Numerical Methods

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Numeriske Metoder / Numerical methods. F09

[ FAQ | F08 home-page ]

NB:

Fristen for projektet i numeriske metoder er Mandag d. 22. Juni kl. 12:00. Projektets mål er at overbevise mig at I kan:
1. redegøre for den teoretiske baggrund for de basale numeriske metoder;
2. analysere simple videnskabelige problemstillinger med hensyn til numerisk løsning;
3. identificere de passende numeriske metode og argumentere for disses anvendelighed i den enkelte problemstilling;
4. anvende de numeriske metoder, der skal benyttes i løsningen af den enkelte problemstilling;
5. argumentere for den numeriske nøjagtighed af resultater.
A template for a project in numerical methods with makefile and latex file and a figure is now available.
To include danish letters, like ø, with UTF8 encoding in a LaTeX file do:
```
\usepackage[utf8]{inputenc}
\usepackage[english,danish]{babel}
```

Schedule:

Lectures: Monday, 08:15, 1520-616; Thursday, 09:15, 1525-~~319~~ 229.

Exercises: Wednesday, 12:15, 1525-319.

Exercises:

Hello World, QR-decomposition, Eigenvalues, Least-Squares fit, Interpolation, RootFinding and Optimization, ODEs, Adaptive integration, Monte Carlo integration, FFT,

Weekly notes:

Introduction: Numerical computations in physics; Unix servers; programming in Unix environment; make utility; programming languages.
[png-1, png-2; pdf] Systems of linear equations: QR-decomposition and back-substitution; determinant of a matrix; matrix inverse.
[png-1, png-2; pdf] Eigenvalues and eigenvectors: Jacobi transformation; QR algorithm; Power method; Inverse iteration.
[png-1, png-2; pdf] Linear least-squares problem; least-squares solution with QR-method; linear least-squares fit; errors and covariance matrix.
[png-1, png-2; pdf] Interpolation: polynomial interpolation; spline interpolation.
[png-1, png-2; pdf] Nonlinear equations and optimization: modified Newton's method; Broyden's method; downhill simplex method.
[png-1, png-2; pdf] Ordinary differential equations: one-step, multi-step, predictor-corrector methods; estimation of errors, adaptive step-size strategy.
[png-1, png-2; pdf] Numerical integration: quadratures with optimized abscissas (Gauss); quadratures with equally spaced abscissas (Newton-Cotes); adaptive algorithm with equally spaced abscissas; Gauss-Kronrod quadratures.
[png-1, png-2; pdf] Monte Carlo integration: plain Monte Carlo; importance sampling; stratified sampling; quasi-random sequences.
[png-1, png-2; pdf] Fast Fourier transform and applications: Danielson-Lanczos lemma and Cooley-Tukey algorithm.
[png-1, png-2; pdf] Krylov subspace methods: Arnoldi and Lanczos methods for finding few eigenvalues of a matrix; GMRES (Generalized Minimum RESidual) method for solving systems of linear equations.
Ideas for projects:
- [ png1, png2, pdf; see also this e-print ] Quantum mechanics: eigen-values and eigen-functions: Feynman path integrals;
- [ png, pdf] Calculations of the complex valued special functions.
- Schrodinger equation: eigen-functions: shooting method;
- Schrodinger equation: eigen-functions: finite-difference method;

Literature:

Numerical Recipes in ... , by W.H.Press et.al., available on-line

Evaluation:

Only project, 7-scale grade. The project should be like a short article with the description of the problem, description of the way you solve it, and discussion of the results.

Useful links:

[ Linuxbog.dk | GNU scientific library | HTMLhelp | Computer Language Benchmarks ]

Computers:

You can use our servers lifa.phys.au.dk which run RedHat. We also have a dedicated server for numerical methods, genryc, which runs Debian. A lot more software is installed on genryc.

Program examples