Numerical Mathematics

Lecture Topics

• Introduction:
  Computer Arithmetic, Error Propagation, General Aims of Numerical Mathematics,
  More about Computer Arithmetics: Floating Point Operations, Rounding Errors,
  Condition of a Problem, Numerical Stability
  Course material: Foils (PDF) , Printer-friendly version: PDF , Postscript
• Direct Methods for Linear Algebraic Systems:
  Linear Systems, Gaussian Elimination, LU Decomposition, Examples,
  Programming Technique, Pivoting, Complexity, Cholesky Decomposition;
  Vector and Matrix Norms, Condition Number of a Matrix, Basic Error Estimates,
  Orthogonal Matrices, Matrices with a Special Structure, Gram-Schmidt Orthogonalisation
  QR Factorization (Householder, Givens), LU Decomposition for Tridiagonal Matrices
  Course material: Foils (PDF) , Printer-friendly version: PDF , Postscript
• Iterative Methods for Linear Algebraic Systems:
  A First Example: The Dirichlet Problem, Convergence of One-Step Methods,
  Jacobi and Gauss-Seidel Iteration, Error Estimations, Relaxation Methods,
  Iterative Schemes at Work, Acceleration of Convergence, Richardson Iteration,
  Krylov-Space Methods, Richardson Iteration, Pre-Conditioning, Generalising,
  CG Methods (CGNR, CGNE, GMRES)
  Course material: Foils (PDF) , Printer-friendly version: PDF , Postscript
  
• Interpolation:
  Introduction, Examples, Problems, Lagrange and Newton Interpolation
  The Choice of node points, Chebychev Polynomials
  Hermite Interpolation, Spline Interpolation: Piecewise Polynomial Interpolation, Natural and B-Splines,
  Bernstein Polynomials and Bezier Curves,
  Course material: Foils (PDF) , Printer-friendly version: PDF , Postscript
  
• Numerical Integration / Quadratur:
  Examples, Newton-Cotes Formulas,
  Richardson Extrapolation: Romberg Integration
  Remarks: Gaussian Quadrature, Error Analysis, Adaption
  Course material: Foils (PDF) , Printer-friendly version: PDF , Postscript
  
• Nonlinear Equations:
  Root-Finding for Nonlinear Equations, Conditioning, Geometric Approach
  (Bisection, Chord and Secant Method, Newton's Method), Fixpoint Iteration, Convergence,
  Systems of nonlinear equations, Newton's method
  Course material: Foils (PDF) , Printer-friendly version: PDF , Postscript
  
• Approximation:
  Gauss Approximation, Gram-Schmidt Algorithmus, Legendre Polynomials
  

Fröhner's Home-Page

zur Seite Vorlesungsskripten / Übungsblätter