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precision and accuracy
*To*: mathgroup at yoda.ncsa.uiuc.edu
*Subject*: precision and accuracy
*From*: uunet!yoda.ncsa.uiuc.edu!keiper%wri (Jerry Keiper)
*Date*: Mon, 26 Mar 90 13:03:08 CST
uunet!renoir.Berkeley.EDU!fateman (Richard Fateman) writes:
The notions of precision and accuracy as used in Mathematica
are, in my view, wrong. I have argued at some length (and
apparently with no effect) for a model in which a floating-point
number's precision can be extended by adding 0's to its
internal representation, if you wish. The mathematica model
is (somewhat crudely) that if you write 0.5 you do not mean
0.500000... but any number between 0.450 and 0.5499.. .
While this latter model may be reminiscent of a physics-lab
approach of carrying significant figures in your measurements,
it is not useful in computation. Numerical analysis texts
talk about relative and absolute error, not precision and accuracy.
I presume he is unhappy with Mathematica's use of significance arithmetic
as opposed to fixed, but arbitrary precision arithmetic. These two models
of arithmetic are indeed very different and they both have their advantages
and disadvantages, but I have seen no convincing evidence that one is
better than the other. Certainly significance arithmetic is very difficult
to implement correctly and Mathematica has not yet achieved perfection in
this matter, but the fact that most elementary numerical texts only discuss
fixed precision arithmetic (note that they don't discuss arbitrary precision
arithmetic either) is not a reason to reject significance arithmetic.
As to increasing the precision of a number, Mathematica allows this to be
done very simply using SetPrecision[] (likewise the accuracy can be
increased with SetAccuracy[].) y = SetPrecision[x, prec] gives y a value
such that y == x is True and such that the precision of y is prec. If the
precision of y is greater than the precision of x the "missing" digits of x
are filled in for y with 0's in binary. This does not mean that they will
appear as 0's in decimal however. e.g. the decimal number 1.2 has a
nonterminating binary expansion 1.0011001100110011001100... Clearly if
we start with a finite approximation to this and attach 0's in binary
we will not get the decimal expansion for 1.200000000... Since it is not
possible to know what number a user has in mind when he wants to increase
its precision it is not possible to give a better approximation; all that
Mathematica can do is give another number with more digits in it.
As to relative and absolute error vs. precision and accuracy, precision
is a nonnegative integer and accuracy is an integer. Relative and absolute
error are small numbers, sometimes complex, but they are never known
exactly (if they were the approximate answer could be corrected.) I
find it more convenient to deal with an accuracy of 27 rather than an
absolute error of 0.000000000000000000000000001, but perhaps not everyone
does.
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