= Absolute and relative error tolerances
{title2=$s_i={\rm atol}_i+{\rm rtol}_i\max(|y_i|,|Y_i|)$}
Absolute tolerances provide a nonzero error scale near zero, while relative tolerances follow component magnitude. A max-norm test $\max_i|\widehat e_i|/s_i\leq1$ controls every scaled estimated component locally. A root-mean-square test has a different componentwise implication. Neither test automatically certifies the complete trajectory's <global error>.
Write $Y_i$ for the candidate accepted component and require $s_i>0$. The <absolute and relative error tolerances> scale in the title produces the dimensionless <maximum norm> test
$$
E=\max_{1\leq i\leq m}\frac{|\widehat e_i|}{{\rm atol}_i+{\rm rtol}_i\max(|y_i|,|Y_i|)}\leq1.
$$
Absolute tolerances have the same units as the corresponding solution components; relative tolerances are dimensionless. Choose separate absolute scales for components measured in different units or with different significant magnitudes. Near zero, a purely relative test can demand a zero allowable error; a positive absolute scale avoids that difficulty. The crossover is $\max(|y_i|,|Y_i|)\approx{\rm atol}_i/{\rm rtol}_i$ when ${\rm rtol}_i>0$. Official solver documentation specifies the analogous blend ${\rm atol}+{\rm rtol}|y|$ and allows component-specific absolute tolerances; the old/new maximum above is the convention used in this example.<SciPy 1.16.2 documentation/scipy.integrate.solve_ivp>[solve_ivp, rtol and atol parameters: local error estimates, additive scaling and component-specific atol]{ref}
A <root mean square> version replaces the <maximum norm> by $E_{\rm rms}=\sqrt{m^{-1}\sum_i(|\widehat e_i|/s_i)^2}$. It permits one scaled component as large as $\sqrt m$ when all others vanish, so passing that test does not mean every component is at most one. State the aggregation rule along with the <absolute and relative error tolerances>. Neither rule is a bound on actual <global error>.
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