Error Definitions

Numerical errors arise from the use of approximation methods to represent mathematical operations and quantities.

• True value = Approximation + Error
• $E_t$ = “true error” = exact value of error = true value - approximation

Definitions

True Percent Relative Error

$${ϵ}_t=\frac{\text{true error}}{\text{true value}}\cdot 100\%$$

This can be used when the true value is known.

Approximate Percent Relative Error

$$\begin{array}{rl}{ϵ}_a& =\frac{\text{approximate error}}{\text{approximate value}}\cdot 100\%\\ & \\ {ϵ}_a& =\frac{\text{current approximation}-\text{previous approximation}}{\text{current approximation}}\cdot 100\%\end{array}$$

This can be used when the true value is unknown. This is also used in iterative methods, where in each subsequent trial, the approximate value gets closer to the true value.

Note that in the above definitions, the absolute value of error values are usually used to avoid positive/negative values.

Absolute Error

$${ϵ}_{t\text{Absolute}}=\mid x_{\text{true}}-x_{\text{Calculated}}\mid$$

This is in units of the value $x$.

Relative Error

$${ϵ}_{t\text{Relative}}=\left|\frac{x_{\text{True}}-x_{\text{Calculated}}}{x_{\text{True}}}\right|$$

This is non-dimensionalized.