[pgorczak]

I guess the issue is having to store the intermediate state somewhere. It’s true for the PID example that numerical integration is easier to compute - if you look at the comment with the closed form solution, you need trigonometric and exponential functions to evaluate it. It’s kind of fascinating that the iterative method approximates the same thing with just addition and multiplication.

[MITSardine]

At the same time, the cos and exp are also computed using addition and multiplication. Unless the values are tabulated, I imagine you'd use a series expansion. Or an even more straightforward argument is computers only know how to do addition and multiplication...

In the case of the exponential, if you're willing to start from a known value (say 0, where exp(0) = 1) and need values until the last, then using the very definition of the exponential is even more straightforward... this is the only function s.t. f' = f and f(0) = 1. In other words, step through it! The most natural definition of the exponential is as an ODE to begin with.

Tangentially related, this is one of my favourite articles "Nineteen Dubious Ways to Compute the Exponential of a Matrix":

https://www.cs.jhu.edu/~misha/ReadingSeminar/Papers/Moler03....