In a straightforward Monte Carlo approach, where a perturbation is evaluated by taking difference of two independent simulations, the relative variance of the difference tends to infinity as the perturbation tends to zero. To address this, the correlated sampling technique [Rie84, Blo83] forces both the perturbed and unperturbed histories to follow the same transition points in phase space. Using the known transition and collision kernels, the actual simulation can be done in either the unperturbed or the perturbed system. A correlated simulation is done by modifying the appropriate weight factors of the other system. This can be understood as if the unperturbed and perturbed particles are migrating in parallel along the same trajectories. This technique forces the responses of the perturbed and unperturbed histories to be strongly correlated. As a result, their difference is expected to have a smaller uncertainty than the corresponding difference in the uncorrelated game. It can be shown that for sufficiently small perturbations, this method leads to finite relative variance of the differential effect.
For the calculation of eigenvalue perturbations, along
with the unperturbed (or perturbed) fission matrix, another fission matrix for
the perturbed (or unperturbed)
system is generated from the correlated histories reacting with the perturbed
(or unperturbed)
system. The difference of the dominant eigenvalues of these two fission
matrices gives the required 
K.
The source of error in this approach is due to the modifications of the appropriate weight factors. These modifications force particles in one system to simulate the physics that is consistent with the other system. Modification of weight factors are needed to account for the distance to collision, scattering event, fission reaction etc.