@Article{CiCP-18-1,
author = {},
title = {Computing the Smallest Eigenvalue of Large Ill-Conditioned Hankel Matrices},
journal = {Communications in Computational Physics},
year = {2015},
volume = {18},
number = {1},
pages = {104--124},
abstract = {<p style="text-align: justify;">This paper presents a parallel algorithm for finding the smallest eigenvalue
of a family of Hankel matrices that are ill-conditioned. Such matrices arise in random
matrix theory and require the use of extremely high precision arithmetic. Surprisingly,
we find that a group of commonly-used approaches that are designed for high
efficiency are actually less efficient than a direct approach for this class of matrices.
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 previously possible, with the potential for extending this work to systems with
greater levels of parallelism. The contributions of this work are in three areas: determination
that a direct algorithm based on the secant method is more effective when
extreme fixed-point precision is required than the algorithms more typically used
in parallel floating-point computations; the particular mix of optimizations required
for extreme precision large matrix operations on a modern multi-core cluster, and the
numerical results themselves.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.260514.231214a},
url = {https://global-sci.com/article/80312/computing-the-smallest-eigenvalue-of-large-ill-conditioned-hankel-matrices}
}