Fast solvers and Sylvester equations: both sides now

We're using the structure of a problem with n/sup 2/ unknowns to reduce the amount of computation from O(n/sup 6/) (using the Cholesky decomposition) to O(n/sup 4/) (exploiting sparsity) and then to O(n/sup 3/) (using the Sylvester structure), a substantial savings when n is large. But knowing just a bit more about the problem's structure allows further reduction, down to O(n/sup 2/ log/sub 2/n) when n is a power of two. Because we're computing n/sup 2/ answers, this is close to optimal, and it illustrates the value of exploiting every possible bit of structure in our problems.