# [SciPy-User] solving large sparse linear system with Laplacian matrix

Emmanuelle Gouillart emmanuelle.gouillart@normalesup....
Fri Oct 30 02:46:48 CDT 2009

```Dear list,

I'm looking for a memory-savvy algorithm in scipy.sparse.linalg, that I
could use to solve a very large sparse linear system. The system can be
written as AX = B, where A is a symmetric band matrix with non-zero
elements on only 4 or 6 upper (and lower) diagonals. The shape of A is
NxN where N is very large (up to 10e6, if possible...), the shape of B
and X is NxM, where M is much smaller (up to 50, say). Unfortunately, A
is symmetric but not positive-definite as the sum of each row and column
is null (A is a Laplacian matrix). B is also sparse (it is actually a
block of the original Laplacian matrix).

What kind of solver implemented in scipy would you recommend, so that I
can solve such a system with N as large as possible (on a bottom-end
computer with only 2Gb RAM)? Up to now I have used
scipy.sparse.linalg.spsolve, and the CSR scipy.sparse format for A and B:
this works fine for N as large as 300, but the memory requirements are
too high for greater values of N on my computer. Should I use an
iterative solver (I'm a complete newbie in linear algebra)? Also, spsolve
requires that the right hand side is a flat vector, so I have to solve as
many systems as the number of columns in B, which must be highly
inefficient... Any way I can solve the "whole" linear system? Any hints
will be much appreciated!

random walker algorithm [1] to segment large 3-D X-ray tomography images.
N=l**3 is the number of voxels in the cubic image, and M is the number of
regions which I would like to segment. I don't require a very good
precision on the elements of the solution X.