The shift-and-invert method is very efficient in eigenvalue computations, in particular when interior eigenvalues are sought. This method involves solving linear systems of the form $(A-\sigma I)z=b$. The shift $\sigma$ is variable, hence when a direct method is used to solve the linear system, the LU factorization of $(A-\sigma I)$ needs to be computed for every shift change. We present two strategies that reduce the number of floating point operations performed in the LU factorization when the shift changes. Both methods perform first a preprocessing step that aims at eliminating parts of the matrix that are not affected by the diagonal change. This leads to $43\%$ and $50\%$ flops savings respectively.