Learning outcomes
The course aims at comforting the application of numerical techniques to problems that are typical of both experimental and theoretical physics. The different methods presented for these problems are demonstrated in class and used in computer-lab sessions.
Content
The course starts with an upgrade enabling a more advanced use of Fortran 90 (use of command lines, compilation with a Makefile, use of scripts, advanced notions of Fortran 90). We revise briefly the limits of a numerical representation of data. We then address the following topics: (1) Resolution of systems of linear equations (2) Numerical interpolations (3) Numerical derivatives (4) Numerical quadratures (5) Linear adjustments (6) Optimization (7) Integration of differential equations. The libraries enabling the resolution of these problems in Fortran 90 and Octave/Matlab are presented.
WARNING : it is absolutely necessary to have taken a class of Fortran 90 in order to consider this course.
Table of contents
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Working with gfortran
Fortran 90 : advanced notions
Representation of numerical data
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Resolution of systems of linear equations
Appendix : the LAPACK library
Appendix : Octave (MATLAB)
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Interpolation methods
Linear interpolation, parabolic interpolation, polynom of Lagrange
Interpolation by spline functions
Interpolation in multiple dimensions
Appendix : the Numerical Recipes
Appendix : interpolation with Octave
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Numerical Derivatives
Asymmetric and symmetric formulas for a first derivative
Romberg's algorithm
Symmetric formula for a second derivative
N-points formulas for arbitrary-order derivatives
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Numerical Quadratures
Trapezoidal rule, Simpson's rule
Generalized Simpson formulas
Adaptative Quadratures (quanc8)
Gauss' method
Appendix : integration with Octave
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Linear adjustements
Least squares method
Generalized linear adjustments by a SVD decomposition
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Optimization methods
Minimization in one dimension (golden search)
Minimization in several dimensions (gradient descend, conjugate gradient)
Monte Carlo method (simulated annealing)
Genetic Algorithms
Appendix : optimization with Octave
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Integration of differential equations
Euler's method, multi-step method, predictor-corrector method, Runge-Kutta method
Fourth-order Runge-Kutta method (rkf45)
Appendix : integration of differential equations with Octave
Exercises description
Exercises consist in writting computer programs in Fortran 90 in order to solve typical problems in Physics.
Teaching methods
Classes are given using a video-projector and a board for additional developments.
Evaluations
The exam consists of two parts: (i) a written exam on the theoretical course (14 points) and (ii) a practical exam in a computer lab (6 points). The final grade accounts for the work done during the practical sessions.
Recommended readings
W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes in Fortran, 2nd edition, Cambridge University Press (Cambridge, 1992).
Language of instruction
French / Français
Location for course
NAMUR
Organizer
Faculté des sciences
Rue de Bruxelles, 61
5000 NAMUR
Degree of Reference
Master's Degree