Brent solver is non-optimal, because it doesn't use the user's guess.
---------------------------------------------------------------------
Key: MATH-156
URL: http://issues.apache.org/jira/browse/MATH-156
Project: Commons Math
Issue Type: Improvement
Reporter: Tyler Ward
Hi guys, I noticed that your brent solver isn't using the initial guess given
by the user. This can often radically improve the performance of the solver. In
my tests, it improved it by roughly 30% or more, with a decent guess.
Basically, we should try the guess first. If that's close enough, rreturn it.
Otherwise, try one of the end points. If that's close enough, return it. Then
if that brackets, go into the main loop. If that doesn't bracket, then we
instead try the other endpoint. If that's close enough, return it. If that
doesn't bracket, throw an exception. If it does bracket, go into the main loop
with all three trial points available, allowing the quadratic interpolation to
be used immediately.
Worst case scenario, the initial guess doesn't bracket. In that case it is
better than the default algorithm only if the user's initial guess is better
than linear interpolation, which I imagine it almost always would be. If the
user can't guess better than linear interpolation, presumably they wouldn't
provide a guess at all, and then nothing would change.
It's a small addition, less than 100 lines of code. I can't send it in due to
legal at work, but from the above ideas, you should be able to put it in
easily. One caveat. You may need to slightly modify the baisic solve(double,
double) method to linearly interpolate a good beginning point and then break
out a six double (solve(x0, y0, x1, y1, x2, y2)) method that both solve(min,
max) and solve(min, max, initial) can use. If you don't do this, then
solve(min, max) will bisect on its first iteration rather than intepolating,
which could cost performance. If you do break it out like so, then this will
always perform better than the current implementation.
As a good case study, here's our situation from work.
We are trying to compute a root, we know it falls in a VERY large interval (in
this case -1.0 to 1.0), but more than 99% of the time it will be between (say)
1.04 and 1.07. We can guess with stunning accuracy, at least 90% of the time we
can come to within 10^-4 for very little cost (a very nice cubic approximation
of the more complicated function), but we really need it to within 10^-8 or
better. In addition, that 1% of the time, it can fall in very strange areas
(0.9, or -0.3 or so), and our guess isn't very good when that happens. So we
can't tighten down the interval, because that would cause spurious errors, and
at the same time your linear interpolation isn't going to be very good at all.
It easily takes you 3 or 4 steps to get as close as our first guess. Using the
first guess in this manner would cut the steps to convergence from about 8-10
to about 3-5. A speed up of around 100%, with no loss in accuracy.
--
This message is automatically generated by JIRA.
-
If you think it was sent incorrectly contact one of the administrators:
http://issues.apache.org/jira/secure/Administrators.jspa
-
For more information on JIRA, see: http://www.atlassian.com/software/jira
---------------------------------------------------------------------
To unsubscribe, e-mail: [EMAIL PROTECTED]
For additional commands, e-mail: [EMAIL PROTECTED]
