Reliability of Sage vs commercial software

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Compared to Mathematica 6.0.1 (the version of mathematica i bought - without paying more, they won't even let me have version 6.0.2 which fixed a bug I encountered), I have found Sage to be more reliable for what I do.

The most important difference is ease of debugging. When you make a mistake, Mathematica plows on. For example, if you forget to define a variable, Mathematica assumes it's symbolic and will keep going until it crashes, hangs, generates a meaningless expression, or you treat it as a list and slice it. if you're lucky Mathematica will give meaningless errors along the way that are hard to trace back to the problem (e.g. they occur 400 lines after the initial problem, and do not mention variable names or line numbers). If you make such a mistake in Sage, it will straight away throw up an error message with a traceback, which makes for MUCH faster debugging.

In my few years' experience using Sage, I have run into several parts that were rough around the edges (e.g. interaction with Mathematica, and to a lesser extent numpy/scipy), and found one bug that silently gave me incorrect answers (returning a vector as a row vector instead of a column vector). But then I also found an issue with Mathematica, where results were qualitatively different depending on whether it was run from a notebook or a command line (which turned out to be 32-bit vs 64-bit).

Still, I find it harder to write code that silently gives the wrong answer in Sage than in Mathematica, and it is also much easier to write code that gives the correct answer in Sage than to do the same in Mathematica, due to the errors and tracebacks that Sage provides. Sage almost always gives errors when something goes wrong, which helps you work around its inconsistencies, unlike mathematica, which hopes for the best and gives no advice as to how to work around its inconsistencies.

I have had to write my own functions and do lots of numerics - both things which Sage (which incorporates Python and SciPy) is very well suited to, and Mathematica is poorly suited to.

Other advantages Sage has, like redo/multiple level undo (provided by your text editor, or to a limited extent in your browser) are just gravy.

I don't have too much experience with symbolics. The one time I tried a full_simplify, I got a better result in Sage than in Mathematica, which surprised me.

But the biggest advantage of Sage's price tag is that you have a perpetual free trial. So give it a go - try reproducing some of your Mathematica results in Sage, and see whether it's up to scratch for your needs.

To address your reliability or power question briefly from a different vantage point, I think it depends a lot on what you are doing. If you need the most symbolic antiderivatives you can, Maple or Mma are probably better; on the other hand, many people in "real work situations" use Maxima as a standalone program very, very effectively for symbolic DEs and such. If you are doing more numerical work, my understanding is that the Numpy/Scipy/Matplotlib stack, or the GSL (both in Sage), are pretty darn effective and not necessarily "buggy" at all. If you are doing graph theory or serious number theory, you shouldn't even be asking the question of which package to use. But either of them (or Maple) will probably do 99% of what you need to do, correctly, in most cases.

Matlab is a different situation, because of the enormous amount of third-party code and drivers available (only) for it. Octave might be a free option, but it would really depend on what you need to do.

In the end, you'll have to figure things out for yourself. If your department head is too cheap (to be crass, which is unavoidable in your situation) to buy you a Mathematica license, then they are really putting their money where their mouth is, and you should use Sage.

There are plenty of bugs in any mathematical software, and a quick internet search will find them for you. You should never ever implicitly trust output from a computer in any case, without some other means of checking it (such as "plugging in the answer"). Even when Euler proved that $\zeta(2)=\frac{\pi^2}{6}$ or that $\gamma$ exists, he also checked it by hand to an insane number of digits. Thankfully, we don't have to do that to check our work.