Integration based solvers for standard and generalized Hermitian eigenvalue problems / von Lukas Krämer. 2014
Inhalt
- Motivation and outline
- Introduction
- Basics: (Computational) Linear Algebra
- Matrices and vectors
- Norms
- Scalar products and orthogonality
- Matrix induced scalar products and norms
- Projectors
- Singular value decomposition
- Computer arithmetic
- Eigenvalues and eigenvectors
- Angles between vectors and subspaces
- Eigenproblems and their numerical solution
- Measures for the quality of an eigensolver
- General theory of contour integration based eigensolvers
- Subspace eigensolvers
- Rayleigh–Ritz-method
- Subspace iteration
- Eigenvalue bounds
- Convergence of Ritz vectors
- Residual based bounds
- Harmonic Rayleigh–Ritz
- A few facts from complex analysis
- Numerical integration
- Eigensolvers based on integration
- Error analysis of integration based eigensolvers
- Introduction
- Error in the integration—Trapezoidal rule
- Error in the integration—Gauß–Legendre
- Choice of integration contour
- Influence of error in linear systems
- Conclusion
- FEAST eigensolver
- Basic algorithm
- Counting eigenvalues and size of search space
- Problems with wrongly chosen mest
- The selection function
- Convergence rate
- Eigenvalues of bu
- Efficient computation of a basis for the search space
- Preprocessing of FEAST
- Alternatives and further discussion
- Numerical experiments
- Numerical integration revisited
- Polynomial approximation
- Introduction
- Chebyshev approximation
- Error estimation
- Error at the boundary of ilambda
- Experiments with Chebyshev-FEAST
- Connection of polynomial degree and convergence rate
- Adaptive choice of polynomial degree
- Generalized problem
- Why Chebyshev? (Other polynomials)
- Transforming the integration region
- Use of integral transformation
- Conformal transformation of integration region
- Numerical experiments and discussion
- Miscellaneous issues
- Linear systems
- Parallelism
- Orthogonality
- Stopping criteria and eigenpair locking
- Integration error/convergence of eigenvalues and subspaces
- Conclusion
- Conclusion and outlook
- Index
- Summary of Notation
- List of Figures
- List of Tables
- List of Algorithms
- Bibliography