 Absolute and relative error tolerances

Absolute tolerances provide a nonzero error scale near zero, while relative tolerances follow component magnitude. A max-norm test ${\max }_i|{\hat{e}}_i|/s_i\le 1$ 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 testAbsolute 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 $\max (|y_i|,|Y_i|)\approx {\mathrm{atol}}_i/{\mathrm{rtol}}_i$ when ${\mathrm{rtol}}_i>0$. Official solver documentation specifies the analogous blend $\mathrm{atol}+\mathrm{rtol}|y|$ and allows component-specific absolute tolerances; the old/new maximum above is the convention used in this example.^[ref]

A root mean square version replaces the maximum norm by $E_{\mathrm{rms}}=\sqrt{m^{-1}{\sum }_i(|{\hat{e}}_i|/s_i{)}^2}$. 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.