Question

What is difference between local error and global error?

Answer 1

Answer on Question #79635 – Math – Quantitative Methods

Question

What is the difference between local error and global error?

Solution

The output of a discrete variable method is a set of points $\{t_n, X_n\}$ and the output of the dynamical system is a continuous trajectory $x(t)$. For the numerical results to provide a good approximation to the trajectory we require that the difference

$$\left| X_N - x(t_N) \right| < \varepsilon$$

where $\varepsilon$ is some defined error tolerance, at each solution point. This difference is called the global error and is the accumulated error over all solution steps.

The local error:

$$\left| \bar{X}_n - x(t_n) \right|$$

at each step where $\bar{X}_n$ is the numerical solution obtained on the assumption that the numerical solution at the previous solution point is exact.

The difference between local error and global error is as follows:

Global error is the accumulated error over all solution steps;

local error is the error at each step of the numerical solution.

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