Difference between revisions of "Solving sparse systems"
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Revision as of 18:57, 15 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 $A x = b$ 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 | 0.78 |
| Bicgstab / Krylov | 19.32 | 0.39 |
