Lecture 24 - Reserve, Old Exams

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Topics: Revision of old exams, see Final Examination Info; questions

 

Check list lecture 18 - 24:

- Eigenvalue
      - Eigenvalue problem
      - What is an eigenvalue and eigenvector
      - Calculating (with) eigenvalues and eigenvectors
      - Rayleigh quotient
      - Power Method (how/why does it work, convergence, ... )
      - Shifted Inverse Power Method (how/why does it work, convergence, ... )
      - Similar matrices
      - QR method for eigenvalue problem 
      
  - Singular value decomposition (SVD)
      - what is it?
      - how can it be computed?
      - properties of the SVD
      - relation between singular values and original matrix
      - low-rank approximation (Example: compression)
  
  - Numerical Differentiation
      - forward/backward/centered difference formula
      - difference formula for higher derivates
      - Computing the order of a difference formula
      - Deriving difference formula
      - error analysis
      - numerical differentiation in matrix form
      
  - Numerical Integration (quadrature)
      - Newton-Cotes formulas (open and closed, Trapezoid(al) rule, Simpson rule, midpoint rule)
      - properties of Newton-Cotes formulas
      - Exactness of a quadrature
      - Error formula
      - Computation of weights
      - Transformation of nodes and weights (change of interval)
      - Composite quadrature
      - Adaptive quadrature
      - Maximal exactness of quadratures
      - Gauss-Legendre quadrature (properties)

  - Fourier Transformation
      - Discrete Fourier transformation
      - Fast Fourier transformation