Spectral problems are present in many mathematical problems in engineering, economy and other areas. These problems arise either as intermediate computations in the solution of many problems, for instance in image processing, or autonomously for studying dynamic properties of mathematical models, for instance in stability analysis. In the last years, high performance computing has taken an outstanding role in the resolution of realistic problems modelled mathematically, both because of the increasing complexity of these models and because of the large dimension of the associated algebraic problems. In order to be able to make use of modern computer architectures in an efficient way, it turns out to be necessary to adapt existing numerical methods or to create new algorithms that suit these architectures. The development of efficient computational kernels for the computation of spectral elements in high performance computing environments such as parallel processing is thus of great interest, currently being a very active research area . Among all the numerical methods for the computation of eigenvalues and eigenvectors, those that are recently receiving more interest belong to the Davidson type, in particular the Jacobi-Davidson method . Members of the Portuguese team of the current proposal have developed, in collaboration with M. Ahues and A. Largillier, a method called PFSI (Perturbed Fixed Slope Iteration). In [3,4] the Portuguese team found and proved necessary and sufficient conditions for the convergence of such a class of methods. The aim is now to carry out a mathematical study in order to establish possible relations with the aforementioned class of methods. Of particular interest is the implementation of these methods in parallel computing environments, sharing the know-how of the Spanish team and making use of the subroutine library developed by them for the solution of eigenvalue problems, SLEPc (Scalable Library for Eigenvalue Problem Computations) [5,6]. After about 5 years of development and 6 public releases, SLEPc has become a well-know tool with increasing acceptance in the computational science community, especially for very large-scale problems. It is a general-purpose library covering many types of eigenvalue problems, and provides a wide range of solvers (see www.grycap.upv.es/slepc for details). The new developments produced in the context of the current proposal will be included in future releases of SLEPc, thus guaranteeing wide dissemination of the results. For the efficient resolution of many realistic problems it is fundamental to take profit of the algebraic characteristics such as the sparsity of matrices and in some cases it is possible for approach the solution of the algebraic problem to make use of information from the mathematical model. The Portuguese team has been developing the method of “defect correction” for spectral computations in singular integral operators, in particular in a problem of computational physics [7,8,9] modelled by an integral equation. The Spanish team has also dedicated a lot of attention to the application of computational kernels for eigenvalues and eigenvectors to the resolution of realistic problems, particularly in the field of computational electromagnetics , and has interest in the study and application of methods of type AMLS (Automated Multi-Level Substructuring) . The study and implementation of such numerical methods in parallel in the context of SLEPc is one of the objectives of this cooperation. From this interaction between the two teams in the development, implementation and application of these techniques, it is expected to obtain as outcome new advancements in the Computational Mathematics scientific area. The Portuguese team, more acquaint with mathematical issues, and the Spanish team, more up to date with computational aspects, are highly complementary. This fact will certainly promote the increase of the scientific results already obtained by each one of the teams. References:  Z. Bai et al. (eds), “Templates for the Solution of Algebraic Eigenvalue Problems”, SIAM, Philadelphia (2000).  G. Sleijpen, Van der Vorst, “A Jacobi-Davidson Iteration Method for Linear Eigenvalue Problems”, SIAM J. Matrix Anal. Appl., vol. 17, no. 2, pp. 401-425 (1996).  M. Ahues, A. Largillier, F. D. d’Almeida, P. B. Vasconcelos, “Spectral Refinement on Quasi-diagonal Matrices”, Linear Algebra and its Applications, 401 pp. 109-117 (2005).  M. Ahues, A. Largillier, F. D. d’Almeida, P. B. Vasconcelos, “Spectral refinement for clustered eigenvalues of quasi-diagonal matrices”, Linear Algebra and its Applications, , 413 pp. 394-402 (2006).  V. Hernández, J. E. Román, V. Vidal, “SLEPc: Scalable Library for Eigenvalue Problem Computations”, Lecture Notes in Computer Science, vol. 2565, pp. 377-391 (2003).  V. Hernandez, J. E. Roman, and V. Vidal, "SLEPc: A Scalable and Flexible Toolkit for the Solution of Eigenvalue Problems", ACM Transactions on Mathematical Software, 31(3), pp. 351-362 (2005).  M. Ahues, F. D. d'Almeida, A. Largillier, O. Titaud, P. Vasconcelos, “An L1 Refined Projection Approximate Solution of the Radiation Transfer Equation in Stellar Atmospheres”, Journal of Computational and Applied Mathematics, 140, 1-2, pp. 13-26 (2002).  M. Ahues, A. Largillier, B. Limaye, “Spectral Computations for Bounded Operators”, Applied Mathematics series, Chapman & Hall (2001).  B. Rutily, “Multiple Scattering Theoretical and Integral Equations”, Integral Methods in Science and Engineering: Analytic and Numerical Techniques, Birkhäuser, 40, pp. 211-232 (2004).  V. Hernández, J. E. Román, “High-Quality Computational Tools for Linear-Algebra Problems in FEM Electromagnetic Simulation”, IEEE Antennas and Propagation Magazine, 46(6), pp. 110-119 (2004).  J. K. Bennighof, R. B. Lehoucq, “An Automated Multilevel Substructuring method for Eigenspace Computation in Linear Elastodynamics”, SIAM J. Sci. Comput., vol. 25, no. 6, 2084-2106 (2004).