Numerical Methods

• Background in matrix theory

• Methods for eigenvalue problems

  • Motivation: spectral clustering
  • Power method
  • Perron-Frobenius theorem
  • Subspace iteration method and QR method
  • Krylov method for Ritz values

• Methods for sparse linear systems

  • General projection methods
  • GG and GMRES
  • Preconditioning strategies
  • Multigrid methods

• Linear multistep methods for ODEs

  • Explicit and implicit Adams methods
  • Local error and zero-stability
  • Convergence
  • Stability, integration method for optimization, and variable step size LMM

Homeworks and labs:

Books of reference:

• E Hairer, G Wanner, S P Nørsett – Solving Ordinary Differential Equations I
• Y Saad – Iterative methods for Sparse Linear Systems – Free Online Version
• Y Saad – Numerical methods for large eigenvalue problems – Free Online Version