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I am trying to implement the backward euler integration (in c++) for the pendulum problem. I have the forward euler implemented, but frankly I don't know where to even start from for the implicit integration. I understand the update expressions for implicit, and of course the pendulum motion (theta'' = -gravity / length * sin (theta) ).

Almost all content online explaining implicit integration give the update expressions and say that you have to solve for x (t+1), but I haven't found any resources that actually explain how to solve them.

I am not looking for libraries that do this already, I would like to learn the process and do it myself (except maybe for matrix inversion etc). If you understand the process of solving for x(t+1) would you please enlighten me? Do I need to find the derivative of the acceleration? Do I need to write the motion function ( -gravity / length * sin (theta) ) in matrix notation?

Very confused.

Thank you so much!

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# Backward Euler / Implicit Integration Implementation

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