Reproducible Power Method for Computing Dominant Eigenvalue and Eigenvector

This article presents an implementation of the power method, enhanced with the ExBLAS library, for accurately computing the dominant eigenvalue of a matrix. The power method is a fundamental algorithm with widespread applications, notably in PageRank for web search and network analysis. Traditional implementations of the power method can suffer from numerical inaccuracies due to the non-associativity in floating-point operations. These inaccuracies are significantly amplified in scenarios with ill-conditioned matrices, such as those with very closely spaced eigenvalues, or in the presence of small perturbations that can significantly affect the computed results. To address these challenges, we leverage ExBLAS to provide bitwise reproducible results, ensuring the reliability and consistency of the computed eigenvalue. This work details the integration of ExBLAS into the power method, analyzes its impact on accuracy, and demonstrates its effectiveness using benchmark matrices from SuiteSparce collection. The reproducible power method offers consistent and reliable results across diverse computing environments, which is critical for applications where eigenvalue calculation is important.