Lecture 10 of 'Scientific Computing' (wi4201)

• The following subjects are discussed:
  
  
  • Definition of Krylov subspace
  • Short intro of the Chebyshev method
  • Conjugate Gradient algorithm
  • first step, use of A-norm, condition A is SPD is crucial
  • optimality property, general form
  • discussion of the algorithm, flops per iteration, memory
  • for SPD matrices, CG is the best method
  • properties of the CG iterates and residual vectors
  • proof by induction
  • after a finite number of iterations CG has converged
  • monotone convergence of the error in the A-norm
  • "lucky" breakdown
  • CG can also be used for SSPD (symmetric, semi-positive definite)
  • example: pressure matrix in Navier Stokes
  • CG converges for every SPD matrix A
  • speed of convergence depens on the 2-norm condition number of A
    
    
• Material is described in pages 96-101 of the lecture notes.