I've been using Octave for Galois Field arithmetic and I'm really happy with it. For some reason the script I wrote most recently takes 1/20th (!) as long to run on Octave than on Matlab. (It operates on BCH codes.) (I'm using Octave on the Mac.)
I transitioned from Excel to Python for running basic calculations and simulations a few months back. And it works great! Tons of great modules(matplotlib and numpy are a must) and more than anything its fast as hell. And by fast I mean 50 million points in a few secs and it can plot them without stuttering too!
True, it's not exactly a simulation tool, you've got to spend a little time thinking about how to run the simulation but once thats out of the way, it's all smooth sailing.
PS: I used PSoC Creator before and the line "which allows users to design their own Programmable System on a Chip and have it manufactured (that part costs)" is misleading. You can buy a generic PSoC 3/4/5 and program it with Creator i.e., it's all firmware(although that term actually encompasses a lot more when looking a PSoC) and no custom manufacturing is involved. The cool thing is you can always move stuff around and reprogram so it's pretty much like a custom SoC.
An alternative like this for excel is very good for the engineers. I always felt that engineers need more and should be able to upgrade it themsleves without calling any customer support or getting into copyright issues. I started using Ubuntu and in last three years it never crashed and so easy to operate and has so much flexibility.
You could add PARI/GP from France for advanced algebraic analyses, it's actively maintained, and also goes well beyond what I understand.
I have a question, though: in 2000, I was using a program that could fit a dataset to a function optimally ( least-squares, outlier reject etc.) by trying hundreds of algorithms, e.g. series, logs, ratios, trancentals, relations among them. Do you know the name of that program?
While there are some great tools in this and the original list, I am left scratching my head as to why LTspice failed to make the list again. I use it very often and is at the top of my list of free tools.
Another nice tool is the free version of Mathcad. When you let the free 30 day trial lapse. You get a very stripped down version that is still quite useful as a math scratchpad that handles units. Very useful for free.
Finally, if you are going to included tools tied to vendor's parts like Cypress' PSoC Creator and TI's WEBENCH, you should included the free versions of Quartus from Altera and ISE WebPack from Xilinx.
It's a bit iffy to classify R as "an alternative to Excel". R is not a spreadsheet, it never has been and never will be...and Excel has nowhere near the data manipulation capabilities of R. There is very little overlap between their capabilities.
As a design engineer, I find Excel useful about once every three to four years, usually for things like BOMs, but I use R almost daily for REAL engineering work. R makes short work of anything that my trusty HP calculator can't easily handle. It was designed for statistical work (it traces its roots back to Bell Labs' 'S" language in the 1970s, and has seen a great deal of refinement over the years, but it's more properly presented (in my opinion) as a "mathematical Swiss Army knife".