Comment by noobermin
4 hours ago
Sorry, re-reading this, I should have said "most". As the other reply mentions, Pade approx. are also well liked for numerical methods.
I personally mostly do my everyday work using taylor expansion (mostly explicit numerical methods in comp. EM because they're cheaper these days and it's simpler to write down) so it's what first comes to mind.
A quick meta-take here: it is hard to assess the level of expertise here on HN. Some might be just tangentially interested, other might have degrees in the specific topic. Others might maintain a scientific computing library. Domains vary too: embedded systems, robotics, spacecraft navigation, materials modeling, or physics simulation. Until/unless people step up and fill the gaps somehow, we have little notion of identity nor credentialing, for better and for worse.*
So it really helps when people explain (1) their context** and (2) their reasoning. Communicating well is harder than people think. Many comments are read by hundreds or more (thousands?) of people, most of whom probably have no idea who we are, what we know, or what we do with our brains on a regular basis. It is generous and considerate to other people to slow down and really explain where we're coming from.
So, when I read "most people use Taylor approximations"...
1. my first question is "on what basis can someone say this?"
2. for what domains might this somewhat true? False?
3. but the bigger problem is that claims like the above don't teach. i.e. When do Taylor series methods fall short? Why? When are the other approaches more useful?
Here's my quick take... Taylor expansions tends to work well when you are close to the expansion point and the function is analytic. Taylor expansions work less well when these assumptions don't hold. More broadly they don't tend to give uniform accuracy across a range. So Taylor approximations are usually only local. Other methods (Padé, minimax, etc) are worth reaching for when other constraints matter.
* I think this is a huge area we're going to need to work on in the age where anyone can sound like an expert.
** In the case above, does "comp. EM" mean "computational electromagnetics" or something else? The paper talks about "EML" so it makes me wonder if "EM" is a typo. All of these ambiguities add up and make it hard for people to understand each other.