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It is even more important in mathematics than in the physical sciences
that the data under investigation are completely reliable.
At first glance it may seem that the increasing reliance of mathematicians
on programs such as
Maple and Mathematica has decreased the need for verification.
Computers very rarely make arbitrary mistakes in arithmetic and
algebra. But all the systems have known and unknown bugs in their
programming. It is therefore imperative that we that we
check our results. So what efforts
did we take to verify our findings?
First of all, we had to make sure that the roots we computed were accurate
to at least 10,000 (resp. 20,000) digits. We computed these roots using
Maple as well as Mathematica, having them compute the roots to an accuracy
of 10,010 digits. We then did two checks on the computed approximation
to . First, we tested that by checking that . Second, we tested that the 10,000th through 10,005th digits
were not all zeros or nines. This ensures that we actually computed the
first 10,000 digits of the decimal expansion of . (We note that
Maple initially did not give us an accuracy of 10,000 digits for all of
the cube roots, so that we had to increase the precision here.)
We then had to make sure that we computed the statistics and probability
values accurately---or at least to a reasonable precision, since we used
asymptotic formulas anyway. We did this by implementing them both in Maple
and in Mathematica and comparing the results. We detected no significant
discrepancy.
We claim that these measures reasonably ensure the reliability of our
experimental results.
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