Rounding Errors
The discussion up to now has ignored the consequences of rounding error. In step n of the Euler method, the rounding error is roughly of the magnitude εyn where ε is the machine epsilon. Assuming that the rounding errors are all of approximately the same size, the combined rounding error in N steps is roughly Nεy0 if all errors points in the same direction. Since the number of steps is inversely proportional to the step size h, the total rounding error is proportional to ε / h. In reality, however, it is extremely unlikely that all rounding errors point in the same direction. If instead it is assumed that the rounding errors are independent rounding variables, then the total rounding error is proportional to .
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.
Read more about this topic: Euler Method
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