I'm looking for a way to speed up my computation that involves solving a moderate size generalised eigenvalue problem (matrices 400x400) large number of times depending on the parameter $m$. The eigenproblem has a particular structure: $$ (A - \lambda B)x = \left ( \begin{bmatrix} 0 & A_{12} - m A_\mathrm{mod} \\ A_{12} - m A_\mathrm{mod} & A_{22} \end{bmatrix} - \lambda \begin{bmatrix} I & 0 \\ 0 & B_{22} \end{bmatrix} \right ) x = 0 $$ where $A_{12}$, $A_{22}$, $A_\mathrm{mod}$, $B_{22}$ are square full-rank matrices and $m$ is a scalar parameter.
The range of values for $m$ is larger than size of A and B (let's say size of A, B is [r $\times$ r]).
The hypothesis is the following: suppose that I calculate eigenvalues and eigenvectors for $r$ values of parameter $m$ with $r$ being the dimension of the system. Is it a complete description of the system? Does anyone have any intuition on whether this enables me to calculate the eigensolution at any value of $m$ (based on the 'kernel' solution)?
Has anyone read/seen a similar problem and could guide me where to look for a possible solution to the problem? I guess it is closely related to model updating problems...
Many thanks!