C ++中的有效大型稀疏块压缩线性方程

Question

everyone. I have little experience about linear algebra equation solver. But now, I need to solve the large sparse block compressed linear equation. ie, Ax=b .

Matrix A is composed of sub-blocks whose size is (nb,nb) . and size of A is (N,N) subblocks. Then the real size of matrix A is (nb*N,nb*N) .

There are only few sub-blocks exists. And A is diagonal dominant.

I have tried Eigen's built-in slovers. ie, SimplicialLLT and BiCGSTAB, the first is slow while the second cannot converge.

Then I tried third-party solvers of Eigen with Intel MKL, ie, PardisoLLT 、PardisoLDLT and PardisoLU. They were slow too.

Finally, I tried third-party solvers of Eigen with SuiteSparse, ie, CHOLMOD and SPQR. The first is slow while the second gave the same value of input, it seems that it didn't work.

I think maybe it is because that I used the solvers in wrong way that I didn't take advantage of feature "block compress" and thus triggered a lot of useless computing.

Could anyone give me some advices?

Answer 1

As long as your matrix size doesn't exceed about 5000 rows/columns, using a sparse solver won't bring any reasonable benefits; for such matrices an ordinary dense solver would be good as well, because it doesn't contains "overheads" related to storing and operating sparse structures, such as linked lists or adjacency structures. On the other hand, if your matrix is very large (say, a million or so rows/columns), an iterative solver only would work. I suggest that you visit the following webpage, which contains description of a problem and solution very similar to what you are working with (including solution of systems with sparse matrices having block structure, as well as image processing): http://members.ozemail.com.au/~comecau/CMA_LS_Sparse.htm