Exit codes for fzero

[Rik]

Jaroslav,

The fzero routine you contributed is very good, but I have a question about
the exit code for certain pathological functions.

I am looking at the function 1/(x - pi) for the range [0, 5].  The zero,
such as it is, is at pi.  On either side of this the function is basically
-/+ infinity.

Running the following code I get an exit code of 1 which seems to indicate
that the algorithm has converged successfully.

[x,fval,info,output] = fzero(@(x) 1./(x-pi), 3)
x =  3.1416
fval =  7.5060e+14
info =  1
output =
{
  iterations =  85
  funcCount =  87
  bracket =

     3.1416   3.1416

  bracketf =

    -1.1259e+15   7.5060e+14

}

When I run with fsolve I get a return code of zero indicating that there
was a problem during the computation of the solution.

 [X, FVEC, INFO, OUTPUT, FJAC] = fsolve (@(x) 1./(x-pi), 3)
X = -2746.1
FVEC = -3.6373e-04
INFO = 0
OUTPUT =
{
  iterations =  401
  successful =  344
  funcCount =  519
}

FJAC = -1.3230e-07

Earlier versions of Octave were even more explicit.  From the 3.0.X branch
I get "error: fsolve: number of function calls exceeded limit" when running
fsolve.

My question is whether fzero could be modified to indicate correctly when
there was a problem with the algorithm.  I suspect in my case that fzero
switched to a bisection algorithm and kept adding one bit of precision with
each function evaluation while simultaneously flipping above and below the
value of pi.  Eventually it collected the 53 bits of precision for IEEE
floating point numbers and then returned.

One thought I had would be a single sanity check at the end of fzero to
verify that fval is somewhere near zero.  That would easily catch this case
where the end value of 7x10^14 cannot be confused with zero.  Is there
another better way to get an error exit code when the algorithm has failed?

--Rik