Calculation Rounding Error/Examples/Arbitrary Example 1
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Example of Calculation Rounding Error
Consider the equation:
- $x = \dfrac 1 {1 - \cos 1 \degrees}$
Evaluating the calculation while rounding to $4$ decimal places gives:
- $x = 5000$
but the true value is $6565.8$ to $1$ decimal place.
Hence the calculation rounding error of this calculation is $1.6565.8$, or some $24 \%$ or $31 \%$ relative error, depending on how the latter is calculated.
Proof
We have:
- $\cos 1 \degrees = 0.999847695 \ldots$
which is $0.9998$ to $4$ decimal places.
So to $4$ decimal places:
\(\ds \dfrac 1 {1 - \cos 1 \degrees}\) | \(=\) | \(\ds \dfrac 1 {1 - 0.9998}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {0.0002}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 5000\) |
To $9$ decimal places:
\(\ds \dfrac 1 {1 - \cos 1 \degrees}\) | \(=\) | \(\ds \dfrac 1 {1 - 0.999847695}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {0.000152305}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 6565.772 \ldots\) |
Hence we calculate the relative error as follows, using the two variants:
\(\ds \dfrac {\size {6565.8 - 5000} } {6565.8} \times 100 \%\) | \(=\) | \(\ds 23.85 \%\) | ||||||||||||
\(\ds \dfrac {\size {6565.8 - 5000} } {5000} \times 100 \%\) | \(=\) | \(\ds 31.32 \%\) |
$\blacksquare$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): error: 1. (in numerical computation) Rounding (or roundoff) errors
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): error: 1. (in numerical computation) Rounding (or roundoff) errors