Content

•  
   Acknowledgments
•  
   Abstract
•  
   Foreword
•  
   Contents
•  
   Motivation and outline
•  
  Introduction
•  
  Fundamentals from numerical linear algebra
•  
   Hermitian and Hermitian positive definite matrices
•  
   Orthogonality and B-orthogonality
•  
   Sparse matrices
•  
   Projection
•  
   Matrix decomposition
•  
   Eigenvalues and eigenvectors
•  
   Hermitian eigenproblems
•  
   Interior eigenproblem
•  
   Spectral projection
•  
  Building blocks for iterative eigensolvers
•  
   Power method
•  
   Krylov subspace methods
•  
   Basic subspace iteration
•  
   Rayleigh–Ritz
•  
  Subspace filtration methods
•  
  Subspace filtration
•  
   Partial spectral projector
•  
   Rational filters
•  
   Contour integration
•  
   Numerical integration
•  
   Composite midpoint rule
•  
   Gauss-Legendre quadrature
•  
   Mapping to a complex contour
•  
   Moments and rational filters
•  
  Rational filter-based eigensolvers
•  
   RFEs with moments
•  
   Convergence of an RFE
•  
   History of the RFE
•  
   Polynomial filters
•  
   Chebyshev polynomials
•  
   Additional algorithmic considerations
•  
   Solving linear systems of equations
•  
   Direct solvers
•  
   Iterative solvers
•  
   Orthogonalization
•  
  A software framework for iterative subspace filtration
•  
   Main algorithmic choices
•  
   Parallelism
•  
   Precision
•  
   Linear solvers
•  
   Conclusion
•  
  Mixed precision
•  
   Introduction
•  
   Background
•  
   Varying precision within a projected subspace iteration
•  
   Algorithmic components
•  
   Methodology
•  
   Numerical experiments
•  
   Precision changes over subspace iterations
•  
   Numerical experiments
•  
   Conclusions
•  
  Optimizing rational filters
•  
   Introduction
•  
   Convergence of an RFE
•  
   Numerical example - subspace size and convergence
•  
  Other rational filters
•  
   Zolotarev
•  
   SliSe and WiSe
•  
   Least-squares filters
•  
   Pole placement and iterative solvers
•  
   Iterative solver convergence
•  
  Kaczmarz sweeps and CG acceleration
•  
   Kaczmarz sweeps
•  
   CGMN
•  
   Parallelization and block multicoloring
•  
   Implementation of CGMN
•  
  Predicting cost
•  
   Analyzing the behaviour of CGMN
•  
   Condition number relationship
•  
   Analyzing the behaviour of GMRES
•  
   Predicting RFE iterations
•  
   Choosing weights
•  
  Optimization
•  
   Cost function
•  
   Visualization of cost function
•  
   Optimization scheme
•  
  Numerical Results
•  
   Numerical results for CGMN
•  
   Numerical results for GMRES
•  
   Comparing to other optimized filters
•  
   Conclusions
•  
   RFEs with multiple moments
•  
  Algorithmic overview
•  
   Extraction of eigenvalues and eigenvectors
•  
   Subspace iteration
•  
   Subspace size
•  
   Multiple moments
•  
   Numerical experiments
•  
   Eigenvalue counting
•  
   Extension of the BEAST framework
•  
   Definition of cost metrics
•  
   Quadrature rule
•  
   Numerical experiments
•  
   Considerations for larger problems
•  
   Rank and orthogonalization
•  
   Solution of linear systems
•  
   Numerical experiments
•  
   Parallelism and quadrature nodes
•  
   Choosing the quadrature degree
•  
   Predicting degree for a single iteration
•  
   Numerical experiments
•  
   Conclusion
•  
   Conclusions and outlook
•  
   Summary of test problems
•  
   List of Figures
•  
   List of Tables
•  
   List of Notations
•  
   Bibliography