Question
Is there a measure of the "indefiniteness" of a matrix? And what constitutes a "highly indefinite" matrix?
Explanation
I have to solve a linear system computationally, obtained from a multiphysics problem, using the Finite Element Method. But in order to correctly setup the solver I want to use (Pardiso), I have to know whether or not the matrix is "highly indefinite" or just indefinite.
The linear system
The equation can be written as $$\mathbf{K}\mathbf{u}=\mathbf{b}$$
where the matrix $\mathbf{K}$ is square and non-singular and can be extended to $$\mathbf{K}=\begin{bmatrix}\mathbf{A}&\mathbf{C}\\\mathbf{C}^t&\mathbf{B}\end{bmatrix}$$
where $\mathbf{A}$ is square and positive-definite, $\mathbf{B}$ is diagonal and negative-definite, and $\mathbf{C}$ is rectangular.
Without preconditioning, the values in $\mathbf{A}$ have an order of about $10^{10}$, the values in $\mathbf{B}$ have an order of about $10^{-9}$ and the values in $\mathbf{C}$ have an order of about $10^1$.
BUT I can perform left/right diagonal preconditioning to significantly improve the condition number of $\mathbf{K}$ (I imagine that the measure of indefiniteness takes the matrix conditioning into account?).