Solving sparse systems: Difference between revisions

Revision as of 12:50, 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 $A = \begin{bmatrix} \ddots & \ddots & \ddots && \\ & 1 & 1 & 1 & \\ && \ddots & \ddots & \ddots$ and $b = \begin{bmatrix} 1 \\ \vdots \\ 1 \end{bmatrix}$ with dimension $n$ has the following timings:

@@ Line 1: / Line 1: @@
 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 $A = \begin{bmatrix} \ddots & \ddots & \ddots && \\ & 1 & 1 & 1 & \\ && \ddots & \ddots & \ddots$ and $b = \begin{bmatrix} 1 \\ \vdots \\ 1 \end{bmatrix}$ with dimension $n$
+has the following timings: