`lsoda` from `odepack`: does relative tolerance override absolute tolerance, or the other way around?

Question

I want to compare the number $x$ with $x' = x + \Delta x$.

If I have selected an absolute tolerance $atol = 0.1$, then:

\begin{equation} |\Delta x| < atol \implies x' \approx x \end{equation}

If I have selected a relative tolerance $rtol = 0.1$, then:

\begin{equation} \left|\frac{\Delta x}{x}\right| <rtol \implies x' \approx x \end{equation}

What happens when I set $rtol = atol = 0.1$? Does the relative tolerance check override the absolute tolerance check?

Answer 1

That very much depends on how it is implemented in a specific software, so you need to check either the documentation or the source code. For lsoda, the documentation states:

The estimated local error in Y(i) will be controlled
so as to be roughly less (in magnitude) than

EWT(i) = RTOL*ABS(Y(i)) + ATOL     if ITOL = 1, or
EWT(i) = RTOL*ABS(Y(i)) + ATOL(i)  if ITOL = 2.

Thus the local error test passes if, in each
component, either the absolute error is less than
ATOL (or ATOL(i)), or the relative error is less
than RTOL.

Use RTOL = 0.0 for pure absolute error control, and
use ATOL = 0.0 (or ATOL(i) = 0.0) for pure relative
error control.  Caution:  Actual (global) errors may
exceed these local tolerances, so choose them
conservatively.

So in this case, whichever tolerance is satisfied overrides the other.