Is there a Method out there to solve a nxn linear EQ?

I am considering using Java on a project I'm doing in school. I need to solve a very large linear equation, or I think it is, 121x121, many many times.
Is there an Method that I can input a matrix A and a vector b and get x:
Ax=b ?
If there is, which I'm sure there is but I can't find it, how will its performance match up with a C++ version (such as lssolve)?
THANKS!!!!
Josh

Have a google search for 'LUP' (or Lower Upper Permutation). If you have to solve 'Ax=b' for many different vectors 'b', a LUP decomposition of your matrix A comes in handy. The general idea is, that any (non singular) matrix A can be written as P*A = L*U, where P is a simple row-permutation matrix and L and U are lower and upper triangular matrixes respectively. Given a problem 'Ax=b', this problem can be rewritten as --
Ax= b -->
PAx= Pb -->
LUx= Pb -->
Ly= Pb -->
Ux= y
Where vector 'y' can be found by simple forward substitution and finally x can be found by a simple backward substitution. This method has the additional advantage of being quite numerically stable (thanks to the partial row permutation matrix P).
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