It's not a new problem..
This is extracted from some maths notes from an MSc module (sorry about the poor formatting from the cut and paste)
R. Skeel. Roundo error and the Patriot missile. Society for Industrial and Applied Mathematics (SIAM) News, 25(4):11, 1992.
1.2 Roundo error and the Patriot missile
Article [5] by Robert Skeel, professor of computer science at the University of Illinois at Urbana-Champaign, in SIAM News, 1992.
\The March 13 issue of Science carried an article claiming, on the basis of a report from
the General Accounting O ce (GAO), that a `minute mathematical error ... allowed an
Iraqi Scud missile to slip through Patriot missile defenses a year ago and hit U.S. Army
barracks in Dhahran, Saudi Arabia, killing 28 servicemen.' The article continues with a
readable account of what happened.
\The article says that the computer doing the tracking calculations had an internal
clock whose values were slightly truncated when converted to floating-point arithmetic.
The errors were proportional to the time on the clock: 0.0275 seconds after eight hours and
0.3433 seconds after 100 hours. A calculation shows each of these relative errors to be both
very nearly 2^(-20), which is approximately 0.0001%.
\The GAO report contains some additional information. The internal clock kept time
as an integer value in units of tenths of a second, and the computer's registers were only
24 bits long. This and the consistency in the time lags suggested that the error was
caused by a fi xed-point 24-bit representation of 0.1 in base 2. The base 2 representation
of 0.1 is nonterminating; for the fi rst 23 binary digits after the binary point, the value is
0.1 x (1 - 2^(-20)). The use of 0.1 x (1 - 2^(-20)) in obtaining a floating-point value of time in
seconds would cause all times to be reduced by 0.0001%.
\This does not really explain the tracking errors, however, because the tracking of a
missile should depend not on the absolute clock-time but rather on the time that elapsed
between two di erent radar pulses. And because of the consistency of the errors, this time
di erence should be in error by only 0.0001%, a truly insigni cant amount.
\Further inquiries cleared up the mystery. It turns out that the hypothesis concerning
the truncated binary representation of 0.1 was essentially correct. A 24-bit representation
of 0.1 was used to multiply the clock-time, yielding a result in a pair of 24-bit registers.
This was transformed into a 48-bit floating-point number. The software used had been
written in assembly language 20 years ago. When Patriot systems were brought into the
Gulf conflict, the software was modi ed (several times) to cope with the high speed of
ballistic missiles, for which the system was not originally designed.
\At least one of these software modi cations was the introduction of a subroutine for
converting clock-time more accurately into floating-point. This calculation was needed in
about half a dozen places in the program, but the call to the subroutine was not inserted
at every point where it was needed. Hence, with a less accurate truncated time of one
radar pulse being subtracted from a more accurate time of another radar pulse, the error
no longer cancelled.
\In the case of the Dhahran Scud, the clock had run up a time of 100 hours, so the
calculated elapsed time was too long by 2^(-20) x 100 hours = 0.3433 seconds, during which
time a Scud would be expected to travel more than half a kilometer.
\The roundoff error, of course, is not the only problem that has been identi ed: serious
doubts have been expressed about the ability of Patriot missiles to hit Scuds."
End of article by Robert Skeel.
Philip
