Partitioned Matrix-Vector Multiplication

Question

Given a very sparse nxn matrix A with nnz(A) non-zeros, and a dense nxn matrix B . I would like to compute the matrix product AxB . Since n is very large, if carried out naively, the dense matrix B cannot be put into the memory. I have the following two options, but not sure which one is better. Could you give some suggestions. Thanks.

Option1. I parition the matrix B into n column vectors [b1,b2,...,bn] . Then, I can put matrix A and any single vector bi into the memory, and calculate the A*b1, A*b2, ..., A*bn , respectively.

Option2. I partition the matrices A and B , respectively, into four n/2Xn/2 blocks, and then use the block matrix-matrix multiplications to calculate A*B .

Which of the above choice is better? Can I say that Option 1 has high performance in parallel calculation?

Answer 1

See a discussion of both approaches, though for two dense matrices, in this document from the Scalapack documentation. Scalapack is the one of the reference tools for distributed linear algebra.