Solving sparse systems

There are many methods available for solving sparse systems. We compare some of them here.

Mathematica has the following methods available (https://reference.wolfram.com/language/ref/LinearSolve.html#DetailsAndOptions)

• direct: banded, cholesky, multifrontal (direct sparse LU)
• iterative: Krylov

Matlab has the following methods:

• direct: https://www.mathworks.com/help/matlab/ref/mldivide.html#bt42omx_head
• iterative: https://www.mathworks.com/help/matlab/math/systems-of-linear-equations.html#brzoiix, including bicgstab, gmres

Eigen has the following methods: (https://eigen.tuxfamily.org/dox-devel/group__TopicSparseSystems.html)

• direct: sparse LU
• iterative: bicgstab, cg

Solving a simple sparse system $A x = b$ with $A = \begin{bmatrix} -2 & 1 & \\ 1 & \ddots & \ddots \\ & \ddots & \end{bmatrix}$ and $b = \begin{bmatrix} 1 \\ \vdots \\ 1 \end{bmatrix}$ with dimension $n$ has the following timings in seconds:

$n = 10^6$	Matlab	Mathematica	Eigen
Banded	0.16	0.28	/
SparseLU	/	1.73	0.69
Bicgstab / Krylov	??	0.39	??

Preconditioners matter a lot.