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Dennis Hendriks commented on MATH631:

I just got back from a 3 week vacation, so I couldn't reply earlier.
The documentation for the RegulaFalsiSolver states: "Unlike the Secant method, convergence
is guaranteed by maintaining a bracketed solution." While this is theoretically true, in this
case it is not so, because (if I understand correctly) only a single bound is updated repeatedly,
and the update is too small to matter (has no effect), due to the double representation.
The change you propose (which is difficult to see as you also change other things in the same
commit) is to modify x0 and f0 if the new value of x and x1 are equal. As I see it, this changes
the algorithm, and it is no longer the Regula Falsi method as known from literature. I'm therefore
against this change.
The problem that is identified in this issue is very similar to the wellknown problem of
the Regula Falsi method: it converges very slowly for certain problems, due to one side being
updated all the time, while the other one stays the same. The Illinois and Pegasus algorithms
solve exactly this problem, and are welldocumented in literature.
I therefore think it would be better if the RegulaFalsiSolver kept it's original implementation,
and for this problem the Illinois or Pegasus method should then be used instead.
The other changes (if statements to switch with default, extracting bound switch statements,
etc) can be kept, if you wish.
The suggestion to add a warning to the Secant and Regula Falsi solvers that this is a possible
problem, and the solution (use Illinois or Pegasus), would indeed be a good idea. In general,
adding a note that the Illinois and Pegasus algorithms perform better, would be a good idea
regardless of this issue.
Once more, to be clear, I don't think this issue is a bug. It is a result of the limited convergence
of the Regula Falsi method combined with the implications of limited double precision. The
limited convergence of the algorithm is a property of the algorithm, and should in my opinion
not be changed. I also don't think that trying to work around the limited double precision
would be a good idea.
> "RegulaFalsiSolver" failure
> 
>
> Key: MATH631
> URL: https://issues.apache.org/jira/browse/MATH631
> Project: Commons Math
> Issue Type: Bug
> Reporter: Gilles
> Fix For: 3.0
>
>
> The following unit test:
> {code}
> @Test
> public void testBug() {
> final UnivariateRealFunction f = new UnivariateRealFunction() {
> @Override
> public double value(double x) {
> return Math.exp(x)  Math.pow(Math.PI, 3.0);
> }
> };
> UnivariateRealSolver solver = new RegulaFalsiSolver();
> double root = solver.solve(100, f, 1, 10);
> }
> {code}
> fails with
> {noformat}
> illegal state: maximal count (100) exceeded: evaluations
> {noformat}
> Using "PegasusSolver", the answer is found after 17 evaluations.

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