🚀 go-pugleaf

         David Hough said:
> Various versions of the Linpack benchmark all have in common
> that the data is taken from a uniform random distribution, producing problems
> of very good condition.
         I believe "very good condition" is an exaggeration.  Problems
of size 1000 appear to typically have condition number between 10**5
and 10**6.  I would not consider this very good.  The condition num-
ber appears to be growing considerably faster than n, does anyone
know the expected asymptotic behavior?
         David Hough said:
>The Linpack benchmark is intended to solve an equation A x = b
>where the elements of A are chosen from a uniform random distribution
>and b is computed so that the correct solution x is close to a vector
>of 1's.  So looking at the size of components of x-1's is supposed to
>indicate how accurate the solution is, although
>the residual b - A x is a more reliable measure of quality.
         Why do you consider the residual a more reliable measure of
quality.  If the condition is bad the residual will still be small
when the answer is garbage.
         I asked:
>>         If you know of a single realistic application where the diff-
>>erence between 64 bit IEEE rounded to nearest and 64 bit IEEE chopped
>>to zero makes a significant difference in the maximum performance ob-
>>tainable I would like to see it.
         David Hough said:
>I doubt you'll find anything like this in normal scientific computation
>- the difference in problem domain between floating-point systems with
>identical exponent and significand fields and rounding schemes
>comparable in cost and care is rarely likely to be extremely
>noticable.  I would expect to notice the difference in problems like
>orbit computations, however - rounding should yield acceptable results
>for longer simulated times than chopping, even if everything else is
>equal, simply because the statistics of rounding are more favorable.  I
>have also heard that the details of rounding are important in financial
>calculations of bond yields.
         I believe IEEE chopped will be easier to implement and perform
better than IEEE rounded.  If rounding almost never matters why require
it?  I would expect the accuracy of orbit computations to be limited by
the accuracy to which the initial parameters are known and by the accur-
acy of the physical model (if I am not mistaken the models of tidal eff-
ects for example are not too accurate).  I find it hard to believe the
rounding method matters in bond calculations given 64-bit arithmetic and
sound methods (I believe there are foolish ways to compute internal rates
of return for example).
                       James B. Shearer