Approximation error ($x_{\mathrm{approx}}-x$)

For a numerical reference value $x$ and a computed value $x_{\mathrm{approx}}$, the signed approximation error is $x_{\mathrm{approx}}-x$. Stating the reference matters: comparison with a measured value assesses agreement with that value, rather than with an unknown exact value.

An error bound is an established upper bound on a specified approximation error, on a stated domain and under stated assumptions. A numerical comparison at selected points can test an approximation but does not by itself establish a bound throughout that domain. An absolute error bound controls $|\tilde{f}-f|$; a relative error bound additionally specifies the reference scale.

The absolute error is the absolute value of the approximation error, $|x_{\mathrm{approx}}-x|$, measured in the same units as the numerical reference $x$. For example, approximating $1/3$ by $0.33$ gives absolute error $|0.33-1/3|=1/300$.

For a nonzero reference value $x$, the relative error is the absolute error divided by the reference's absolute value:It is dimensionless and expresses the size of the error compared with the quantity being approximated. Approximating $1/3$ by $0.33$ has absolute error $1/300$ and relative error $1/100=1\%$. The definition is unavailable when the reference is zero; use an absolute error or a separately stated scale in that case.

The signed relative error is $\epsilon =(x_{\mathrm{approx}}-x)/x$ for a nonzero reference $x$. Its absolute value is the relative error. For a positive reference, a negative signed relative error means underestimation, and a positive one means overestimation. For a negative reference that interpretation reverses, so retain the displayed definition when interpreting its sign.