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> Python libraries let me replicate everything I wanted to do with Mathematica: Matplotlib for graphics, SymPy for symbolic math, NumPy and SciPy for numerical calculations

Are the Python libraries precise enough for professional mathematicians? And do they deal with mathematical 'edge cases', a variety of inputs (formats, notations, etc.), etc.?

On one hand, I could say 'the author uses them therefore they must be sufficiant'. On the other, I've seen plenty of cases where the professionals were not careful about the tools they use (e.g, spreadsheets running critical, large-scale financial operations).

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SageMath (which I started in 2004) is in fact a Python library targeted at professional mathematicians (mainly research in pure math), and is much stronger than Mathematica in many areas, including number theory, algebraic combinatorics and algebraic dynamics. It is weaker than Mathematica in symbolic calculus.

People tend to underestimate the extent of Mathematica's libraries. Take for example process control [1]. Not only does Mathematica have a pretty thorough set of functions for solving process control problems, it works with both linear and non-linear systems, and can find symbolic solutions. When I search for python equivalents, I find abandoned or incomplete-looking projects that are much more limited in scope (e.g. linear systems only) and trying to just provide some of Matlab's functionality (i.e. no symbolic analysis)

[1] https://reference.wolfram.com/language/guide/ControlSystems....

in particular I'm curious if sympy is really as good as Mathematica. I haven't used Mathematica since doing physics hw as an undergrad but it's symbolic manipulation was amazing most of the time

It's not, but for most people it may not matter. Mathematica seems to have a much larger set of integrals, differential equations, special functions, etc. that it can recognize. So as much as I dislike the language itself, I do keep Mathematica installed because in many cases SymPy cannot handle the transformation.

On the other hand, open source tools are catching up. Part of the SymPy project is SymEngine which is a reconstruction in C++. SymEngine is not as feature-filled as SymPy yet, but it flies. It's much faster than SymPy, and also Mathematica. SymEngine.jl works very well since by using the symbolic types in side of Julia you get both the speed (via function specialization) while getting a lot of free features through generic code. For example, Julia code is generic over number types, so if you call inv on a matrix of SymEngine symbols the built-in Base version will compile a fast version for SymEngine symbols and use it. So you have all of Julia's Base available along with Julia packages (yeah, you can put them in a neural net if you wanted to). So SymEngine.jl is the only thing that works when I need that speed, but I do have to keep Mathematica installed for its special handling of specific equations.

And I hate to say it but... the Mathematica notebook is really pretty. You put the exponents up, the fractions on top of each other...

>And I hate to say it but... the Mathematica notebook is really pretty. You put the exponents up, the fractions on top of each other...

yup. for quick and dirty type setting i preferred it to latex

> On the other hand, open source tools are catching up.

SymPy is doing nicely, but it's decades behind Mathematica when it comes to symbolic computation.

By the way, if you want to invert a symbolic matrix, chances are that generic Gaussian elimination working with symbolic expressions isn't the best algorithm (it might even give wrong answers if zero testing is done incorrectly), and compiled code isn't going to change that. The state of the art in symbolic linear algebra uses specialized algorithms like evaluation-interpolation, modular computation, and all kinds of low level optimizations.

It was a bunch of small matrices so it worked out well, but that's good to know. For future reference, what library implements these symbolic linalg routines? When I was looking around I could only find LinBox (http://www.linalg.org/) but the barrier to entry was a little high for me to dig in.

In Julia, there is Nemo.jl for high performance exact and symbolic linear algebra. At least that is the goal; it doesn't have the best algorithms yet in all cases, but it has some of them (and more are being developed). You can see some examples here: http://nemocas.org/benchmarks.html

I really like that library for other reasons, but I couldn't find documentation for anything symbolic. Could you point to the right places? Feel free to follow up on other forums since this is getting pretty off topic.

You have to work with multivariate polynomials (possibly over number fields). If you have symbolic expressions in another format, you would have to manually convert them to that representation first, possibly after introducing extra variables for non-algebraic constants (which may or may not be a good idea depending on the specific circumstances). Having that kind of rewriting done automatically would certainly be a good thing!

Re: Speed

Mathematica is undergoing a compiler re-vamp, which may change that.

one pretty clearly inspired by Julia's approach, from what I can understand.

Well I think Mathematica has an interesting spot here. If you consider their symbolic language as instead an AST for mathematical functions with tons of available transformations to perform simplifications, then that can allow for a very nice compiled output. Julia, Modelica, TensorFlow, Mathematica, etc. all seem to be heading in this same direction of codifying and then simplifying mathematical structures to receive better runtime code. There's a lot of convergent development going on here.

https://www.youtube.com/watch?v=jxl6IvDvGHU

That's a video where they lay out how they want it to work, if you're interested and haven't seen it.

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