Lecture 11 of 'Scientific Computing' (wi4201)

• The following subjects are discussed:
  
  
  • CG as a polynomial method
  • CG started as a direction solution method in 1950
  • due to rounding errors CG was not used until 1970, then it is used as an iterative method
  • use eigenvalues and eigenvectors, together with the polynomial approach to analyse the convergence in more detail
  • CG converges in m iterations if there are m different eigenvalues
  • effective condition number
  • Ritz matrix, Ritz values and Ritz vectors
  • properties of Ritz values
  • superlinear convergence
  • if smallest Ritz value is close to the smallest eigenvalue this eigenvalue is no longer determining the convergence rate
  • sketch of the proof for this property
  • preconditioner to decrease the condition number
  • required properties of the preconditioner M
  • choices of a preconditioner
  • derivation of the Preconditioned CG (PCG) method
  • work per iteration and memory
  • properties of PCG
  • take M as a diagonal matrix
    
    
• Material is described in pages 101-108 of the lecture notes. Recommended exercises: 7.1.1 - 7.1.6