From: Josef 'Jupp' SCHUGT <jupp@...>
Date: 2004-01-30T05:58:33+09:00
Subject: Re: ruby-math and "why is ** not abelian?"

Hi!

* Van Jacques:
> Why is " Neither (Float, +) nor (Float, *)" a group?

The associative law is violated. Using machine numbers

    (a + b) + c = a + (b + c)

does not hold for all machine numbers a, b, c. Proof is by
counterexample.

Simply Assume a = 1.0 and b, c = 0.6 * eps where eps is the smallest
number that added to 1.0 results in something different from 1.0.

(a + b) results in 1.0 because b is smaller than eps and so (a + b) +
c results in 1.0 as well.

(b + c) results in a number larger than eps so a + (b + c) is larger
than 1.0.

Similar holds for multiplication (it is more complicated so I only
sketch the proof): If a is a huge number and b and c are tiny numbers
the product of b and c can be zero so that a * (b * c) is zero as
well while a * b can still be a number that multiplied by c does not
result in zero.

To put it that way: The most challenging tasks in numerical
mathematics is providing an error estimate. Unfortunately it is one
of the most important as well.

It is often important to know how precisely a value is known. What
happens if one falsely assumes a precision that is not present shows
the Mars Polar Impactor.

Josef 'Jupp' SCHUGT
-- 
http://oss.erdfunkstelle.de/ruby/    -     German comp.lang.ruby-FAQ
http://rubyforge.org/users/jupp/     -     Ruby projects at Rubyforge
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