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→Regarding the section "Rough estimation": relationship between under-estimates and over-estimates |
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:The estimates chosen were the ones which minimized the absolute value of the relative error (see [[Methods of computing square roots#Convergence]]) in the worst cases while preserving the simplicity of the method by limiting the estimate to less than one significant digit. If we used <math>1 \cdot 10^n </math> for the entire range <math>10^{2n - 1} .. 10^{2n + 1} </math> instead of breaking it into two parts, then the relative error in the worst case would be 2.16 rather than 1.0 for the method with 2 or 6. This would require roughly one extra iteration of the Babylonian method to fix.
:One should ''not'' expect the estimates used with binary to be the same as those used with decimal since the ranges being approximated are not the same ranges. [[User:JRSpriggs|JRSpriggs]] ([[User talk:JRSpriggs|talk]]) 08:26, 14 July 2013 (UTC)
::I have reworked the decimal and binary cases, skipping the coarse estimate over the entire range based on your comment. Now it should be clear to the reader how the estimates are calculated and how they can be modified if required. By the way, why is the absolute value not contained in the relative error formula? [[User:Isheden|Isheden]] ([[User talk:Isheden|talk]]) 21:48, 14 July 2013 (UTC)
:::Your most recent changes have made the section more consistent and clear. Thank you.
:::I did not put an absolute value into my version of the definition because if I had, then ''x''<sub>''n''</sub> could not be expressed as a function of ε<sub>''n''</sub> (unless we assume that ''x''<sub>''n''</sub>≥√''S'') which would interfere with deriving the formula for ε<sub>''n''+1</sub>.
:::Also, <math>\vert \frac{x_n}{\sqrt{S}} - 1 \vert</math> is the wrong way to equate over-estimates with under-estimates. Better would be <math>\vert \ln \frac{x_n}{\sqrt{S}} \vert</math>, but I did not want to use the transcendental function ln in an article targeted to a middle school audience. Notice that if ''y''=''S''/''x'', then (''y''+(''S''/''y''))/2=(''x''+(''S''/''x''))/2. So, if ''x'' were an under-estimate of √''S'', then the equivalent over-estimate is ''S''/''x''. If we replace that in the formula for relative error, we get <math>\varepsilon_n = \frac{\frac{S}{x_n}}{\sqrt{S}} - 1 = \frac{\sqrt{S}}{x_n} - 1 </math>. So we could use <math>\varepsilon_n = \max\left(\frac{x_n}{\sqrt{S}},\frac{\sqrt{S}}{x_n}\right) - 1</math> except for the fact that this function is not invertible. [[User:JRSpriggs|JRSpriggs]] ([[User talk:JRSpriggs|talk]]) 08:54, 15 July 2013 (UTC)
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