I want to solve the usual $A x = b$ system. In block form:
$$ \begin{bmatrix} B & c \\ c^{T} & 0 \end{bmatrix} \begin{bmatrix} x' \\ x_{n+1} \end{bmatrix} = \begin{bmatrix} b' \\ b_{n+1} \end{bmatrix}$$
where
- $B \in \mathbb{R}^{n \times n} $ is a positive definite matrix
- $c \in \mathbb{R}^{n}$
- $x,b \in \mathbb{R}^{n+1}$, so $x',b' \in \mathbb{R}^{n}$ and $x_{n+1},b_{n+1} \in \mathbb{R}$
Matrix $A$ is neither positive definite nor positive semidefinite.
I am not aware of well-known methods such as Cholesky or $LDL^T$ to solve this. Is there an efficient method to tackle this problem?