[Home] Numeriske Metoder. Pensum.

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1. Integration of functions.
   • Classical formulas with equally spaced abscissas. Closed and open formulas.
   • Adaptive algorithms with error estimates.
   • Gaussian quadratures and orthogonal polynomials.
   • Gauss-Kronrod formulas.
   • Two-dimensional integrals.
   • Obligatory exercise: Adaptive integration
2. Linear algebraic equations.
   • Gaussian elimination and backsubstitution.
   • LU decomposition and backsubstitution.
   • QR decomposition (modified Gram-Schmidt method) and backsubstitution.
   • Obligatory exercise: QR decomposition and backsubstitution
3. Linear least-squares fit
   • Linear least-squares problem and χ² fit with QR-decomposition.
   • Obligatory exercise: Linear regression
4. Digonalization of matrices (Eigensystems).
   • Jacobi transformation of a real symmetric matrix.
   • Lanczos tridiagonalization of a real symmetric matrix.
   • QR algorithm with explicit shifts.
   • Obligatory exercise: Matrix diagonalization by Lanczos+QR
5. Ordinary differential equations.
   • Runge-Kutta methods with step-size control.
   • Multistep methods. Predictor-Corrector methods.
   • Obligatory exercise: Runge-Kutta and predictor-corrector with adaptive step-size.
6. Nonlinear equations.
   • Modified Newton's method in multidimensions.
   • Obligatory exercise: Modified Newton´s method.
7. Minimization of functions.
   • Simplex method in multidimensions.
   • Obligatory exercise: Simplex method.
8. Monte Carlo integration.
   • Plain Monte Carlo integration.
   • Importance sampling.
   • Stratified sampling.
   • Quasi-random (Low-discrepancy) sampling.
   • Obligatory exercise: Plain Monte Carlo with pseudo- and quasi-random sampling.
9. Interpolation and Extrapolation.
   • Polinomial interpolation. Lagrange interpolating polynomial.
   • Rational function interpolation.
   • Spline interpolation. Search in an ordered table.
   • Derivation and integration of the spline function.
   • Obligatory exercise: Lagrange interpolating polynomial and quadratic spline.
10. Simulated annealing methods.
    • Metropolis algorithm.
    • Simplex method with simulated annealing.

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"Copyleft" © 2004 D.V.Fedorov (fedorov at phys dot au dot dk)