[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 + QAG (or similar)
   • Examination problem: Adaptive integration of a given function
2. Linear algebraic equations.
   • Gaussian elimination and backsubstitution.
   • LU decomposition and backsubstitution.
   • QR decomposition (modified Gram-Schmidt method) and backsubstitution.
   • Obligatory exercise: QR decomposition of an arbitrary nxm matrix and backsubstitution
   • Examination problem: Solution of a given system of linear equations by QR-decomposition
3. 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: Quadratic or Linear spline
   • Examination problem: Spline (linear,quadratic ot cubic) interpolation of a given data set
4. Linear least-squares fit
   • Linear least-squares problem and χ² fit with QR-decomposition.
   • Obligatory exercise: Linear regression
   • Examination problem: Linear least-squares fit of a given set of basis functions to a given set of data
5. Ordinary differential equations.
   • Runge-Kutta methods with step-size control.
   • Multistep methods. Predictor-Corrector methods.
   • Obligatory exercise: Runge-Kutta second-third or predictor-corrector.
   • Examination problem: Integration of a given system of ordinary differential equations
6. 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
   • Examination problem: Matrix diagonalization
7. Nonlinear equations.
   • Modified Newton's method in multidimensions.
8. Minimization of functions.
   • Simplex method in multidimensions.
   • Simulated annealing methods.
9. Monte Carlo integration.
   • Plain Monte Carlo integration.
   • Importance sampling.
   • Stratified sampling.
10. Dynamic (Green's function) Monte-Carlo methods in quantum physics.

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"Copyleft" © 2001 D.V.Fedorov (fedorov@ifa.au.dk)