The eigenvalue problem arises in many scientific areas, e.g., when 
solving partial differential equations, in model reduction or computing matrix 
functions. If the problems are of moderate size (less than 10^6) they can be 
addressed with direct methods, such as the QR algorithm or the 
divide-and-conquer method. However, if the problem is large-scale and only few 
solutions are required compared to the size of the problem, iterative methods 
can be more efficient. Furthermore, iterative methods can provide better 
speedups in high-performance architectures such as distributed memory parallel 
platforms, where there are several computational nodes with private memory 
address spaces and they can only share information and synchronize by message 
passing.

	This thesis addresses the implementation of Davidson methods, 
specifically the Generalized Davidson and Jacobi-Davidson methods, a class of 
iterative methods that can be competitive in particularly difficult cases such 
as calculating eigenvalues inside the spectrum or when the matrix factorization 
is prohibitive or inefficient, and only incomplete factorizations are 
available. The implementation is developed in SLEPc (Scalable Library for 
Eigenvalue Problem Computations), a free software library for solving 
large-scale eigenvalues problems, quadratic eigenvalue problems and singular 
value problems, among others.

	As a result of this thesis, SLEPc includes an implementation of 
Generalized Davidson and Jacobi-Davidson for standard and generalized problems, 
both Hermitian and non-Hermitian, the latter being a remarkable feature 
considering that non-Hermitian problems are not supported in other free 
parallel libraries with Davidson methods. In addition to incorporating 
techniques to improve the convergence of the methods, such as the harmonic 
Rayleigh-Ritz subspace extraction, other optimizations were realized to improve 
the performance, such as avoiding the complex arithmetic when the problem 
matrices are real (because the eigenpairs may have imaginary part) or 
rearranging operations in blocks to take advantage of the cache memory.

	Moreover, we present a new method to expand the subspace, called GD2, 
which provides better results compared with Generalized Davidson when the 
preconditioner is very far from ideal.

	The numerical stability and the computational performance of the 
implementation are evaluated with a battery of problems from real applications, 
and compared with other libraries like PRIMME or Anasazi.

	Finally, we present the integration of the implementation in two 
scientific applications. On the one hand, Jacobi-Davidson improves the 
performance of the calculations of eigenvalues with significant impact in the 
multiparametric studies that appear in GENE, a code that solves the gyrokinetic 
equations for calculating micro-instabilities in fusion plasma. On the other 
hand, Generalized Davidson obtains good results addressing large-scale problems 
that come from the discretization of the Schrödinger equation, in the context 
of computing the electronic configuration of atoms.