by Dr. John C . Nash
Operations on Floating-point Numbers
In last month's column, we presented the concept of the machine precision-the smallest positive that can be added to 1 .0 to give a result in the computer still greater than 1 .0.
The machine precision gives us a measure of how precisely we can represent numbers In the machine, before we even begin to calculate with them. I ndeed, It is frequently at this stage-before calculations begin-that some of the most serious errors creep into our results. We have, for Instance:
errors caused by the limitations of the measurement process, be it physical or statistical; others caused by the Incorrect recording or entry of the data, (e.g. digit transposition In writing down or keying data); and others that occur in converting the numbers to machine representable form.
This last class of errors, that of Input conversion, should be at most one unit in the last place (ulp) If the conversion method is correct. Sadly, a number of popular computers have sloppy compilers or Interpreters with errors in the Input/
output conversion routines . that may occasionally cause a disaster for our calculations. Fortunately, many micro
computers use decimal floating-point representations, which effectively avoid this difficulty. The IEEE Draft Floating-Point Arithmetic Standard prescribes a binary representation, but specifies that conversion of decimal Input shall be such that a number is represented by the nearest representable number to it on the real scale. Let us hope that the Implementations of this standard (e.g. the Intel 8087) perform to specification.
For the purposes of discussion, we will assume that our input is exact up to the moment we enter it. Then there is a possible error proportional to the machine precision, which we will denote 8, as in the previous column. In a number X, this error will be such that:
A8S(X - fi(X)) � A8S(x) * 8
That is, the absolute relative error may be as large as 8, the machine precision. Once again; fl ( ) will denote the floating
point representation of the quantity within the parentheses.
In machines that round, 8 can be replaced by 8/2 in the above expression, since a smaller error will be possible.
We now wish to consider operations with numbers. Even if these are exactly represented, operations with them will create some rounding or truncation error. For example, in a four digit decimal machine:
fl(1 000 + 0.4999) = 1 000
In a rounding machine, we make an error of at most 0.5 in the last place in the result (half an ulp). Since 8 should be 1 ulp relative to 1 .0, we write:
A8S(fi(X + V) - (X + V)) � A8S(X + V) * ( 1 + 0.5*8) It is not too hard to show that: ·
A8S(fi(X • V) - X • V)) � A8S(X • V) * ( 1 + 0.5*8)
where • represents any of + , - , * , or /. However, this only describes the Individual operations and not their combinations.
Most students of engineering or physics are aware of the rules for combining error bounds on quantities already known to be subject to error or uncertainty. To review these, consider that X Is subject to an error as large as e, V to one as large as f. That is, the true values of the quantities we are dealing with lie in the intervals:
[X - e,X + e] and [V - f,V + f].
It is fairly easily seen that the true value of (X + V) lies in an Interval:
[(X + V) - (e + f), (X + V) + (e + f)].
Also that the true value of (X - V) lies In the interval:
[(X - V) - (e + f), (X - V) + (e + f)].
In this case, we see that while the quantities X and V are subtracted, the absolute errors are added. (See figure 1 .)
X
e e
v f f
r
e + f
\X + V e + f
X - V e + f e + f
Figure 1 . Addition and subtraction of numbers having errors.
Multiplication and division are slightly more complicated.
To simplify matters, we take all quantities positive. (In a strict analysis, one keeps absolute value symbols throughout.) For multiplication, the new interval is:
[ XV - Ve - Xf - ef, XV + Ve + Xf + ef ].
Scaling this by XV (i.e. dividing through) gives:
(XV) * [ 1 - e/X - flY - (ef)/(XV), 1 + e/X + flY + (ef)/(XV) ].
I f w e Ignore e f as being small relative t o 1 (this i s reasonable if X and V are greater than 1 and e and f are of a magnitude similar to the machine precision), we see that the deviation in the true XV is described by the sum of the relative errors (e/X) and (flY). A similar approach shows that the deviation in a quo
tient is described by the sum of the relative errors in its parts.
To summarize: 1 ) the deviation In a sum or difference is given by the sum of the absolute deviations in the numbers;
and 2) the relative deviation in a product or quotient is given by the sum of the relative deviations in the numbers.
Note that relative errors can be converted to absolute errors. In the product example:
r = (e/X) + (flY)
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which is the relative error in XY, gives the absolute error in this product as r(XY). What about a function of X? Can we estimate the error in say:
Z = g(X)
where X has an error bounded by e? Clearly, Z has an error as large as d , where d is given by:
d = max (g(t)) - min (g(t))
for all t in the interval [X - e, X + e]. Usually, one is not this strict, and assumes that the function g(X) is relatively smooth, so that it may be approximated by its tangent in the region of interest. The tangent is described by the first derivative of
Figure 2. Deviation In a function given a deviation In Its argument x.
The interval analysis we have been considering can be automated, but it has proven very expensive and difficult to carry out for real-world calculations. It is but one example of forward error analysis, which follows a calculation step by step to obtain error bounds on the results. This is very useful when one must know the limits on the results, but usually bounds obtained in this way are too conservative. That is, they usually indicate that very large deviations are possible in an answer, when the probability of ever having the errors combine in such a perverse way is infinitesimally small. In reality, errors may cancel each other, and the computer answer is frequently quite close to the exact result desired.
The difficulty, of course, is that we cannot be sure how close we are to the wanted result without a lot of analysis.
However, with the introduction of directed rounding (that is, up or down or both) in the Draft Floating-Point standard , it is likely that a lot more interval analysis will be perfqrmed.
Instead of looking at the estimation of errors in this forward analysis, one can consider a backward error analysis, which looks at the results and tries to find a neighboring problem that would give the same results if exact arithmetic were used throughout. This type of analysis is usually easier to carry out than the forward variety, but requires more attention to the mathematics.
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