Solving sparse systems

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

Figure 1: Matrix of the discretized PDE.

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 $A x = b$ for steady space of heat equation in 1d with $n$ nodes, results in a matrix shown in Figure Figure 1.

The following timings of solvers are given in seconds:

$n = 10^6$	Matlab	Mathematica	Eigen
Banded	0.16	0.28	0.04
SparseLU	/	1.73	0.82
BICGStab / Krylov	0.33	0.39	0.53

Incomplete LU preconditioner was used for BICGStab. Without the preconditioner BICGStab does not converge.