structural:reformatting, structural:scholar_edition
Rounding errors

In step n {\displaystyle n} of the Euler method, the rounding error is roughly of the magnitude ε y n {\displaystyle \varepsilon y_{n}} where ε {\displaystyle \varepsilon } is the machine epsilon. Assuming that the rounding errors are independent random variables, the expected total rounding error is proportional to ε h {\textstyle {\frac {\varepsilon }{\sqrt {h}}}}. Thus, for extremely small values of the step size the truncation error will be small but the effect of rounding error may be big. Most of the effect of rounding error can be easily avoided if compensated summation is used in the formula for the Euler method.