10.1: Absolute and Relative Errors

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Consider an algorithm that tries to find the value of $x^{\star}$. If the algorithm actually returns the value $x$, then there will be some error. The absolute error is \[|x - x^{\star}|\]

and the relative error is \[\left|\frac{x-x^{\star}}{x^{\star}}\right|.\]

Often, the percent error is helpful as well, which is just the relative error times 100.

Example

Consider an algorithm that returns $x=0.0153$ and the actual answer is $x^{\star}=0.0150$. Find both the absolute and relative errors.

The absolute error is $|0.0153-0.015|= 0.0003$ and the relative error is \[\left|\frac{0.0153-0.015}{0.015}\right|=0.02\] or 2\%.

We can do this is julia as follows. Here is the absolute error:

xstar = 0.0150;
x = 0.0153
abs(x-xstar)

0.0002999999999999999

and the relative error is

xstar = 0.0150;
x = 0.0153
abs((x-xstar)/xstar)

0.019999999999999997

which is the same results as above.

Exercise

Find the relative, absolute and percent error if $x^{\star} = 130.32$ and $x=130$.

# fill in code here.