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# Meaning of 'Error' for Standard integrator in LMS Amesim

Experimenter

Hi, I am using LMS Amesim very frequently and I am happy to know the opening of LMS Amesim forum.

The subject is about the meaning of 'Error' for Standard integrator in LMS Amesim.

I have always wondered what the 'Error' exactly means for numerical solution like LMS Amesim.

Of course there are two types of error; relative error and absolute error. But the point of my questions is about the target of the 'Error'. Error can get any meaning when the target is known.

In general numerical integration, error means the difference between exact solution and numerically calculated solution. If the error means the difference between exact solution and numerically calculated solution, does it mean that LMS Amesim already has exact solution for the system?

# Re: Meaning of 'Error' for Standard integrator in LMS Amesim

Siemens Experimenter

Hi,

When doing integration, one need numerical methods that are based on approximation formulaes, meaning that when one evaluates some quantity by the mean of formulae, the formulae is exact to a certain accuracy (or order of magnitude).

By error, here, we thus consider this residual term, i.e. the amount by which the exact solution of the differential system fails to satisfy the used formulae for the integration.

E.g., think to the trapezoidal formulae, where the integral of a function is approximated by the area of the trapeze defined by both extreme values. The error comes from the fact that here, higher order terms of Taylor series  (e.g. 2nd order term) are not taken into account in the formulae.

Doing this, one get some error due to this approximation and it can be proven with some few mathematics that this error is bounded to  and approximated by a higher residual term (thus no need to own the exact solution at a given time, this higher error term being computed from previous approximations).

Note that for dynamical system, it's very important to master this error, as we need to provide the computed solution within a certain accuracy range ( we test that the error  can be controlled and bounded according to provided tolerance).

Hope this answers your question. If this not clear enough, I'll be happy to develop on this.

Thank you for the interesting question!