastating Roundoff Error." Math. Horizons 13, No. 4, 11, Apr. 2005.

European Space Agency. "Ariane 501 Inquiry Board Report." Paris, 19 July 1996. https://ravel.esrin.esa.it/docs/esa-x-1819eng.pdf.

McCullough, B. D. and Vinod, H. D. J. Economic Lit. 37, pp. 633-665, June 1999.

Quinn, K. "Even Had Problems Rounding Off Figures? This Stock Exchange Has." The Wall Street Journal 202, No. 91, p. 37, Nov. 8, 1983.

Skeel, R. "Roundoff Error and the Patriot Missile." SIAM News 25, 11, Jul. 1992.

The Toronto Star. November 19, 1983.

United Stated General Accounting Office. "GAO/IMTEC-92-26 Patriot Missile Software Problem." 1992. https://www.fas.org/spp/starwars/gao/im92026.htm.

Wilkinson, J. H. "Modern Error Analysis." SIAM Rev. 13, 548-568, 1971.

Wilkinson, J. H. Rounding Errors in Algebraic Processes. New York: Dover, 1994.