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Every mathematician has only a few tricks (2020)

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Re: Every mathematician has only a few tricks (2020)

#51

Post by amirhirsch »

ghtbircshotbe wrote: post #39
sigmoid10 wrote: post #30

Earlier quoted context omitted.

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.

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.

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.

Re: Every mathematician has only a few tricks (2020)

#52

Post by ErroneousBosh »

I do various things where for example I need to convert a linear movement of a control knob to a non-linear value.

Quite often I'll look at it and go "aha, it's exponential subtracted from a constant", and then go "aha, that doesn't quite work, it's a polynomial", and then fiddle with that for twice as long.

Eventually, "meh, sod it, I'll just sketch out a graph of what I want on paper, turn it into a lookup table, and LERP for the correct value".

It's amazing how quickly you get a sense of "I just need to bend that in a little so that if these are both up full it doesn't go absolutely mental", and plan your tables accordingly. Also ROM is cheap these days.

Re: Every mathematician has only a few tricks (2020)

#53

Post by sigmoid10 »

seanhunter wrote: post #37
sigmoid10 wrote: post #30

Earlier quoted context omitted.

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 proble…

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.

Re: Every mathematician has only a few tricks (2020)

#54

Post by 19f191ty »

nabla9 wrote: post #21
Animats wrote: post #18

Feynman wrote in his autobiography that much of his success came from having different mathematical tricks than most of his peers. So when they were stuck, he could sometimes make progress.

He had many tricks in limited area. All his his tricks were limited to classical calculus and nineteenth-century mathematics. He didn't do anything fancy. His style was always the same, he just mastered it really well.

That is the actual trick. To master one trick so well that you can spot anywhere its hiding. Unfortunately, this takes a long time, which is why most mathematicians can only do it for a couple of tricks in their lifetime

Re: Every mathematician has only a few tricks (2020)

#55

Post by ComplexSystems »

bsoles wrote: post #50
seanhunter wrote: post #19

My dad (an engineer not a mathematician) would use Newton-Raphson[1] to solve basically any problem that wasn’t very obviously linear. When I was a kid, some of my first programming memories were my dad getting me and my brother to implement Newton-Raphson in basic on an HP85a, getting me to implement Newton-Raphson in rpn on an HP calculator, debugging my dad’s (genuinely revolting) basic program[2] which wouldn’t r…

The most brilliant software developer (an EE PhD) that I have ever worked with has been using the singular value decomposition (SVD) to solve an enormous number of linear algebra and numerical computing problems in engineering software. The SVD seems to be useful for many engineering computations if you know how to apply it.

What are some interesting things he has used the SVD to solve?

Re: Every mathematician has only a few tricks (2020)

#57

Post by ghtbircshotbe »

bell-cot wrote: post #40
ghtbircshotbe wrote: post #39

Earlier quoted context omitted.

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.

Re: Every mathematician has only a few tricks (2020)

#58

Post by golem14 »

TeMPOraL wrote: post #17
tibbar wrote: post #11

Earlier quoted context omitted.

This reminds me of my further theory that everyone needs one 'heavy' and one 'light' technique. The 'light' technique is something that often works well as a heuristic and can be an effective unit of iteration. The 'heavy' technique is something that you can fall back on in difficult cases, something that can reliably solve hard problems, even if it's slow. Sometimes the heavy technique is: just ask someone else. ;)

> Sometimes the heavy technique is: just ask someone else. ;) For a lot of people I know, this is the light technique!

for those, llms are a godsend

Re: Every mathematician has only a few tricks (2020)

#59

Post by bee_rider »

bsoles wrote: post #50
seanhunter wrote: post #19

My dad (an engineer not a mathematician) would use Newton-Raphson[1] to solve basically any problem that wasn’t very obviously linear. When I was a kid, some of my first programming memories were my dad getting me and my brother to implement Newton-Raphson in basic on an HP85a, getting me to implement Newton-Raphson in rpn on an HP calculator, debugging my dad’s (genuinely revolting) basic program[2] which wouldn’t r…

The most brilliant software developer (an EE PhD) that I have ever worked with has been using the singular value decomposition (SVD) to solve an enormous number of linear algebra and numerical computing problems in engineering software. The SVD seems to be useful for many engineering computations if you know how to apply it.

It’s the optimal low-rank approximation of a matrix in some norm-2 sense, it has those nice beautiful orthogonal factors on both, and it is often quite easy to compute. Why wouldn’t you use it, right? If you are going to have one trick, truncated SVD is a good pick.

Does he like Halko, Martinsson, and Tropp’s randomized SVD? It is pretty slick.

Re: Every mathematician has only a few tricks (2020)

#60

Post by bee_rider »

forinti wrote: post #34
seanhunter wrote: post #19

My dad (an engineer not a mathematician) would use Newton-Raphson[1] to solve basically any problem that wasn’t very obviously linear. When I was a kid, some of my first programming memories were my dad getting me and my brother to implement Newton-Raphson in basic on an HP85a, getting me to implement Newton-Raphson in rpn on an HP calculator, debugging my dad’s (genuinely revolting) basic program[2] which wouldn’t r…

My dad was also an Engineer. He was also a Fortran fan. Once he asked me to explain OOP. After I explained the basics he said it was useless and never looked back.

Fortran has OO features these days. It is nice.

I am kind of dumb and old-school so I wrote a bunch of code using macros to handle multiple precisions. If I could go back in time, I’d definitely just use object oriented code. In either case, though, “we can try a mixed precision implementation, I automatically generated single precision” is an incredibly liberating thing to be able to say as a computational scientist!

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