APPLICATION OF LINEAR ALGEBRA METHODS TO OPTIMISE QUERIES IN DATABASE MANAGEMENT SYSTEMS

Abstract

The article investigates the use of linear algebra methods for query optimisation in modern database management systems. The relevance of the study is due to the growth of data volumes and the need for more efficient methods of their processing. The proposed approach is to formalise tabular data structures as numerical matrices, which allows us to interpret relational operations (selection, projection, join) as linear algebraic transformations, which opens up opportunities for the use of powerful mathematical tools that optimise computations. The main methods of linear algebra for query optimisation are considered: matrix factorisation (LU, QR decomposition) for structuring complex calculations; singular value decomposition (SVD) for reducing dimensionality and identifying hidden dependencies; matrix modelling of JOIN operations as an efficient block multiplication; systems of linear equations for formalising data filtering. It is noted that matrix factorisation helps to optimise the execution of complex analytical queries, SVD is effective in aggregation and dimensionality reduction tasks, the matrix representation of JOIN provides high performance when processing large amounts of data, and linear systems unify the filtering logic with numerical processing. To empirically verify the proposed methods, an experimental study was conducted on a synthetic dataset simulating a transactional table with a million records. The experiment was carried out using the NumPy and PyTorch libraries with the use of GPU acceleration. We evaluated such indicators as query execution time, RAM usage, and scalability. The results showed that JOIN matrix modelling was the fastest, which is especially important for parallel computing. LU/QR factorisation demonstrated stability and versatility, while SVD proved to be resource-intensive but valuable for data compression. Linear equation methods showed balanced performance. It is concluded that linear algebra is a powerful formalisation framework for next-generation DBMS architectures capable of providing mathematical consistency, computational efficiency and scalability.