Sparse matrix vector product

Hi,

I have to evaluate the following expression thousands of times (maybe up on the millions):

M=v1'*A*v2 + v2'*B*v2 + v2'*A*v1

where v1 and v2 are column vectors (typically six columns) with a large number of rows filled with zeros and only a few with real values. v1 and v2 have been stored as full vectors. A and B are sparse square symmetric matrices of the same size. Like v1 and v2, A and B also have a few blocks of non-zero values and have a large number of rows and columns filled with zeros. Finally, this expression is inside a parfor loop.

I am looking for suggestions as to how I can speed up this operation?

All help is gratefully appreciated.

Thanks in advance.

José

[John]

Sparse matrix-vector multiplication:

- Since A and B are sparse matrices, you can utilize MATLAB's sparse matrix operations to perform efficient matrix-vector multiplications.

- Replace A*v2 with A*sparse(v2) and B*v2 with B*sparse(v2) to take advantage of sparse matrix-vector multiplication.

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Transpose and reuse intermediate results:

- Instead of computing v1'*A*v2 and v2'*A*v1 separately, you can compute v1'*A*v2 and then reuse the result for v2'*A*v1 by taking its transpose.

- Replace v2'*A*v1 with (v1'*A*v2)'.

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Preallocate memory:

- Preallocate memory for the output variable M before the loop to avoid repeated memory allocations.

- Use M = zeros(1, num_iterations) to preallocate memory for M, where num_iterations is the number of iterations in the parfor loop.

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Vectorize the loop:

- If possible, try to vectorize the parfor loop to reduce the overhead of parallel computation.

- Consider restructuring the loop to operate on multiple iterations simultaneously, if feasible.

[David Goodmanson]

Hello Jose, do any of v1, v2, A or B stay the same as the expresson is evauated maybe millions of times?