Generally speaking, there are 4 types of error:
Modeling Error
To abstract (simplify) a mathematical model from practical
problems, there exists an error between the model and the
actual problem.Measurement Error
The values of the parameters in the model are obtained
through measurement, and the observations find errors.Truncation Error
Using numerical methods to obtain the approximate solution of
the model, there exists an error between the approximate solution
and the exact solution.Roundoff Error
Machine word length is limited, resulting in errors in
representing data in computers.
In this course, we mainly focus on the latter 2 types of error.
Definition of Variables
| Sign | Significance |
|---|---|
| $x$ | Exact Value |
| $x^*$ | Estimated Value |
| $p$ | Digits to be preserved after rounding off |
| $R_n$ | Truncation Error |
| $E$ | Roundoff Error |
| $e$ | Error |
| $\epsilon$ | Absolute Error |
| $e_r$ | Relative Error |
Truncation Error
Formula: $R_n = x - x^*$
Example :
Suppose that we want to calculate the value of $e^x$ with first 4 terms of Taylor Expansion at point 0.
$x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + … + \frac{x^n}{n!} = \Sigma_{n = 0}^{n \to \infty}\frac{x^n}{n!}$
$x^* = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!}$
$R_n = x^* - x = - \Sigma_{n = 4}^{n \to \infty}\frac{x^n}{n!}$
Roundoff Error
Formula: $E = x - x*$ Please note: the order is reversed.
There are two types of Roundoff Error, but the formula is the same. Assume $x = \frac{2}{3}$ and $p=4$
Example:
Round-by-chop
Only preserve the value before $p$ and abandon rest digits.
$$
x^* = 0.6666\space E = x^* - x = -0.000066666……
$$
Round-to-nearest
Rounded the estimated value at digit $p$. If the number at digit $p$ is greater or equal to 5, abandon it; otherwise add the $p-1$ digit with value 1.
$$
x^* = 0.6667\space E = x^* - x = 0.0000333333……
$$
Total Error
Total Error = Truncation Error + Roundoff Error
Absolute Error & Error
$\epsilon(x) = |x^* - x|$
$e(x) = x^* - x$
Relative Error
$$
e_r(x) = \frac{e}{x} = \frac{x^* - x}{x}
$$
