On Monday, April 25, 2016 at 4:49:40 PM UTC-4, Jonas Kersulis wrote: > > Can someone explain why qrfact is faster and requires less memory than > lufact for square, sparse matrices? LU is less computationally complex than > QR, and with decent pivoting LU should be able to preserve sparsity better > than QR, so I'm confused by what I'm seeing in practice. >
50% nonzeros is not sparse enough to exploit sparsity effectively, and even 10% nonzeros is probably too much. None of the sparse algorithms are really going to work particularly well in such cases; you are better off with dense matrices. I'm not sure comparing the sparse LU and QR algorithms in such cases is particularly meaningful. A more realistic application of sparse matrices would be something like discretizing a PDE on a 2d mesh. A 100x100 grid with nearest-neighbor interactions is a 10000x10000 matrix, with about 5*10000 nonzero entries, corresponding to 5/10000 = 0.05% sparsity. (There is a famous Wilkinson quote: "A sparse matrix is any matrix with enough zeros that is worthwhile to take advantage of them." You don't have enough zeros to be worth exploiting.)
