Re: [Reliable Computing] abs[x] for intervals? (reason for a standard)
> Alan Eliasen wrote:
> >
> > * Which definition do you find most appropriate for converting
> > real-valued algorithms to use intervals?
On 4/14/09 7:33 AM, "Dan Zuras Intervals" <intervals08@xxxxxxxxxxxxxx> wrote:
As for the question of how an algorithm designed for the
Reals maps onto the intervals in this case, I will leave
that for those better qualified to answer.
To my mind, that is a very real danger of efforts to promote wider use of intervals, whether this standardization effort or other efforts. If we are successful in convincing others that intervals are a good idea, a VERY natural first response is, "Great, how do I convert my real-valued algorithms to use intervals?"
After 30 years of searching for a responsible answer, I have not found one. The best advice is, "Don't. Start over." That's usually a show-stopper.
You do not get the guaranteed bounds for the solution to an ODE by changing double to interval in a Runge-Kutta (or whatever you like) code. If you change double to interval in a good linear solver, you will be VERY disappointed with the results. An intervalized trust region or line search pseudo-Newton optimizer is nonsense. Similarly for PDEs and any other problem I know in scientific computation.
As "we" all know, good interval algorithms are different from good approximate algorithms. They often involve
1. Handling uncertain parameters
2. Approximate, then validate
3. Contractive maps
4. Enclosures of remainder terms
5. Three-valued logic
and other interval "design patterns".
Even double pi = 3.1415...; changed to interval pi = 3.1415...; lies, unless the compiler is smart enough to widen all literals. Yes, I tell my students pi = 4 * arctan(1); but I know how many of them remember that after the course final exam :-(
If we are successful and achieve an interval standard, it is very likely we will put intervals into the hands of well-intentioned people, who will believe our message, change double to interval as Alan asked, find their results do NOT enclose the true answer to their problem, and conclude interval advocates have lied.
I hate to be elitist, but I think intervals are only usable by experts. Experts can develop tools which allow T.C. Mits (The Celebrated Man In The Street - S. I. Hiakowa, linguist and former US Senator) to exercise interval algorithms without even knowing he has done so. Expecting T.C. Mits to write algorithms yielding guaranteed bounds is unrealistic. A standard is essential for experts, too, but we must beware the Law of Unintended Consequences.
That is another reason why I argue for SIMPLICITY and as much care as we can muster for warnings on miss-use. We cannot assume the programmer knows what (s)he is doing.
Dr. George F. Corliss
Electrical & Computer Engineering
Haggerty Engineering #296
Marquette University
P.O. Box 1881
1515 W. Wisconsin Ave.
Milwaukee WI 53201-1881
George.Corliss@xxxxxxxxxxxxx
414-288-6599; -288-4400 (GasDay); -288-5579 (Fax)
Www.eng.mu.edu/corlissg