Approximation error

In the mathematical field of numerical analysis, the numerical stability of an algorithm indicates the extent to which errors in the input of the algorithm will lead to large errors of the output; numerically stable algorithms do not yield a significant error in output when the input is malformed and vice versa.

The correct reading being 6 mL, this means the percent error in that particular situation is, rounded, 16.7%.

First, relative error is undefined when the true value is zero as it appears in the denominator (see below).

[5]: 34 We say that a real value v is polynomially computable with absolute error from an input if, for every rational number ε>0, it is possible to compute a rational number vapprox that approximates v with absolute error ε, in time polynomial in the size of the input and the encoding size of ε (which is O(log(1/ε)).

Analogously, v is polynomially computable with relative error if, for every rational number η>0, it is possible to compute a rational number vapprox that approximates v with relative error η, in time polynomial in the size of the input and the encoding size of η.

But, if we assume that some positive lower bound on |v| can be computed in polynomial time, e.g. |v| > b > 0, and v is polynomially computable with absolute error (by some algorithm called ABS), then it is also polynomially computable with relative error, since we can simply call ABS with absolute error ε = η b.

An algorithm that, for every rational number η>0, computes a rational number vapprox that approximates v with relative error η, in time polynomial in the size of the input and 1/η (rather than log(1/η)), is called an FPTAS.

In most indicating instruments, the accuracy is guaranteed to a certain percentage of full-scale reading.

are n-dimensional vectors, by replacing the absolute value with an n-norm.

Graph of (blue) with its linear approximation (red) at a = 0. The approximation error is the gap between the curves, and it increases for x values further from 0.
Best rational approximants for π (green circle), e (blue diamond), ϕ (pink oblong), (√3)/2 (grey hexagon), 1/√2 (red octagon) and 1/√3 (orange triangle) calculated from their continued fraction expansions, plotted as slopes y / x with errors from their true values (black dashes)