Cholesky Decomposition

This is not the canonical way to solve a batch of states. It's just a middle man.

This is one way, doing a sparse Cholesky decomposition followed by forward and backward passes.

A Cholesky Decomposition decomposes a matrix into a lower-triangular representation and its transpose

$$\mathbf{A}=\left[\begin{array}{ccc}4& 12& -16\\ 12& 37& -43\\ -16& -43& 98\end{array}\right]=\left[\begin{array}{ccc}2& 0& 0\\ 6& 1& 0\\ -8& 5& 3\end{array}\right]\left[\begin{array}{ccc}2& 6& -8\\ 0& 1& 5\\ 0& 0& 3\end{array}\right]$$

To do a Cholesky Decomposition, its pretty easy, a pattern forms that allows you to compute it in linear time.