Difference between revisions of "Solving sparse systems"

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

Revision as of 14:00, 16 March 2017

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 $A x = b$ for steady space of heat equation in 1d with $n$ nodes. has the following timings in seconds:

$n = 10^6$ Matlab Mathematica Eigen
Banded 0.16 0.28 0.04
SparseLU / 1.73 0.69
BICGStab / Krylov 0.33 0.39 0.53

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