This component belongs to the category of LinearSolver. The role of the SparseLUSolver is to solve the linear system assuming that the matrix is symmetric and sparse.

The Cholesky decomposition (https://en.wikipedia.org/wiki/Cholesky_decomposition) is a numerical method that solves a linear system by factorizing the matrix of the system as . By doing so, we only need to solve two triangular systems to compute the solution. It is only applyable on **symetric** matrices but is roughtly twice as efficient as the LU solver. The decomposition is heavily related to the Cholesky decomposition.

## Sequence diagram

The SparseCholeskySolver **requires** the use (above in the scene graph) of an integration scheme, and (below in the scene graph) of a MechanicalObject storing the state information that the SparseCholeskySolver will access.

## Data

There is one data that change the behaviour of the solver, **permutation**, that allows three choices : **-None**, no permutation, nor on the rows nor on the columns, is applied **-SuiteSparse**, use the SuiteSparse library as intended for a symmetric matrix and apply a fill reducing permutation on both the columns and the rows (those two permutations are the inverse of each other), **-METIS**, use the METIS library to compute a fill reducing permutation and apply it on both the lines and the columns.

It is not currently possible to change the type of permutation applied during a simulation.

Applying a fill reducing permutation aims at minimizing the number of non-null values in the decomposition, which would reduce the time spent on solving the triangular systems.

As the impact of the use of fill reducing permutations on the performances is highly influenced by the repartition of the nodes used to model an object, we advise the users to test which type of permutation is the best suited for their simulations.

## Example

This component is used as follows in XML format:

or using SofaPython3:

An example scene involving a SparseCholzskySolver is available in *examples/Components/linearsolver/FEMBAR-SparseCholeskySolver.scn*

Last modified: 23 May 2022