Numerical Methods for Eigenvalue Problems

Abstract

This investigation centers on tactics for overseeing eigenvalues, merging both established and current methodologies. In various fields like engineering and science, eigenvalues serve an essential purpose, impacting crucial functions such as structural evaluations and quantum studies. Approaches like the Power Method and QR method run into obstacles, mainly when managing large or sparse matrices. The primary eigenvalue is disclosed through the power method, but the QR method does compute eigenvalues and may find larger matrices to be problematic. This paper delves into iterative techniques that enhance efficiency for extensive systems through Lanczos and Arnoldi, all while reducing size. Additionally, it will examine strategies aimed at accelerating operational processes, highlighting parallel execution and proactive conditioning, to enhance the effectiveness of iterative methods. In the near past, the landscape of quantum computing has transformed, alongside the progress of quantum systems that incorporate machine learning to handle eigenvector complications. An elaborate investigation into progressive eigenvalue calculation methods could aid professionals in attaining their targets with superior dexterity and capability.