问题
I'm trying to figure out how to efficiently solve a sparse triangular system, Au*x = b in scipy sparse.
For example, we can construct a sparse upper triangular matrix, Au, and a right hand side b with:
import scipy.sparse as sp
import scipy.sparse.linalg as sla
import numpy as np
n = 2000
A = sp.rand(n, n, density=0.4) + sp.eye(n)
Au = sp.triu(A).tocsr()
b = np.random.normal(size=(n))
We can get a solution to the problem using spsolve, however it is clear that the triangular structure is not being taken advantage of. This can be demonstrated by timing the solution and comparing it to the solve method in splu. (Here using iPython's %time magic)
%time x1 = sla.spsolve(Au,b)
CPU times: user 3.63 s, sys: 79.1 ms, total: 3.71 s
Wall time: 1.1 s
%time Au_lu = sla.splu(Au)
CPU times: user 3.61 s, sys: 62.2 ms, total: 3.67 s
Wall time: 1.08 s
%time x2 = Au_lu.solve(b)
CPU times: user 25 ms, sys: 332 µs, total: 25.4 ms
Wall time: 7.01 ms
As Au is already upper-triangular the call to splu really shouldn't be doing much of anything, however, as n gets large this call becomes prohibitively expensive (as does the use of spsolve), while the solve time remains small.
Is there any way of using superLU's triangular solver without first calling splu? Or is there a better way of doing this altogether?
回答1:
I'm afraid this isn't terribly instructive, but have you tried changing the column permutation? When I use "NATURAL", I get huge speedups.
%time x1 = sla.spsolve(Au, b, permc_spec="NATURAL")
CPU time: user 46.7 ms, sys: 0 ns, total: 46.7 ms
Wall time: 49 ms
For me, it's not quite as fast as using the solve methods of the splu function output, but it seems to get reasonably close (and avoids calling splu). Perhaps that will suffice? Scipy Docs
来源：https://stackoverflow.com/questions/18602800/solve-sparse-upper-triangular-system