Error Definitions

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Numerical errors arise from the use of approximation to represent exact mathematical operations and quantities. These include truncation errors, wich result when approximations are used to represent exact mathematical procedures, and round-off errors, wich result when the numbers having limited significant figures are used to represent exact numbers. For both types, the relationship between the exact, or true, result and the approximation can be formulated as

True value = approximation + error (1)

By rearranging (1), we find that numerical error is equal to the discrepancy between the truth and the approximation, as in

Et = true value - approximation (2)

where Et is used to designate the exact value of the error. The subscrit t is included to designate that this is the "true" error. This is contrast to other cases, as described shortly, where an "approximte" estimate of the error must be employed. A shortcoming of this definition is that it takes no account of the order of magnitude of the valued under examination. One wa to account for the magnitudes of the quantities being evaluated is to normalize the error to the true value, as in

True fractional relative error = true error / true value (3)

where, as specified by (3), error = true value - approximation. The relative error can also be multiplied by 100 percent to express it as.

et = (true error / true value) * 100

where et designates the true percent relative error.