Improving the performance of solving linear equations

[tvn]

Hi, I am wondering if there's some way to speed up the process of solving linear equations for a lot unknowns (200+). For Current I just apply solve() directly over 200+ linear equations and let it solve for the unknowns, e.g., solve([2c_0 + 2c_1 + 3= 0 ,  4.2c_0 + 5c_1 + 1 = 0  ... ] , solution_dict=True).  This takes lots of time when I have hundreds of equations with hundreds of unknowns c_i.    

I do know that most (over 70%) of the unknown coefficients will be 0 and most of the non-zero coefficients will be related.  That is,  among the non-zero coefficients, some of them will get independent values r_i and other non-zero coefficients will depend on these. For example, c_10 = r1 , c_9 = 2*r1+3 , c_8=r2 , c_1 = 1/2*r2 .  Given these information, is there something that I can do to speed up the process ?   Could I use some special data structure, sparse matrix or something ?  

[Volker Braun]

Don't use the general-purpose solve() for linear equations. Write your system of linear equations in matrix form and then use the solve_left() / solve_right() method of matrices. Solving something in the order of 200 equations should be almost instantaneous.

[tvn]

I can convert my data to a matrix M but M.solve_left() will give 0 to *all* coefficients in my matrix. This is not the results given by solve(), which is something like c_10 = r1 , c_9 = 2*r1+3 , c_8=r2 , c_1 = 1/2*r2.  I have tried using the reduced row echelon form and M.rref() seems to give what I want (equiv to the answer from sol(). The problem is that this method rref() also takes long time, seem to be even longer than calling solve()  itself.

[Volker Braun]

As usual, the most general solution is a particular solution plus something from the kernel.

Whats the base ring of the matrix that you are constrcuting?