I'm currently working with large (15000x15000), sparse matrices that are not triangular nor banded, but are real, symmetric, Hermitian, and have all positive values. Following the flow chart for the mldivide function for sparse matrices, it would appear that the Cholesky solver would be used. I solve Ax=b for multiple b's, so I thought that if I pre-computed the Cholesky factorization once, that then solving the equation would be faster, so I decided to do some tests. These were the times I got for each step, and comparison to the usual mldivide time
>> tic; A_factor = chol(A, 'lower'); toc;
Elapsed time is 51.273020 seconds.
>> tic; x = A_factor\b; toc;
Elapsed time is 0.186439 seconds.
>> tic; x = A\b; toc;
Elapsed time is 3.275133 seconds.
It is clear that pre-computing the factorization helps (0.18s < 3.28s), but my question is why does the initial pre-computation of the Cholesky factorization take so long (~50s) when the mldivide call without the pre-computation takes only 3s and would be performing the same factorization. Wouldn't we expect the Cholesky factorization to take < 3s? Any ideas?