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When I use the newton's method or hybrid solver in the GSL package to deal with 1-D or multidimensional root solving problems, the code frequently crashes when the solver requests function value outside its domain of definition. For example, to solve $f(x)=0$ where $f(x)$ is only defined at $x\leq0$, even when I start with an initial guess very close to the root, the solver may still ask to evaluate the $f(x)$ at a negative $x$. Bisection method may be helpful in the 1-D problem, but it won't work for a multidimensional problem.

I typically try to solve the problem by arbitrarily define the $f(x)$ in the whole domain. But in some situation, especially in the case of complicated multidimensional root solving, I feel it's hard to extend the domain of the function and make sure it roughly maintain the general trend of the original function.

So I'm wondering if there is a method to restrict the region that solver would evaluate the function.

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  • $\begingroup$ Possibly related question. Can you take a look at the tricks described there and evaluate them for the root-finding usage? $\endgroup$ – Anton Menshov♦ Sep 20 '19 at 9:00
  • $\begingroup$ @AntonMenshov Do you suggest defining the function to NaN outside the domain? I believe the root solver will stop when the function doesn't return a real number. Or do you suggest to construct a fake function outside the domain? If so, do you have any trick on how to construct such a function? $\endgroup$ – HD189733b Sep 20 '19 at 9:30
  • $\begingroup$ Can you provide an example where Newton's method is evaluating your function outside of the domain? $\endgroup$ – nicoguaro♦ Sep 21 '19 at 14:00
  • $\begingroup$ @nicoguaro I think I figured out the reason. I tried to reproduce my problem with a simple function, but I failed. My multidimensional function has some discontinuity in one dimension. The problem happens when the solution is very close to a discontinuity point. I temporarily solve the problem by checking the location of the discontinuity point before the root solving. If it is too close to the potential solution, I modify the function to avoid that. $\endgroup$ – HD189733b Sep 25 '19 at 5:45
  • $\begingroup$ Newton method works for continuous functions, so I would say that what you mention makes sense. If you provide your function (or simplified one that reproduces the problem) you could write an answer on how you solved it. $\endgroup$ – nicoguaro♦ Sep 25 '19 at 12:10
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Although I do not know about the GSL package, I can give some tips for root-finding concerning domain issues.

In the 1-D case, you can guarantee convergence with a bracketing method such as Newt-safe or bisection as you've suggested. This requires a bracketing interval given by two points where the function changes signs.

In the general case, I recommend decreasing the step size of the iteration (e.g. by halving) until you reenter the domain of your function. This should hopefully prove to be more reliable.

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