THE GARDEN

Error Taxonomy (Numerical Methods)

Updated 15 Sep 2026

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  What it is

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    the standard set of error measures used to evaluate a numerical approximation

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  Types

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    absolute error — $E_a = |x - \hat{x}|$

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    true relative error — $\varepsilon_t = \left|\dfrac{x-\hat{x}}{x}\right| \times 100\%$; requires knowing the true value $x$

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    approximate relative error — $\varepsilon_a = \left|\dfrac{\hat{x}_k - \hat{x}_{k-1}}{\hat{x}_k}\right| \times 100\%$; usable in iterative methods where the true value is unknown, by comparing successive approximations

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    Scarborough criterion — $\varepsilon_s = (0.5 \times 10^{2-n})\%$, the stopping threshold guaranteeing at least $n$ correct significant figures once $\varepsilon_a$ drops below it

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  Worked example: comparing $\pi$ approximations

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    $\hat{\pi}_1 = 22/7$: true relative error $\approx 0.0402\%$

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    $\hat{\pi}_2 = 355/113$: true relative error $\approx 8.49 \times 10^{-6}\%$ — about 4 orders of magnitude more accurate

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  Worked example: Scarborough criterion

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    for 3 correct significant figures: $\varepsilon_s = 0.05\%$; for 6 figures: $\varepsilon_s = 0.00005\%$

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  Worked example: Babylonian method for $\sqrt{7}$ (iterating $x_{k+1} = \frac{1}{2}(x_k + 7/x_k)$, starting at $x_0=2.0$)

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    $x_1=2.750000$ ($\varepsilon_a = 27.27\%$) → $x_2=2.647727$ ($3.86\%$) → $x_3=2.645752$ ($0.0746\%$) → $x_4=2.64575131$ ($0.000026\%$, below the $0.01\%$ target) — convergence in 4 iterations

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  Related

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    Chopping vs Rounding

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