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.
The plots below compare computation time, memory allocation, and garbage collection time for the two factorization methods. I generated them by factorizing sprand sparse matrices. The top plot shows results for matrices with 10% nonzeros; the bottom plot shows results for matrices with 50% nonzeros. The methods seem to converge in memory performance as density increases, but LU loses to QR in both cases. I have looked through the paper describing the multifrontal QR algorithm Julia calls, but I don't understand it fully. Before I spend more time studying it, I figured I would see if someone here knows the secret sauce that helps it beat LU. <https://lh3.googleusercontent.com/-EIfk6ZvRVY8/Vx6AN44Se4I/AAAAAAACOgI/hIKvJGOuRGc45IGlAJ_zobqoMUquBFeCwCLcB/s1600/2016-04-25-lu-vs-qr-01.png> <https://lh3.googleusercontent.com/-agWjcrtP4iE/Vx6ASu0AnBI/AAAAAAACOgM/sOyXToRy2BEuW1gjaIcHNsbEDLYtp-LtACLcB/s1600/2016-04-25-lu-vs-qr-05.png>
