Solving sparse systems

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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:

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 $A = \begin{bmatrix} 1 & -2 & \\ 1 & \ddots & \ddots \\ & \ddots & \end{bmatrix}$ 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 2.86 /
/ 1.73 0.78
Bicgstab / Krylov 19.32 3.92