SparseCholeskySolver

This component belongs to the category of LinearSolver. The role of the SparseLUSolver is to solve the linear system $\mathbf{A}x=b$ assuming that the matrix $\mathbf{A}$ is symmetric and sparse.

The Cholesky decomposition (https://en.wikipedia.org/wiki/Cholesky_decomposition) is a numerical method that solves a linear system $mathbfAx=b$ by factorizing the matrix of the system as ${\mathbf{LL}}^{\mathbf{T}}$. 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 ${\mathbf{LDL}}^{\mathbf{T}}$ decomposition is heavily related to the Cholesky decomposition.

$$\{\begin{array}{l}\mathbf{A}x=b\\ \mathbf{A}={\mathbf{LL}}^{\mathbf{T}}\end{array}⟺\{\begin{array}{l}\mathbf{L}y=b\\ {\mathbf{L}}^{T}x=y\end{array}$$

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.