What happens if I set both RelTol and AbsTol for ODE45 and/or both TolX and Tolfun for FSOLVE?

I'm solving an eigenvalue problem via a combination of ODE45 and FSOLVE, and I'm not sure what happens if I set both RelTol and AbsTol for the former and both TolX and Tolfun for the latter. Do these functions refine steps, etc. until both requirements are satisfied or only one of them?

Answer 1

[Torsten]

Do these functions refine steps, etc. until both requirements are satisfied or only one of them?

Both.

Read about error control for ode45 here

https://uk.mathworks.com/help/matlab/ref/odeset.html

and for fsolve here

https://uk.mathworks.com/help/matlab/ref/optimset.html

But you should start with the default settings - a reasonable specification is sometimes subtle, especially if two solvers interact.

[Eugene Benilov]

Thank you, Torsten.

The ref. on ODESET was particularly helpful: it explains that AbsTol and RelTol serve different purposes:

AbsTol is a threshold below which the value of the solution becomes unimportant.

RelTol controls the number of correct digits in all solution components, except those smaller than the absolute tolerance AbsTol.

Thus, both should be set (unless you are happy with their default values).

As for OPTIMSET, its description just explains the meaning of TolX and TolFun, but does NOT explain whether the stopping criteria of FSOLVE are "either/or" or "both". Could you please recall where exactly you read about this issue?

[Eugene Benilov]

I might add that I ran some experiments (by solving the equation x^2-1=0 with various values of TolX and TolFun) -- and the results suggest it's either/or, but I'd like to be sure.

[Torsten]

I might add that I ran some experiments (by solving the equation x^2-1=0 with various values of TolX and TolFun) -- and the results suggest it's either/or, but I'd like to be sure.

I agree (see the text in italics):

From the documentation:

TolFun — Termination tolerance on function value

1e-4 (default) | nonnegative scalar

Termination tolerance on the function value, specified as the comma-separated pair consisting of 'TolFun' and a nonnegative scalar. Iterations end when the current function value differs from the previous value by less than TolFun, relative to the initial function value.

TolX — Termination tolerance on x, the current point

1e-4 for fminbnd and fminsearch, eps for fzero, 10*eps*norm(c,1)*length(c) for lsqnonneg (default) | nonnegative scalar

Termination tolerance on x, the current point, specified as the comma-separated pair consisting of 'TolX' and a nonnegative scalar. Iterations end when the current point differs from the previous point by less than TolX, relative to the size of x.