Computing the so-called quasispecies in Eigen's quasispecies model is tantamount to the computation of the dominating eigenvector of a highly structured large scale matrix. In particular, all of the involved matrices are data-sparse matrices whose dimension grows exponentially with the practically important model parameter. As an immediate consequence, standard solvers suffer from the curse of dimensionality and are incapable of reaching practically relevant dimensions. In order to overcome this issue, it was shown that the required matrices allow for a Kronecker product representation which directly implies that highly efficient implicit matrix vector routines can be provided. Based on such efficient matrix vector products, the power iteration serves as simple tool to compute the dominating eigenvector also for problems at a relevant scale. The severe drawback of the power iteration is its inferior convergence speed in case of a bad separation of the dominating eigenvalue. In this thesis we show how to overcome this issue by employing more subtle shift-and-invert methods in order to provide efficient solvers also in case of a bad separation of the dominating eigenvalue---as it has to be expected in practice. Shift-and-invert methods require the solution of a linear system in every iteration. The iterative solution of linear systems is in principle a well-developed subject and a myriad of solvers, e.g., Krylov subspace methods, exist. The actual problem in this context is to provide a good preconditioner such that only a considerably small number of iterations are required. In particular, standard preconditioning approaches fail in the context of the problems considered in this thesis due to the high dimensionality and structure of the respective matrices. Therefore, the high level aim of this thesis is to provide suitable preconditioners for the problem at hand, in order to allow for an efficient computation of the quasispecies also in cases of a bad separation of the dominating eigenvalue. In the course of that, we elaborate on different fields: In terms of existing work we discuss Krylov subspace methods and the theory and application of the Kronecker product each with respect to the considered extreme scale eigenvalue problem. The central novel results presented in this thesis consist of a comprehensive theory of Hamming distance-based matrices, i.e., matrices whose (i,j)-th element solely depends on the Hamming distance between the indices i and j (appropriately interpreted as finite strings). These novel results cover among others algebraic and algorithmic aspects, as well as the structure of this family of matrices in general. Based on the theory of Hamming distance-based matrices, we eventually derive efficient preconditioners for the problem at hand. Moreover, we provide experimental data of the resulting solvers which depict overall performance gains by at least an order of magnitude.