I am developing a simulation software which involves solving around 10^8 simultaneous linear equations. Equations are sparse. I already made a prototype in which I implemented LU decomposition myself. But my implementation can only address less than 10^4 equation due to performance and memory constraints. I never worked with any CAS systems. Can anyone suggest some good CAS system from where I can start? Even after a long web search I was not able to get a good comparison of existing CAS systems in terms of the size of data it can process and also a performance comparison.
Development environment I am using is C# with .Net 4.0, but I can use any C++ , C or C# libraries in my system.
I found many CAS systems like Mathematica, Boost, Sage, Intel's MKL library. But i am not able to sort out a comparison from that. And how much data they can handle with a significant performance.
I am expecting answers like "I have solved 1,00,000 equations in 1,00,000 variables with xxx Library with in xxxx time in a machine with x cores and xGB Ram using xThreads"