Computational Methods for Astronomy

by: Sebastian Bustamante 2014

This course is intended for students of Astronomy at the Universidad de Antioquia and will cover some numerical methods commonly used in science and specially in astronomy. These topics will be addressed from a formal context but also keeping a practical and computational approach, illustrating many useful applications in problems of physics and astronomy.

The practical component will be almost entirely developed in Python and slightly less in C (when computational performance is required). However students with knowledge in other programming languages (except privative languages like MatLab, Mathematica) are also aimed to use them.

In this repository it can be found all the related material of the course, including the detailed program, notes and presentations, examples (ipython notebooks) and homeworks. (This repository may be subject to changes continuously as the course advances).

SYLLABUS: detailed description of the program of the course, including a brief motivation and presentation, topics to be covered, evaluation and bibliography.

Contents

1. Python (2 weeks)

Topics

• Overview of python
• Basic scripting
• Implementation of scientific libraries
• Plotting with matplotlib
• Ipython notebooks

Tasks

• Task01: Solve the problems in the section Miscellanea. (due to: Oct 31).
• Task02: Solve the two activities proposed in here. (due to: Nov 08).

2. Mathematical Preliminaries (1 week)

Topics

• Computer arithmetics and round-off methods
• Algorithms and convergence

Tasks

• Task03: Solve the two activities proposed in here. (due to: Nov 15).

3. One Variable Equations (2 weeks)

Topics

• Bisection method
• Fixed-point iteration
• Newton-Raphson method
• Secant method

Tasks

• Task04: Solve the three activities proposed in here. (due to: Nov 22).
• Task05: Solve the activity discussed during class. (due to: Nov 29).

4. Interpolation Techniques (2 weeks)

Topics

• Linear interpolation
• Lagrange polynomials
• Divided differences
• Hermite interpolation

Test 1 (25%)

This test is divided in two parts. A theoretical part that will be evaluated during class time (50% of the test). And a computational activity due to: Jan 23, Midnight (50% of the test).

5. Numerical Calculus (2 weeks)

Topics

• Numerical differentiation
• Numerical integration
• Composite numerical integration
• Adaptive quadrature methods
• Improper integrals

Tasks

• Task06: Solve the activity proposed in here. (due to: Feb 1).
• Task07: Solve the activity proposed in here. (due to: Feb 15).

6. Linear Algebra (2 weeks)

Topics

• Linear systems of equations
• Gaussian elimination
• Pivoting strategies
• Matrix inversion
• Determinant of a Matrix
• LU factorization

Tasks

• Task08: Solve the activities proposed in here. (due to: Feb 22).

Test 2 (25%)

This test is divided in two parts. A theoretical part that will be evaluated during class time (50% of the test). And a computational activity due to: Mar 1, Midnight (50% of the test).

7. Differential Equations (2 weeks)

Topics

• First order methods
• High order methods
• Two-Point boundary value problems

Tasks

• Task09: Solve the activities proposed in here. (due to: Mar 8).

Test 3 (25%)

This test is entirely comprehended by a computational activity due to: Mar 17, Midnight (100% of the test).