# Sparse versus dense Cholesky factorization benchmark

## Alec Jacobson

## September 18, 2013

Here's a very informal benchmark comparing the **sparse** to the **dense** Cholesky factorizations as implemented by MATLAB. The problem I'm considering is a Laplace/Poisson equation on an n by n regular grid (x-axis is N = n^{2}). Here's the code I used:

```
td = [];
ts = [];
N=[];
for n = 1:30
[V,F] = create_regular_grid(n,n,0,0);
L = cotmatrix(V,F);
FL = full(L);
s = @() chol(-L(2:end,2:end));
d = @() chol(-FL(2:end,2:end));
ts = [ts(:);timeit(s,3)];
td = [td(:);timeit(d)];
N=[N(:);size(V,1)];
loglog(N,ts,N,td,'LineWidth',4);
drawnow;
end
```

And here is the **log-log** plot.

On my machine, an iMac 3.4 Intel Core i7 with 16GB ram, I see the cross-over point at about N = 200. Shortly after that **sparse** wins handily.