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Relatedly, I've found Newton-Raphson is a great example of an algorithm where Knuth's "I have only proven it correct, not tried it" rears its head very prominently. The obvious implementations can work flawlessly on toy examples and then fail miserably on real-world examples.

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There's a reason why numeric analysis is still actively studied by research mathematicians. If we could just throw something as simple as newton's method at any nonlinear problem, we'd only need people to learn this once in school and everyone could solve everything.

Yeah for that you need Euler’s method… I mean of course Runge-Kutta. … By that I’m of course referring to rk4. … I mean, you have a point.

Joking aside I think having a few basic numerical methods like say Newton, rk4, brent root, monte carlo simulation etc in your general toolbox of techniques you know how to do can make you unreasonably effective in a wide range of situations. Just yesterday I had to solve a problem in a relatively small space so I first used a brute force method to check all the feasible solutions and having got the full list of actual solutions out, figured out the analytical solution. It meant I could be very confident that my analytical solution was correct when I had it.

If you know some basic composition methods, Euler, RK and all the higher order methods can be easily constructed as you need them. But there are still many applications where you for example want to use symplectic methods instead. If you know about those+composition, you can at least solve more or less all of classical newtonian dynamics. But if you go to quantum mechanics or field theory (or quantum field theory), you need to enter a whole other world of numerics to get real world usage out of it.

PDEs are PDEs, regardless of where they come from Newtonian or quantum. Would you care to elaborate why you think quantum requires a new kind of numerical analysis?

For field theory, you can still go some way using normal finite-difference approaches, but you have entered a huge can of worms regarding stability. For quantum physics, the problem starts well before you even get to writing a solver, since (at least for QFT) you are actually dealing with operator-valued distributions rather than normal fields - and that in extremely high (even infinite) dimensional spaces. That means you actually need to solve a path integral instead of PDEs if you want to do any sort of actual numerics, which comes with its very own can of worms. And even if the numeric discretisation is at least mathematically valid, you still need to solve the damn thing over a huge configuration space (depending on your lattice size). Even with purely statistical methods and modern supercomputing, you're quickly running into the limit of what can be achieved in reasonable time for comparatively simple systems. But nobody in e.g. lattice QCD uses normal PDE solvers.

The challenges you mentioned, and techniques to address them, are not unique to quantum physics. I am still not understanding how quantum physics require "new" kind of numerical analysis. And what are these new kinds of techniques you hint at? Could you give me some examples of unique techniques that arose from quantum physics and are not used elsewhere?

They are pretty unique. Many of them were actually developed for it. But I wouldn't even know where to begin to explain this to a layperson without knowing their level of education first. Like, how much do you know about Dyson series? Path integrals? Wick rotations? You're basically asking for a graduate level lecture on numerical methods in quantum field theories. If you really want to know the details, a quick google will probably net you some books or lecture notes.

But I'm not sure I'd recommend going into the field. There's something demoralizing about doing research on something which already has dozens of valid and successful methods, of which you are trying to create a slightly more optimized version.

It's not for a field for everyone.

But that "slightly more optimized version" may mean "one that does not quietly produce disastrously incorrect results for some input values".

I should probably qualify my statement more. There are certainly new and interesting problems in numerics. And even going from say O(n) convergence to O(n log n) convergence can lead to whole new classes of problems you can solve. I don't want to discourage anyone who loves numerics.

What I was trying to say was that as a graduate student you might be given a problem that already has many really good and smart solutions and be essentially told to find a better solution than all of these. How this goes will depend a lot on the specific problem, your advisor, etc.

I am so into optimizing fast polynomial multiplication I assure you there is nothing that will demoralize me from creating a slightly more optimized version.

>something which already has dozens of valid and successful methods, of which you are trying to create a slightly more optimized version

If you ever get a PhD, you'll find that this is pretty much all of academia.

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