Approximation error

The approximation error in some data is the discrepancy between an exact value and some approximation to it. An approximation error can occur because

In the mathematical field of numerical analysis, the numerical stability of an algorithm in numerical analysis indicates how the error is propagated by the algorithm.

One commonly distinguishes between the relative error and the absolute error.

Given some value v and its approximation v_approx, the absolute error is

where the vertical bars denote the absolute value. If ${\displaystyle v\neq 0,}$ the relative error is

and the percent error is

In words, the absolute error is the magnitude of the difference between the exact value and the approximation. The relative error is the absolute error divided by the magnitude of the exact value. The percent error is the relative error expressed in terms of per 100.

These definitions can be extended to the case when ${\displaystyle v}$ and ${\displaystyle v_{\text{approx}}}$ are n-dimensional vectors, by replacing the absolute value with an n-norm.

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