Note that truncation error would be incurred even if computers performed floating-point arithmetic operations to infinite accuracy. Unfortunately, computers do not perform such operations to infinite accuracy. In fact, a computer is only capable of storing a floating-point number to a fixed number of decimal places. For every type of computer, there is a characteristic number, , which is defined as the smallest number which when added to a number of order unity gives rise to a new number: i.e., a number which when taken away from the original number yields a non-zero result. Every floating-point operation incurs a round-off error of which arises from the finite accuracy to which floating-point numbers are stored by the computer. Suppose that we use Euler's method to integrate our o.d.e. over an -interval of order unity. This entails integration steps, and, therefore, floating-point operations. If each floating-point operation incurs an error of , and the errors are simply cumulative, then the net round-off error is .
The total error, , associated with
integrating our o.d.e. over an -interval of order unity is (approximately)
the sum of the truncation and round-off errors. Thus,
for Euler's method,
The value of depends on how many bytes the computer hardware uses to store floating-point numbers. For IBM-PC clones, the appropriate value for double precision floating point numbers is (this value is specified in the system header file float.h). It follows that the minimum practical step-length for Euler's method on such a computer is , yielding a minimum relative integration error of . This level of accuracy is perfectly adequate for most scientific calculations. Note, however, that the corresponding value for single precision floating-point numbers is only , yielding a minimum practical step-length and a minimum relative error for Euler's method of and , respectively. This level of accuracy is generally not adequate for scientific calculations, which explains why such calculations are invariably performed using double, rather than single, precision floating-point numbers on IBM-PC clones (and most other types of computer).