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Numerical Methods

A practical introduction to numerical and linear algebra methods in Python, with implementations and tests for core algorithms used in computer science.

Course Overview

This repository contains exercises, homework-style implementations, and supporting scripts from the Numerical Methods course at the University of Zurich (Fall Semester 2025).

Grade: 5.25/6.0 | ECTS: 6

Topics Covered

• Scientific Computing Basics: Working with NumPy, vectorized operations, and numerical workflows
• Linear Systems: Gaussian elimination, pivoting, consistency checks, and structured problem solving
• Matrix Factorization: LU decomposition with permutation matrices and determinant computation
• Linear Algebra Foundations: Basis transformations, subspace checks, and orthonormalization
• Eigenvalue Methods: Power method and inverse power method
• Data Approximation: Polynomial interpolation and linear least squares
• PCA (2D): Principal direction estimation from centered data
• Numerical Solvability Concepts: Rank, invertibility, and model reconstruction

Repository Structure

├── Exercise 0/
│   └── exercise_00.py
├── Exercise 1/
│   ├── backend.py
│   └── backend_test.py
├── Exercise 2/
│   ├── backend.py
│   ├── backend_test.py
│   ├── exercise_02.py
│   └── frontend.py
├── Exercise 3/
│   └── Exercise_3_Practice/
│       ├── backend.py
│       ├── backend_test.py
│       ├── exercise_03.py
│       └── frontend.py
├── Exercise 4/
│   └── Exercise_04_Practice/
│       ├── backend.py
│       ├── backend_test.py
│       ├── exercise_04.py
│       └── frontend.py
├── Exercise 5/
│   └── Exercise_05_Practice/
│       ├── backend.py
│       ├── backend_test.py
│       ├── exercise_05.py
│       └── frontend.py
└── Scripts/
    ├── 1.4 Interpolation.ipynb
    ├── 2.5 LU Example.ipynb
    └── 2.9 Solvability.ipynb

Format

Materials are provided as Python modules and Jupyter notebooks:

• backend.py files for algorithm implementations
• backend_test.py files for automated validation
• frontend.py files for visualizations
• Notebook scripts for conceptual demonstrations and experimentation

Key Learning Outcomes

• Implement stable numerical algorithms for solving linear algebra problems
• Analyze matrix properties (rank, solvability, decomposition quality)
• Apply eigenvector-based techniques to extract dominant directions from data
• Use least squares and interpolation for data fitting and approximation
• Build tested, modular scientific Python code

Notes

This repository reflects hands-on coursework with focus on correctness, numerical stability, and reproducibility.

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Course instructors: Renato Pajarola and team