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Numerical Linear Algebra Experiments

This repository collects small numerical experiments in applied linear algebra and numerical analysis. The notebooks are based on topics from AMATH 352 (Applied Linear Algebra & Numerical Analysis, UW Seattle) and extended into self-contained mini projects.

Each notebook explores a specific numerical question: conditioning, stability of algorithms, empirical time complexity, and basic regression modeling.

Contents

  • Gaussian Pairwise Angles and Linear Transforms
    gaussian_pairwise_angles_and_linear_transforms.ipynb
    Studies the distribution of angles between random Gaussian vectors in ℝ¹⁰ and how a structured banded matrix (A), and its powers (A^5) and (A^{10}), distort those angle distributions. Illustrates how linear transforms can create strong alignment and anti-alignment in high dimensions.

  • Gram–Schmidt vs Modified Gram–Schmidt
    gram_schmidt_vs_modifiedgram_schmidt_stability.ipynb
    Compares classical Gram–Schmidt (CGS) and modified Gram–Schmidt (MGS) on correlated data. Uses angle histograms and the off-diagonal entries of (Q^\top Q) to quantify loss of orthogonality and show the improved numerical stability of MGS.

  • Empirical Complexity of Vector and Matrix Operations
    operation_timing_complexity_vec_mat.ipynb
    Implements manual versions of vector–vector, matrix–vector, and matrix–matrix products, then times them for increasing sizes. Fits slopes on log–log plots to empirically recover (O(n)), (O(n^2)), and (O(n^3)) behavior.

  • Vandermonde Least-Squares: Normal Equations vs QR vs lstsq
    vandermonde_least_squares_method_comparison.ipynb
    Fits noisy polynomial models using three solvers: normal equations (via LU), QR factorization, and numpy.linalg.lstsq. Compares relative errors across degrees and noise levels, and relates performance to the conditioning of (A) vs (A^\top A).

  • Wine Quality Linear Regression
    wine_quality_linear_regression.ipynb
    Builds a linear model for wine quality from physicochemical features using least squares with an intercept term. Reports train/test relative error and visualizes predicted vs true quality scores.

Technology

All notebooks use:

  • Python 3
  • NumPy
  • SciPy (for some linear solves and factorizations)
  • Matplotlib

You can run them in a standard scientific Python environment (e.g. conda, venv, or JupyterLab).

How to Run

  1. Clone the repository:

    git clone https://github.com/<your-username>/<your-repo-name>.git
cd <your-repo-name>

  2. Create and Activate an enviornment (optional but recommended) python -m venv .venv source .venv/bin/activate # on macOS/Linux

.venv\Scripts\activate # on Windows

  1. Install dependencies: pip install numpy scipy matplotlib jupyter
  2. Launch Jupyter: jupyter notebook
  3. Open any .ipynb file and run the cells.