Solving sparse systems: Difference between revisions
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* iterative: bicgstab, cg | * iterative: bicgstab, cg | ||
Solving a simple sparse system $A x = b$ with | 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: | has the following timings in seconds: | ||
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| 0.16 | | 0.16 | ||
| 0.28 | | 0.28 | ||
| | | 0.04 | ||
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! SparseLU | ! SparseLU | ||
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| 0.69 | | 0.69 | ||
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! | ! BICGStab / Krylov | ||
| ?? | | ?? | ||
| 0.39 | | 0.39 | ||
| | | 0.53 | ||
|} | |} | ||
Incomplete LU preconditioner was used for Eigen BICGStab. | |||
Revision as of 11:25, 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:
- 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. 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 | ?? | 0.39 | 0.53 |
Incomplete LU preconditioner was used for Eigen BICGStab.