This thesis addresses two key problems related to stability and security studies of nuclear reactor cores, where it is necessary to solve very-large sparse linear equations systems efficiently. The first one is related to Lambda modes equation problem of the neutron diffusion equation applied to a realistic test case (Ringhals--I nuclear reactor, which is a boiling water reactor or BWR type), and it constitutes a generalized eigenvalue problem. The second one is related to the solution of very-large sparse linear equation systems that arise from the temporal discretization of the neutron diffusion equation applied to another realistic test case (Leibstadt nuclear reactor, which is a BWR type) and they must be solved in different time steps. 

In order to solve the very--large sparse linear equations systems associated with the Lambda modes problem, in this thesis we have carried out a numerical study of the sequential and parallel performance of direct, iterative and Krylov--based iterative methods. This study has been done using sequential and parallel free distribution libraries. With the numerical results from this study we have identified all those methods and libraries that solve efficiently the sparse linear equation systems for the test case.

On the other hand, to solve the sparse linear equation systems related to second problem, in this thesis we have proposed second--degree iterative methods to speedup their solution. These methods have been implemented using sequential and parallel free distribution libraries. Several experiments carried out with second--degree iterative methods have showed a significant speedup of the solution process and they can be an option to carry out more complex simulations. 

The algorithms which solve the problems presented in this thesis and their parallel implementations have been evaluated with respect to execution time, speedup and efficiency in two distributed memory parallel platform. Numerical experiments have showed that parallel implementations reduce significantly sequential execution times and overall parallel performance has been satisfactory.