Numerical Methods for Physicists I
SS 2006
H. Kroha and D. N. Chigrin
Lectures: Wednesday 10-12, HS IAP
Tutorials: Every second Friday 10-12, C-POOL, AVZ
Recommended Literature:
- Engeln-Muellegs G. and Uhlig H., "Numerical Algorithms with C," Springer 1996
There are also older German Editions.
- Stoer J. and Bulirsch, R., "Introduction to Numerical Analysis," Springer 1993
There are also newer German Editions.
- Hoffman J. D., "Numerical Methods For Engineers and Scientists," Marcel Dekker 2001
- Dubin D., "Numerical and Analytical Methods For Scientists and Engineers Using Mathematica," Wiley 2003
- Golub G. H. and M., Ortega J., "Scientific Computing and Differential Equations: An Introduction to Numerical Methods," Academic Press 1992
Topics
Lecture 1 (05.04.2006) (Chigrin): Introduction. Computer Numbers. Error Analysis.
(Script pdf)
Lecture 2 (12.04.2006) (Chigrin): Roots of a Single-Variable Equation.
(Script pdf)
- Iteration procedure
- General convergence theorems
- Newton's method
- Secant methods
Lecture 3 (19.04.2006) (Chigrin): Roots of a Single-Variable Equation. Roots of Polynomials.
(Script pdf)
- Bisection method
- Regula Falsi methods
- Newton's method for multiple roots
- Horner method
- Muller method
Lecture 4 (26.04.2006) (Chigrin): Systems of Linear Equations. Direct Methods. (Script pdf)
- Gauss elimination method
- LU-decomposition method
- Matrix determinant
- Matrix Inversion
Lecture 5 (03.05.2006) (Chigrin): Systems of Linear Equations. Direct Methods. (Script pdf)
- Sparse systems
- Tridiagonal systems
- Cyclic tridiagonal systems
- Cholesky decomposition method
- Iterative improvements
- Errors and conditioning
Lecture 6 (10.05.2006) (Kroha): Systems of Linear Equations. Direct Methods.
- Singular value decomposition method
Lecture 7 (17.05.2006) (Kroha): Eigenvalue Problem.
- Jacobi method
- Housholder reflection method
Lecture 8 (24.05.2006) (Kroha): Eigenvalue Problem.
- Given rotation method
- QR decomposition method
- Flow equation method
Lecture 9 (31.05.2006) (Kroha): Numerical Fourier Transform.
- Discrete Fourier Transform
Lecture 10 (31.05.2006) (Kroha): Numerical Fourier Transform.
- Fast Fourier Transform
Lecture 11 (28.06.2006) (Chigrin): Functions Approximation and Data Fitting. (Script pdf)
- Least squares method
- Legendre approximation
- Fourier series as least square approximation
- Error analysis for least squares method
Lecture 12 (12.07.2006) (Kroha): Functions Interpolation.
- Polynomials interpolation
- Cubic splines
Tutorials
Tutorial 1 (21.04.2006): Roots of a Single-Variable Equation.
The task is to program 3-7 methods and to determined their convergence rate p experimentally.
- Newton's method (for a nonlinear equation of your choice)
- Secant methods (for a nonlinear equation of your choice)
- Bisection method (for a nonlinear equation of your choice)
- Regular Falsi method (for a nonlinear equation of your choice)
- Horner method (to evaluate polynomial at a given point x)
- Muller method (to find all roots of at least 4th order polynomial of your choice)
- Muller method combined with any from 1)-4) (to find all roots of at least 4th order polynomial of your choice with higher accuracy)
Tutorial 2 (05.05.2006): Systems of Linear Equations. Direct Methods.
The task is to program 3-6 methods and find solution of a systems of linear equations.
- Gauss elimination method (for arbitrary system of size large than 20x20)
- LU- decomposition method (for arbitrary system of size large than 20x20)
- Tridiagonal system (for tridiagonal system of size large than 20x20)
- Cyclic tridiagonal (for cyclic tridiagonal system of size large than 20x20)
- Cholesky decomposition method (for symmetric system of size large than 20x20)
- Matrix inversion using LU method (for arbitrary matrix of size large than 20x20)
- Matrix determinant using LU meth
- Discrete Fourier Transform
od(for arbitrary matrix of size large than 20x20)
Tutorial 3 (19.05.2006): Eigenvalue Problem.
- Jacobi method
- Housholder reflection method
Tutorial 4 (16.06.2006): Numerical Fourier Transform.
- Discrete fourier transform
- Fast fourier transform
Tutorial 5 (30.06.2006): Least squares method.
- Using singular value decomposition method
- Using housholder transformation method