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Krylov subspace iterative methods provide an effective tool for reducing the solution of large linear systems to a size for which a direct solver may be applied. However, the problems of limited storage and speed are still a concern. Therefore, in this dissertation work, we present iterative Krylov subspace algorithms for non-Hermitian systems which do have fixed memory requirements and have favorable convergence characteristics. This dissertation describes three projects. The first project concerns short-term recurrence Krylov subspace methods for nearly-Hermitian linear systems. In 2008, Beckermann and Reichel introduced a short-term recurrence progressive GMRES algorithm for nearly-Hermitian linear systems. However, we have found this method to be unstable. We document the instabilities and introduce a different fixed-memory algorithm to treat nearly-Hermitian problems. We present numerical experiments demonstrating that the performance of this algorithm is competitive. The other two projects involve extending a strategy called Krylov subspace recycling, introduced by Parks and colleagues in 2005. This method requires more overhead than other subspace augmentation methods but offers the ability to recycle subspace information between cycles for a single linear system and recycle information between related linear systems. In the first project, we extend subspace recycling to the block Krylov subspace setting. A block Krylov subspace is a generalization of Krylov subspace where a single starting vector is replaced with a block of linearly independent starting vectors. We then apply our method to a sequence of matrices arising in a Newton iteration applied to fluid density functional theory and present some numerical experiments. In the second project, we extend the methods of subspace recycling to a family of linear systems differing only by multiples of the identity. These problems arise in the theory of quantum chromodynamics, a theory of the behavior of subatomic particles. We wish to build on the class of Krylov methods which allow the simultaneous solution of all shifted linear systems while generating only one subspace. However, the mechanics of subspace recycling complicates this situation and interferes with our ability to simultaneously solve all systems using these techniques. Therefore, we introduce an algorithm which avoids this complication and present some numerical experiments demonstrating its effectiveness. / Mathematics

Ultra-relativisitic heavy ion collisions produce quark gluon plasma-a hot and dense soup of deconfined quarks and gluons akin to the early universe. We study two models in the context of these collisions namely, Polyakov Quark Meson Model (PQM) and Hadron Resonance Gas Model (HRGM).The PQM Model provides us with a simple and intuitive understanding of the QCD equation of state and thermodynamics at non zero temperature and baryon density while the HRGM is the principle model to analyse the hadron yields measured in these experiments across the entire range of beam energies. We study the effect of including the commonly neglected fermionic vacuum fluctuations to the (2+1) flavor PQM model. The conventional PQM model suffers from a rapid phase transition contrary to what is found through lattice simulations. Addition of the vacuum term tames the rapid transition and significantly improves the model’s agreement to lattice data. We further investigate the role of the vacuum term on the phase diagram. The smoothening effect of the vacuum term persists even at non zero . Depending on the value of the mass of the sigma meson, including the vacuum term results in either pushing the critical end point into higher values of the chemical potential or excluding the possibility of a critical end point altogether. We compute the fluctuations(correlations) of conserved charges up to sixth(fourth) order. Comparison is made with lattice data wherever available and overall good qualitative agreement is found, more so for the case of the normalised susceptibilities. The model predictions for the ratio of susceptibilities approach to that of an ideal gas of hadrons as in HRGM at low temperatures while at high temperature the values are close to that of an ideal gas of massless quarks. We examine the stability of HRGMs by extending them to take care of undiscovered resonances through the Hagedorn formula. We find that the influence of unknown resonances on thermodynamics is large but bounded. We model the decays of resonances and investigate the ratios of particle yields in heavy-ion collisions. We find that extending these models do not have much effect on hydrodynamics but the hadron yield ratios show better agreement with experiment. In principle HRGMs are internally consistent up to a temperature higher than the cross over temperature in QCD; but by examining quark number susceptibilities we find that their region of applicability seems to end even below the QCD cross over.