Computing the smallest eigenvalue of large ill-conditioned Hankel matrices

TL;DR

This paper introduces a parallel algorithm for computing the smallest eigenvalue of large, ill-conditioned Hankel matrices, utilizing high precision arithmetic and combining message passing with shared memory for scalable performance.

Contribution

It develops a novel parallel approach that efficiently handles high-precision computations for large ill-conditioned matrices, surpassing previous methods in scalability and size capability.

Findings

Achieved near-perfect scalability in parallel implementation

Successfully computed eigenvalues for larger matrices than before

Demonstrated efficiency despite high arithmetic costs

Abstract

This paper presents a parallel algorithm for finding the smallest eigenvalue of a particular form of ill-conditioned Hankel matrix, which requires the use of extremely high precision arithmetic. Surprisingly, we find that commonly-used approaches that are designed for high efficiency are actually less efficient than a direct approach under these conditions. We then develop a parallel implementation of the algorithm that takes into account the unusually high cost of individual arithmetic operations. Our approach combines message passing and shared memory, achieving near-perfect scalability and high tolerance for network latency. We are thus able to find solutions for much larger matrices than has been previously possible, with the potential for extending this work to systems with greater levels of parallelism.