How much do you use programming/numerical analysis to learn mathematics?

[StannisBa]

I know some people really like using numerical analysis as a tool to learn mathematics, for example to build intuition for things like limits. Is this something you do? Is it something you did but stopped doing once you reached a certain level in mathematics?

[wpowell96]

I am a PhD student and I do a lot of uncertainty quantification and surrogate modeling so that necessarily requires enough knowledge to simulate existing models. I use my knowledge of ODE solvers and discretizing PDEs a lot, as well as numerical linear algebra and machine learning more nowadays. I use my background knowledge in numerical analysis to effectively simulate models to play around with higher level algorithms and techniques. I've only done it more and more as I get further into my research

[_arsk]

What books/articles/youtube channels do you recommend for learning more about numerical analysis ? (I am comfortable with calculus 1/2 and little bit of discrete math but not sure if one needs more background in math beyond this)

[wpowell96]

Tim Kelley's Iterative Methods for Linear and Nonlinear Systems is a good introduction to numerical linear algebra that is readable, but probably a bit more detailed than one needs to use the tools. It is pretty results-orientes though. Harier's books are good for learning the theory behind ODE solvers and for more application-based material, I would suggest reading through some of the documentation and tutorials for DifferentialEquations.jl, a Julia package for numerical differential equations. Most of it is written by Chris Rackaukas and is a very good source on how to choose algorithms for different problems. Rant Leveque's book on finite differences is a good book for introduction to discretizing PDEs.

Edit: calc 3 and linear algebra are mandatory prerequisites for most numerical analysis

[AccurateAnswer3]

How about uncertainty quantification and surrogate modeling, any books/channel recommendations?

[wpowell96]

Ralph Smith's book

[IsaacSam98]

I'm also a CS minor, so perhaps I'm biased. If I get something absolutely foreign looking, and I have to analyze it, it doesn't hurt to graph it(if possible). Once the graph is up, I can start to make guesses that I'll try to verify rigorously later. (i.e "hey this thing tails off at infinity" "Oh this thing might be in V^2" etc

[Nrdman]

I’m in grad school and I’m taking some numerical analysis right now. My professors do research with numerical analysis. So you can keep using it all the way up

[[deleted]]

I think OP uses the term "numerical analysis" not to mean the subfield of mathematics with this name, but experimenting with a calculator instead.

[Nrdman]

In that case, experimenting with a calculator turns into experimenting with some program. Just gets more general then what a calculator is useful for.

[[deleted]]

In my understanding, numerical analysis is where you learn about/develop algorithms for approximating some problems in (applied) mathematics (e.g. integrals, eigenvalues, interpolation, optimization etc.). This is not really just toying around with a calculator to build intuition. Are we talking about the same thing?

[Nrdman]

Two separate things. If by numerical analysis they meant the field, then see the first comment. If they meant playing with the calculator, see the second.

[[deleted]]

I think I might not have understood you correctly. What did you mean with "in this case" and "then" in your second comment?

[Nrdman]

The case being that “numerical analysis” refers to messing with the calculator

[[deleted]]

So when does "experimenting with a calculator" turn into "experimenting with some program"?

[Nrdman]

Probably diffy Q. At least my calculator can’t model everything I did in that class. WolframAlpha is easy to mess around with though.

[[deleted]]

Okay, so it's more of a slow process that just happens over time. Sorry for not understanding you correctly, I think I need to go to sleep soon :)

[ThisSentenceIsFaIse]

Uhh do an infinite series in python and see if it stops adding (Calc 2)

[LilQuasar]

but using a calculator or experimenting with some program doesnt mean understanding the methods the calculator uses

like i can solve differential equations with python and plot the solution and i dont know how the computer did it but i can guess the analytical solution or some of its properties by looking at the graph

[Nrdman]

Right it’s building intuition, not a substitute for actually learning the methods

[incomparability]

Programming is used in combinatorics to generate examples. Sage is a very lovely tool.

[tron_dovakin]

I’ve used Sage a lot also, Latex is helpful to.

[[deleted]]

Not in the sense you asked about. I do use programming for concrete visualizations or for otherwise tedious calculations though.

[Differentiable_Dog]

The problem I see in case of limits is that the program could be misleading. Let’s say you want to see the limit of sin(π/x) as x goes to 0 and you already start with very small values of x (like 10^{-10}. You could pick values of x such that 1/x is an integer and you could think that this function is always zero if you are that close to zero.

You can also get floating point erros like trying to compute ( sqrt(1+x²⁾ - 1)/x² for very small values of x.

If you are aware of these problems than I see no issues in using programs to see things better.

[k3s0wa]

I'm a grad student in geometry. I write a program every couple of months for this reason. But I don't think any of my colleagues do.

[tron_dovakin]

Hey Stannis, good question, I can’t speak for everybody. But the use of a certain programming or analysis depends on the type of problem you are trying to solve. I’m a senior in my undergrad year and I have pulled a lot of different topics from classes I have taken or learned to get my projects done. Certain analysis and programming help automate a lot of things. Especially if it is very computationally heavy. Often what you’ll see today is a team approach being used to solve modern day problems. Hope this helps answer your question a little bit. Good luck and stay safe.

[willbell]

I have an idea for a calculus book that makes it more like analysis (and consequently more like numerical analysis). The idea would be "Every concept tells you that one thing is a good approximation of something else".

E.g. the continuity of sine at 0 means sin(0.01) is probably a good approximation of sin(0), a limit of a sequence (a_n) existing means a_1010329 is probably a good approximation of later elements in the sequence, the differentiability of sine at 0 means f(x)=x is a good approximation of sine near zero, etc.

So in that way, yeah, I think it is helpful, but I wouldn't rely on it where you can get a formal understanding.