Stopping Criterion

For iterative solutions, we need a stopping criterion to determine the number of iterations. This criterion is based on the difference between the new estimate of the value $x_{i+1}$ and the current value $x_i$. When this difference is less than a specified tolerance (TOL), we say the solution has converged.

Absolute Error Stopping Criterion

$$\mid x_{i+1}-x_i\mid <\text{TOL}$$

This is similar in concept to absolute error; as long as error is below a certain tolerance, it’s fine.

Relative Error Stopping Criterion

$$\left|\frac{x_{i+1}-x_i}{x_{i+1}}\right|<\text{TOL}$$

Unlike the absolute stopping criterion above, this one is relative to the current estimate.

For definitions of absolute and relative error, see Error Definitions.

In some cases, the solution may be close to zero, which leads to the possibility of division by zero. In this case, a hybrid version is needed:

Hybrid Stopping Criterion

$$\frac{\mid x_{i+1}-x_i\mid }{\text{Max}(\mid x_{i+1}\mid ,\theta )}<0$$

where $\theta >0$ is a problem-specific value.