Backslash for solving very big sparse matrix in Ax=b

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Alireza Moghaddam on 16 Jul 2015
Answered: Royi Avital on 9 Aug 2015
In my code at each time step sparse matrix A is built as follows:
1< DX <120000
A = sparse(i,j,s,DX,DX,6*DX)
b = (1,DX)
The problem that I am dealing is a sort of discretization problem. I have maximum 120000 nodes. Each of these nodes are having special charachters and I choose only the ones that meet my criteria. the number of these chosen ones is DX and is completely dependent on the physical process.
I am using backslash in x = A\b. But as the size of A could become quite big, the computational time rises drastically (more than 10e5 time steps are having DX > 6e4). As far as I know, backslash operation is already well optimized in MATLAB but I would like to know:
1. Would it make sense to use codegen and convert the code to C?
2. Does any one know an alternative method instead of backslash, so that the compuattional time decreases (maybe an iterative method?)?

Sean de Wolski on 16 Jul 2015

John D'Errico on 16 Jul 2015
NO. The solver in backslash is ALREADY compiled and highly optimized. Why do you think that converting the call to C would help? NOT.
You can try an iterative method. They are designed for problems that are too large to solve in memory. Don't be surprised if you need to learn about preconditioning matrices.
Alireza Moghaddam on 17 Jul 2015
Thanks! I will try the iterative methods.

Royi Avital on 9 Aug 2015
If you're memory limited the best solution I know of is Conjugate Gradient.
MATLAB has Preconditioned Conjugate Gradient (The function ` pcg`).
You can use it without a preconditioner to have classic Conjugate Gradient.
If you want it to be faster, a suitable preconditioner to your problem can do wonders.
The beauty of this method is you never multiply the whole matrix by the vector. You only use do products as needed.
Its speeds depends on the condition number of the Matrix, hence with a good preconditioner things gets much better.