Relative errors
- We normally like to use relative errors because we can set a desired relative error tolerance and have it work for solutions of arbitrary size.
- For a single variable $x$: $$\epsilon_r = \frac{x_k-x_{k+1}}{x_{k+1}}.$$
- Here, we are assuming some iterative calculation of $x$. If we knew the solution $x$, we could write the error in our current approximation $\hat{x}$ as $\epsilon_r = (\hat{x}-x)/x.$ Since we don’t know the solution, we approximate $x$ as $x_{k+1}$ and $\hat{x}$ as $x_k$.
- We normally want the magnitude of the error, so either take the absolute value, or the square root of the square:
$$\epsilon_r = \left|\frac{x_k-x_{k+1}}{x_{k+1}}\right| = \sqrt{\left(\frac{x_k-x_{k+1}}{x_{k+1}}\right)^2}.$$
- Now, suppose that $x$ is a vector of $n$ unkowns that we are solving together.
- We need some measure of the relative error.
- We can either think in terms of components, or we can think in terms of the vector.
- In terms of components, we want some average relative error.
- In terms of the vector, we want some relative length of the error vector.
Relative error in components
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Mean relative error: $$\epsilon_1 = \frac{1}{n}\sum\left|\frac{x_k-x_{k+1}}{x_k}\right|.$$
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Root mean square relative error: $$\epsilon_2 = \sqrt{\frac{1}{n}\sum\left(\frac{x_k-x_{k+1}}{x_k}\right)^2}.$$
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Both of these two equations are measures of the average relative error in a solution component.
Relative error vector
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Here, we’ll use the 2-norm as the length of a vector (square root of sum of components).
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This is the length of the error vector divided by the length of the solution vector: $$\epsilon_3 = \frac{\sqrt{\sum(x_k-x_{k+1})^2}}{\sqrt{\sum x_{k+1}^2}}.$$
- This can be written in python as
np.linalg.norm(xold-x)/np.linalg.norm(x) - This is not a very good choice.
- It assumes some measure of consistency among the variables.
- Suppose we have $x[0] = T$ (K), and $x[1] = m$ (kg). Comparing and combining terms doesn’t make sense. The magnitudes may be way different, and the units don’t even match.
- This can be written in python as
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A better equation is
$$\epsilon_4 = \sqrt{\sum\left(\frac{x_k-x_{k+1}}{x_{k+1}}\right)^2}.$$- This is effectively the length of the relative error vector. Each component error is scaled by that component’s magnitude. (Any units would cancel.)
- In Python, you can write this as
np.linalg.norm((xold-x)/x)