Numerical Methods

Solutions to two university assignments in MATLAB / Octave, covering six independent problems — from solving linear systems iteratively to digital signal processing and matrix factorisation. Each homework has its own directory and a README explaining the method behind every task.

Contents

Homework 1

Task	Problem	Method
Markov is coming	Probability of escaping a maze from every cell	Markov chain solved by Jacobi iteration, sparse matrices
Linear regression	Predicting house prices from mixed numeric and categorical features	Gradient descent, normal equation with conjugate gradient, Ridge and Lasso
MNIST 101	Recognising handwritten digits	Feedforward neural network trained by backpropagation

Homework 2

Task	Problem	Method
Numerical music	Synthesising and analysing audio	FFT, spectrograms with Hann windowing, frequency-domain filtering, convolution reverb
Robotzii	Interpolating a trajectory through waypoints	Vandermonde polynomials vs. cubic splines with C² continuity
Recommendations	Recommending items from a sparse rating matrix	Truncated SVD, cosine similarity

Methods covered

• Linear systems = Jacobi iteration, conjugate gradient, convergence criteria
• Sparse matrices = storage and why it matters for graph-like problems
• Optimisation = gradient descent, regularisation, the bias–variance trade-off
• Signal processing = FFT, windowing, spectrograms, filtering, convolution
• Interpolation = Vandermonde systems, conditioning, cubic splines
• Matrix factorisation = SVD and latent feature extraction

Running

The code targets MATLAB and also runs in GNU Octave. Each task directory is self-contained: add it to the path and call the functions from there.

cd homework-1/markov-is-coming
Labyrinth = parse_labyrinth('labyrinth.txt');
Adj = get_adjacency_matrix(Labyrinth);

The datasets, audio files and test harnesses were provided with the assignments and are not included in this repository.

Note on third-party code

homework-1/mnist-101/fmincg.m is Carl Rasmussen's conjugate gradient minimiser, distributed with the course materials. It is included because the training code calls it, but it is not my work. Everything else here is.