Error Taxonomy (Numerical Methods)

Updated 01 Oct 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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Stopping criteria
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  the Scarborough threshold is $\varepsilon_s=(0.5\times10^{2-n})%$ .
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  applying it to successive-iterate differences is a practical stopping heuristic, not a general proof of $n$ correct significant figures.
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  small updates can also result from stagnation, poor scaling, or floating-point rounding.
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  combine an appropriate step test, residual test, iteration limit, and any available error bound.
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Relative error
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  fractional relative error and percentage relative error differ by a factor of $100$
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  relative-error expressions are undefined when their denominator is zero; use an absolute or combined tolerance near zero
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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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Error Taxonomy (Numerical Methods)

What it is

Types

Stopping criteria

Relative error

Worked example: comparing π\pi approximations

Worked example: Scarborough criterion

Worked example: Babylonian method for 7\sqrt{7} (iterating xk+1=12(xk+7/xk)x_{k+1} = \frac{1}{2}(x_k + 7/x_k), starting at x0=2.0x_0=2.0)

Related

Worked example: comparing $\pi$ approximations

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