Numerical Methods

This is my solutions to the exercises in the course "Numerical Methods" in the spring 2008. All programs are written in Fortran 95.

Hello World

Writes "Hello World" on the screen.

QR-decomposition

Solve system of linear equations, calculate determinant and inverse of a matrix.

Matrix diagonalization

Diagonalise a matrix, to find it's eigenvalues.

Please note that the file matrix.f95 has been changed slightly from the last exercise.

Least Squares Fit

Performs a Linear Least Squares Fit to a given set of datapoints.

Please note that the file matrix.f95 once again has been changed slightly from the last exercise.

The program produces the following plots, showing both a linear fit and a parabola fit:

Interpolation

Interpolate a tabulated function (x_i,y_i) using a polynomial and quadratic spline interpolation.

The program produces the following plot, showing both a polynomial and quadratic spline-interpolation:

RootFinding and Optimization

Find roots of a system of non-linear equations using the modified Newton method. Minimize a function in n variables using the downhill simplex algorithm.

Ordinary differential equations

Solve a system of ordinary differential equations using an adaptive-step-size Runge-Kutta routine and using the Prediction-Correction method.

Numerical Integration

Numerical integration with adaptive quadratures and reuse of points.

Monte Carlo integration

Numerical Monte Carlo integration with pseudo-random, quasi-random and stratified sampling.

Fast Fourier Transformation

Implementation of a Fast Fourier Transformation (FFT) which applies the Danielson-Lanczos lemma recursively. The FFT is then used to calculate amplitude-spectra and do simple filtering.

Project: Fast Fourier Transformation in astrophysics

The project can be found here:

Project

Some of the figures produced: To understand what these show, read the project.