Difference between revisions of "Solving sparse systems"
From Medusa: Coordinate Free Mehless Method implementation
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* iterative: bicgstab, cg | * iterative: bicgstab, cg | ||
| − | Solving a simple sparse system $A x = b$ with $A = \begin{bmatrix} 1 & | + | 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: | + | has the following timings in seconds: |
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
| + | ! $n = 10^6$ | ||
! Matlab | ! Matlab | ||
! Mathematica | ! Mathematica | ||
! Eigen | ! Eigen | ||
|- | |- | ||
| − | ! | + | ! Banded |
| − | | | + | | |
| − | | | + | | 2.86 |
| + | | / | ||
|- | |- | ||
| − | ! | + | ! |
| − | | | + | | / |
| − | | | + | | 1.73 |
| + | | 0.78 | ||
| + | |- | ||
| + | ! Bicgstab / Krylov | ||
| + | | 19.32 | ||
| + | | 3.92 | ||
| + | | | ||
|} | |} | ||
Revision as of 19:55, 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 $A = \begin{bmatrix} 1 & -2 & \\ 1 & \ddots & \ddots \\ & \ddots & \end{bmatrix}$ 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 |
